Morfismos, Vol 4, No 2, 2000

VOLUMEN 4NÚMERO 2

JULIO A DICIEMBRE DE 2000 ISSN: 1870-6525

MORFISMOSComunicaciones EstudiantilesDepartamento de Matematicas

Cinvestav

Editores Responsables

• Isidoro Gitler • Jesus Gonzalez • Isaıas Lopez

Consejo Editorial

• J. Rigoberto Gabriel • Onesimo Hernandez-Lerma• Francisco Hernandez Zamora • Maribel Loaiza Leyva• Raquiel Lopez Martınez • Raul Quiroga Barranco

• Enrique Ramırez de Arellano

Editores Asociados

• Ricardo Berlanga • Samuel Gitler• Emilio Lluis Puebla • Guillermo Pastor• Vıctor Perez Abreu • Carlos Prieto

• Carlos Renterıa • Luis Verde

Secretarias Tecnicas

• Roxana Martınez • Laura Valencia

Morfismos puede ser consultada electronicamente en “Revista Morfismos”de la direccion http://www.math.cinvestav.mx. Para mayores informes dirigirseal telefono 57 47 38 71.Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamentode Matematicas del Cinvestav, Apartado Postal 14-740, Mexico, D.F. 07000 opor correo electronico: [email protected].

VOLUMEN 4NÚMERO 2

JULIO A DICIEMBRE DE 2000ISSN: 1870-6525

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Morfismos

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Morfismos

Contenido

Algebraic K-theory and the η-invariant

Jose Luis Cisneros Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Average optimal strategies in Markov games under a geometric drift condition

Heinz-Uwe Kuenle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Little cubes and homotopy theory

Dai Tamaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Hipergrupos y algebras de Bose-Msner

Isaıas Lopez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Sincronizacion de parejas de automatas celulares

J. Guillermo Sanchez Saint-Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Morfismos, Vol. 4, No. 2, 2000, pp. 1–14

Algebraic K-theory and the η-invariant ∗

Jose Luis Cisneros Molina 1

Abstract

The aim of this paper is to present the main results of J. D. S. Jonesand B. W. Westbury on algebraic K-Theory, homology spheresand the η-invariant [6], giving the basic definitions and prerequi-sites to understand them.

1991 Mathematics Subject Classification: 18F25, 19D06, 53C27, 58J28.Keywords and phrases: Algebraic K-theory, Quillen’s +-construction,spin geometry, Dirac operator, η-invariant.

1 Introduction

In [6] J. D. S. Jones and B. W. Westbury constructed elements in K3(C),the 3rd algebraic K-theory group of the field of complex numbers, usinghomology 3-spheres endowed with a representation of their fundamentalgroup. They also computed the image of such elements under the reg-ulator map, using the η-invariant. The aim of this paper is to presentthe main results of J. D. S. Jones [6], giving the basic definitions andprerequisites to understand them.

The paper is divided in four parts. In section 2 we define the alge-braic K-groups of a ring using Quillen’s +-construction. We also explainhow homology n-spheres equipped with a representation of its funda-mental group in the general linear group over a ring R define elements inthe K-group Kn(R). In section 3 we give the definition of the η-invariantof a self-adjoint elliptic operator on a closed manifold and its variations.In section 4 we describe the Dirac operator which is a very importantexample of this kind of operators and the one which we are interested

∗Invited article1Supported by a scholarship from DGAPA, UNAM.

1

2 JOSE LUIS CISNEROS MOLINA

in. Finally in section 5 the main results by Jones and Westbury arepresented.

2 Algebraic K-Theory

In this section we define the algebraic K-groups and we describe howto construct elements in this groups using homology spheres equippedwith a representation of its fundamental group.

2.1 Classifying space of a group

Any discrete group G has a classifying space BG which is a pointedspace (i.e. it has a base point ∗) unique up to homotopy equivalencesuch that:

π1(BG) = G and πi(BG) = 0 for i = 1 i.e. BG is an Eilenberg-Mac Lane space K(G, 1).

From its definition, the universal covering of BG, denoted by EG iscontractible. The covering EG→ BG is called the universal bundle forG and the space BG satisfies the following universal property:

If EG→ BG is a universal bundle for G and X is of the homotopytype of a CW-complex with base point x0 (e.g. manifold). Then wehave the following one-to-one correspondences

[X,BG]←→ Hom(π1(X,x0), G)←→ FG(X)

where [X,BG] denotes the homotopy classes of maps from X to BG,Hom(π1(X,x0), G) denotes the homomorphisms from π1(X,x0) to Gand FG(X) the equivalence classes of principal (flat) G-bundles over X.

Note that in the case when G = GLN (C), Hom(π1(X,x0), G) isprecisely the set of representations of π1(X,x0) on CN .

2.2 Quillen’s +-construction

In order to define the algebraic K-theory groups of a ring R, we needthe +-construction due to Daniel Quillen in the early 1970’s, for which,among other reasons, he was awarded the Fields Medal in 1978.

Theorem 2.2.1 (Quillen). Let X be a connected CW-complex withbase point x0. Let A ⊂ π1(X) be a perfect normal subgroup (i.e. A =[A,A] and A = [π1(X), A], where [ , ] is the commutator). Then thereis a space X+ (depending on A) and a map i : X → X+ such that:

ALGEBRAIC K-THEORY AND THE η-INVARIANT 3

(a) The map i induces an isomorphism

i : π1(X)/A → π1(X+).

(b) For any π1(X+)-module L one has

i∗ : H∗(X, i∗L)∼=→ H∗(X

+, L).

(c) The pair (X+, i) is determined by a) and b) up to homotopy equiv-alence.

Let R be a ring with 1. Consider the group GLN (R) of invertibleN ×N matrices over R. The elementary group EN (R) is the subgroupof GLN (R) generated by the elementary matrices (see [12, 11, 9] fordefinition).

We have inclusions GLN (R) ⊂ GLN+1(R) which restrict to inclu-sions EN (R) ⊂ EN+1(R) and we can define

GL(R) =!

N

GLN (R)

E(R) =!

N

EN (R).

Let X = BGL(R). Then π1(X) = GL(R) and A = E(R) is per-fect. Then applying the +-construction we get BGL(R)+. Define thealgebraic K-groups of the ring R by

Kn(R) = πn(BGL(R)+) for n ≥ 1.

This definition may seem artificial, the reason is because originallythe first three groups K0(R), K1(R) and K2(R) were given by algebraicdefinitions2 and for a while seemed to be no good way to define the“higher K-functors” Ki, i ≥ 3, until Quillen’s work appeared, for a niceaccount of this facts see [12].

2In the present definition we are not including K0(R), in this case is called the“reduced” algebraic K-theory of R


2.3 Homology spheres

It is well known that the homology of the n-sphere Sn is given by

Hq(Sn) =

!Z q = 0, n

0 q = 0, n.

A homology n-sphere as its name indicates it, is a path-connectedspace (say with the homotopy type of a CW-complex) with the samehomology groups as Sn (n ≥ 3).

Let Σ be a homology n-sphere, since

0 = H1(Σ,Z) = π1(Σ)/[π1(Σ),π1(Σ)]

π1(Σ) can have no abelian quotients and so is perfect. Given a represen-tation α : π1(X) → GLN (R), let f : Σ → BGLN (R) be the map whichinduces α on π1 (by the universal property of classifying spaces). Com-posing this map with the inclusion BGLN (R) → BGL(R) and applyingQuillen’s +-construction we get

Sn ! Σ+ → BGL(R)+,

since the +-construction is functorial by its universal properties. Here! denotes homotopy equivalence. The homotopy class of this map givesus the element in K-theory

[Σ,α] ∈ Kn(R) = πn(BGL(R)+).

2.4 The regulator

There is a homomorphism

e : K2n+1(C) → C/Z

called the regulator map which satisfies the following properties

(i) It is an isomorphism on K1(C) ∼= C∗ → C/Z.

(ii) The homomorphism e gives an isomorphism of the torsion sub-group of K2n+1(C) with Q/Z.

The aim now is to compute the image of the elements [Σ,α] ∈ K3(C)under the regulator map. One way to do this is using the η-invariant.


3 The η-invariant

Let X be a closed (compact without boundary) Riemannian manifoldand let E be a smooth vector bundle over X with an inner product.We denote by C∞(X,E) the space of smooth sections of E and we canendow it with an inner product ⟨ , ⟩ using the inner product on E andintegration. Let A : C∞(X,E) → C∞(X,E) be an elliptic differentialoperator and assume that A is self-adjoint, that is

⟨s1, As2⟩ = ⟨As1, s2⟩

for every s1, s2 ∈ C∞(X,E). Then A has a discrete spectrum with realeigenvalues λ and we define the η-series of A by

η(s;A) =!

λ =0

(signλ)|λ|−s

where the sum is taken over the non-zero eigenvalues of A. This seriesconverges for ℜ(s) sufficiently large. By results of Seeley [13] extends byanalytic continuation to a meromorphic function on the whole s-planeand is finite at s = 0.

The number η(0;A) is called the η-invariant of A and is a spectralinvariant which measures the asymmetry of the spectrum of A.

We also define a refinement of the η-series which takes into accountthe zero eigenvalues of A

ξ(s;A) =h+ η(s;A)

2

where h is the dimension of the kernel of A or in other words, themultiplicity of the 0-eigenvalue of A.

Now consider a representation α : π1(X) → GLN (C). Then α definesa flat bundle Vα over X in the following way. Let X be the universalcover of X. Then Vα = X ×π1(X) CN i.e. Vα is X × CN modulo theaction of π1(X), where π1(X) acts on the first factor with the canonicalaction of π1(X) on the universal cover and via the representation α onthe second factor. The bundle Vα also has a canonical flat connection∇α given by the exterior derivative as follows.

A connection is a first order linear differential operator

C∞(X,Vα)∇α

−−→ Ω1(X,Vα)


which satisfies the Leibnitz rule

∇αfs = df ⊗ s+ f ⊗∇αs

for every f ∈ C∞(X,R) and every s ∈ C∞(X,Vα).By the previous construction of the bundle Vα we have that C∞(X,Vα) ∼=

C∞(X,CN )αand Ω1(X,Vα) ∼= Ω1(X,CN )α, where the spaces C∞(X,CN )

α

and Ω1(X,CN )α are, respectively, the sections and 1-forms which areequivariant under the action of π1(X) via the representation α. On theother hand, the exterior derivative

C∞(X,CN )d→ Ω1(X,CN )

sends invariant sections to invariant 1-forms. Hence the connection ∇α

is given by

C∞(X,Vα) ∼= C∞(X,CN )α ∇α=d−−−−→ Ω1(X,CN )α ∼= Ω1(X,Vα).

Using this connection we can couple the operator A to Vα to get anoperator

Aα : C∞(X,E ⊗ Vα) → C∞(X,E ⊗ Vα)

and as before we define the functions3

η(s;α, A) = η(s;Aα), ξ(s;α, A) = ξ(s;Aα)

and their reduced forms

η(s;α, A) = η(s;α, A)−Nη(s;A), ξ(s;α, A) = ξ(s;α, A)−Nξ(s;A)

where N is the dimension of the representation α.Once more, following [2, Section 2] we can see that the functions

η(s;α, A) and ξ(s;α, A) are finite at s = 0 and if we reduce modulo Zthen

η(α, A) = η(0;α, A) ∈ C/Z, ξ(α, A) = ξ(0;α, A) ∈ C/Z

3The operator Aα is not self-adjoint any more, unless the representation α isunitary. Nonetheless, Aα has self-adjoint symbol and that allows us to define the ηand ξ functions, see [2, p. 90].


are homotopy invariants of A. The reason for regarding values in C/Zand not just in C is that if we vary A continuously the dimension ofkerA is not a continuous function of A. However the jumps of ξ(s;A)are due to eigenvalues changing sign as they cross zero and thereforethe jumps are integer jumps.

Note that if we fix the manifold X and the operator A, the invariantξ(α, A) only depends on the representation α of the fundamental groupof X or equivalently on the flat bundle Vα aver X.

4 The Dirac operator

In this section we describe a particular example of a self-adjoint ellipticdifferential operator called the Dirac operator which is the one we shalluse to compute C/Z-valued invariants of elements of the K-groups of anysubring of C. The Dirac operator is very important by itself and plays acentral role in the Atiyah-Singer Index Theorem, in the Seiberg-Wittentheory and many other things. The main references for the material inthis section are [7, 1].

4.1 Clifford algebras

Let V be a finite dimensional real vector space with a non-degenerate,symmetric bilinear form q : V ⊗ V → R. Let e1, . . . , en be an orthog-onal basis for V then the Clifford algebra Cl(V, q) is the algebra over R,with unit, generated by the ei, subject to the relations

e2i = −q(ei, ei)

eiej = −ejei i = j.

For the special case when V = Rn and q is the standard innerproduct we denote the algebra Cl(V, q) by Cln and its complexificationby ClCn = Cln ⊗ C.

Example 4.1.1.

Cl0 = R with basis 1

Cl1 = C with basis 1, e1

Cl2 = H with basis 1, e1, e2, e1e2


The group Spin(n) is defined as a subgroup of the group of units ofCln and it is the non-trivial double covering of SO(n) and for n > 2 itis its universal covering.

Now lets restrict ourselves to odd dimensional vector spaces, in thiscase, the complexified Clifford algebra ClCn has two inequivalent irre-ducible complex representations and when they are restricted to Spin(n)they give isomorphic irreducible complex representations of Spin(n). Wedenote such a representation space by S.

4.2 Spin structures

LetX be an odd dimensional oriented closed Riemannian manifold. TheRiemannian metric and the orientation give a reduction of the structuregroup of the tangent bundle TX of X to SO(n). A spin structure on Xis a lift of the structure group SO(n) of TX to Spin(n).

A spin structure on X provide us with a principal Spin(n)-bundleQ which is a double cover of the principal SO(n)-bundle P associatedto the tangent bundle TX. The restriction to the fibre of this doublecover ϖ : Q → P is the double covering Spin(n) → SO(n).

Now consider the spin representation S of Spin(n) and let

S(X) = Q×Spin(n) S

be the vector bundle over X associated to the principal Spin(n)-bundleQ. The bundle S(X) is called the spinor bundle of X and its sec-tions are called spinor fields. We denote the space of spinor fields byC∞(X,S(X)).

Let Cl(T ∗X) be the bundle over X whose fibre at x is Cl(T ∗xX),

the Clifford algebra of the cotangent space at x with the inner productgiven by the Riemannian metric.

There is a pairing

C : Cl(T ∗X)⊗ S(X) → S(X)

which is called Clifford multiplication. If we consider the inclusionT ∗X → Cl(T ∗X) then we get a pairing

T ∗X ⊗ S(X) → S(X).

4.3 The Dirac operator

The Riemannian structure of X provides us with the Riemannian con-nection on the tangent bundle. This connection can be seen as a 1-form


β on the principal SO(n)-bundle P with values in the Lie algebra so(n).Since Spin(n) and SO(n) have the same Lie algebra, the double cov-ering ϖ : Q → P given by the spin structure on X gives us a 1-formϖ∗(β) which defines a connection on Q called the spin connection. Thisconnection induces a covariant derivative

∇ : C∞(X,S(X)) → C∞(X,T ∗X ⊗ S(X))

on spinor fields.Composing ∇ with Clifford multiplication

C : C∞(X,T ∗X ⊗ S(X)) → C∞(X,S(X))

we obtain the Dirac operator

D = C ∇ : C∞(X,S(X)) → C∞(X,S(X)).

It is a self-adjoint, first order, elliptic partial differential operator.As in the previous section, a representation α : π1(X) → GLN (C)

defines a bundle Vα with a flat connection ∇α. In this case we can definethe twisted Dirac operator Dα by the composition

C∞(X,S(X)⊗ Vα)∇⊗Id+Id⊗∇α

−→ C∞(X,T ∗X ⊗ S(X)⊗ Vα)C⊗Id−→ C∞(X,S(X)⊗ Vα)

where ∇⊗∇α is the product connection on the bundle S(X)⊗ Vα andId is the identity map.

5 The results of Jones and Westbury

The relation between the value of the regulator map on the classes[Σ,α] ∈ K3(C) and the η-invariant of the Dirac operator of the homologysphere Σ is given by the following theorem:

Theorem 5.1.1 (Jones–Westbury).

e([Σ,α]) = ξ(α, D)

where D is the Dirac operator on Σ.


In [6] Jones and Westbury give a formula to compute e[Σ,α] when Σis a Seifert homology sphere. Let (a1, . . . , an) be an n-tuple of pairwisecoprime integers. The Seifert homology 3-sphere Σ(a1, . . . , an) is a 3-manifold which admits an action of the circle S1 which is free exceptfor n exceptional orbits which have isotropy groups Ca1, . . . , Can whereCm ⊂ S1 is the cyclic subgroup of order m embedded in S1 as the mthroots of unity.

In order to give the aforementioned formula we need to know a bitabout the fundamental group of Σ(a1, . . . , an). Let T (a1, . . . , an) be thegeneralised triangle group which is defined by the following generatorsand relations

T (a1, . . . , an) = ⟨x1, . . . , xn|xa11 = · · · = xann = x1 . . . xn = 1⟩.

This group is perfect and it has a universal central extensionT (a1, . . . , an) which fits into an exact sequence

1 → C∗ → T (a1, . . . , an) → T (a1, . . . , an) → 1

where C∗ is an infinite cyclic group, except for the case of T (2, 3, 5)where C∗ ∼= Z2.

In terms of generators and relations

T (a1, . . . , an) = ⟨h, x1, . . . , xn| [xi, h] = 1, xa11 = h−b1 , . . . , xann = h−bn ,

x1 . . . xn = h−b0⟩where h is the generator of the centre of T (a1, . . . , an).

The bi satisfy the relation

a1 . . . an

!−b0 +

b1a1

+ · · ·+ bnan

"= 1

and we have that

π1(Σ(a1, . . . , an)) = T (a1, . . . , an).

Let α : π1(Σ(a1, . . . , an)) → GLN (C) be a representation, since thegroup π1(Σ(a1, . . . , an)) is perfect every complex representation α musthave image in SLN (C). We shall consider only those representations inwhich the central element h acts as a scalar multiple of the identity, forinstance, that is the case when α is irreducible and in general for anydecomposable representation.


Suppose α(h) = λhI where λh is a scalar, then, since α(h) ∈ SLN (C)

λh = ζrhN

is a Nth root of unity. Here ζd = e2πi/d ∈ C is the standard primitivedth root of unity. Now consider the matrices

α(xj), j = 1, . . . , n.

In view of the relations xajj = h−bj the eigenvalues λ1(j), . . . ,λN (j)

satisfy the equation

λk(j)aj = λ

−bjh .

There are aj roots of this equation and we define sk(j) by

λk(j) = ζNsk(j)−bjrhNaj

.

We refer to the numbers

sk(j), 1 ≤ j ≤ n, 1 ≤ k ≤ N

as the type of the representation α.Now we have

Theorem 5.1.2 (Jones-Westbury). Let α : π1(Σ(a1, . . . , an)) →SLN (C) be a representation of the fundamental group of the Seiferthomology sphere Σ(a1, . . . , an) in which the central element h acts as ascalar multiple of the identity. Let

sk(j), 1 ≤ j ≤ n, 1 ≤ k ≤ N

be the type of the representation α; then

2Nℜ(e[Σ(a1, . . . , an),α]) = −n!

j=1

N!

k=1

N!

l=1

a(sk(j)− sl(j))2

2a2j

where a = a1 . . . an.

This formula was obtained using the fact that the invariants ξ(α, D)are cobordism invariants, so it is enough to compute them on a sim-pler manifold which is cobordant to the Seifert homology sphere (see[6]). The cobordism invariance follows from the index theorem for flatbundles in [2].

Using the previous theorem they also prove the following results


Theorem 5.1.3 (Jones-Westbury). Every element in K3(C) of fi-nite order is of the form [Σ(p, q, r),α] for some representation

α : π1(Σ(p, q, r)) → SL2(C).

Now let Z[ζd] be the ring of algebraic numbers in the cyclotomic fieldQ(ζd). Then combining the results of Borel [3], Merkurjev and Suslin[10] and Levine [8] we have that

K3(Z[ζd]) = Z/w2(d)⊕ Zr2

where

w2(d) = lcm(24, 2d)

and r2 is the number of complex places of Q(ζd). In particular note thatif (6, d) = 1 the torsion subgroup of K3(Z[ζd]) is exactly Z/24d.

Theorem 5.1.4 (Jones-Westbury). If (6, d) = 1 there exists arepresentation α : π1(Σ(2, 3, d)) → SL2(Z[ζd]) such that the element[Σ(2, 3, d),α] ∈ K3(Z[ζd]) is a generator of the torsion subgroup.

Example 5.1.5. The Seifert homology sphere P = Σ(2, 3, 5) is calledthe Poincare 3-sphere. Its fundamental group, known as the binaryicosahedral group, is a subgroup of SU(2) and the matrices which oc-cur in this subgroup can all be chosen to have coefficients in the ringZ[ζ5]. This gives a representation α of π1(P ) in SL2(Z[ζ5]), and usingtheorem 5.1.2 we get

e[P,α] =1

120.

From this we deduce that the generator of the torsion subgroup ofK3(Z[ζ5]) is given by [P,α] where α is the natural representation ofπ1(P ).

6 Further research and progress

One could try to compute ξ(α, D) directly from its definition, withoutusing the fact that it is a cobordism invariant and expect an improvedformula which works for all the representations of π1(Σ) and which alsogives the imaginary part. I established a first step in this direction in[4, 5] computing ξ(α, D) directly from its definition for the Poincare


sphere, which is the only homology 3-sphere with finite fundamentalgroup. The method not only works for the Poincare sphere but for anyquotient S3/Γ of the 3-sphere by a finite subgroup Γ (3-dimensionalspherical space forms), so we get a formula to compute the η-invariantof the Dirac operator of S3/Γ twisted by any representation of Γ, whereΓ is any finite subgroup of S3.

AcknowledgementI would like to thank Professor John Jones for his comments on this

work.

Jose Luis Cisneros MolinaInstituto de Matematicas, UNAM,Unidad Cuernavaca,Apdo. Postal #273-3,Adm. de Correos #3, C.P. 62251,Cuernavaca, Morelos, [email protected]

References

[1] M. F. Atiyah, R. Bott, and A. Shapiro. Clifford modules. Topol-ogy, 3, Suppl. 1, (1964), 3–38.

[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asym-metry and riemannian geometry III. Mathematics Proceedings ofthe Cambridge Philosophycal Society, 79 (1976), 71–99.

[3] A. Borel. Stable real cohomology of arithmetic groups. AnnalesScientifiques de l’Ecole Normale Superieure, 7 (1974), 235–272.

[4] J. L. Cisneros Molina. The η-invariant of twisted Dirac operatorsof S3/Γ. To appear in Geometriae Dedicata, (2000).

[5] J. L. Cisneros-Molina. The regulator, the Bloch group, hyper-bolic manifolds, and the η-invariant. PhD thesis, University ofWarwick, Coventry, England, September (1998).

[6] J. D. S. Jones and B. W. Westbury. Algebraic K-theory, ho-mology spheres, and the η-invariant. Topology, 34(4) (1994),929–957.

[7] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. Spin Geom-etry. Princeton University Press, Princeton, New Jersey, (1989).


[8] M. Levine. The indecomposable K3 of fields. Annales Scien-tifiques de l’Ecole Normale Superieure, 22 (1989), 255–344.

[9] E. Lluis-Puebla. Algebra homologica, Cohomologıa de Grupos yK-Teorıa Algebraica Clasica. Addison-Wesley Iberoamericana,(1990).

[10] A. S. Merkurjev and A. A. Suslin. On K3 of a field. LOMIpreprint M-2-07, (1987).

[11] J. W. Milnor. Introducton to Algebraic K-Theory. Study 72.Princeton University Press, Princeton, New Jersey, (1971).

[12] J. Rosenberg. Algebraic K-Theory and Its Applications. Gradu-ate Texts in Mathematics 147. Springer-Verlag, (1994).

[13] R. T. Seeley. Complex powers of an elliptic operator. In Sin-gular Integrals, Proc. Symp. Pure Math., volume 10, pages 288–307. American Mathematical Society, Providence, Rhode Island,U.S.A., (1967).


Average optimal strategies in Markov gamesunder a geometric drift condition ∗

elneuKewU-znieH

Abstract

Zero-sum stochastic games with the expected average cost crite-rion and unbounded stage cost are studied. The state space is anarbitrary Borel set in a complete separable metric space but theaction sets are finite. It is assumed that the transition probabili-ties of the Markov chains induced by stationary strategies satisfya certain geometric drift condition. It is shown that the aver-age optimality equation has a solution and that both players haveoptimal stationary strategies.

1991 Mathematics Subject Classification: 91A15Keywords and phrases: Markov games, Borel state space, average costcriterion, geometric drift condition, unbounded costs

1 Introduction

In this paper two-person stochastic games with the expected averagecost criterion are studied. The state space is a standard Borel space,that is, an arbitrary Borel set in a complete separable metric space.The action sets of both players are finite. Such a stochastic game canbe described in the following way: The state xn of a dynamic systemis periodically observed at times n = 1, 2, . . .. After an observation attime n the first player chooses an action an from the action set A(xn)and afterwards the second player chooses an action bn from the actionset B(xn) dependent on the complete history of the system at this time.The first player must pay cost k1(xn, an, bn), the second player must pay

∗Invited article

15

16 HEINZ-UWE KUENLE

k2(xn, an, bn), and the system moves to a new state xn+1 in the statespace X according to the transition probability p(· | xn, an, bn).

Stochastic games with Borel state space and average cost criterionare considered by several authors. Related results are given by Maitraand Sudderth [7], [8], [9], Nowak [13], Rieder [15] and Kuenle [6] inthe case of bounded costs (payoffs). The case of unbounded payoffs istreated by Nowak [14] and Kuenle [4]. The assumptions in this paperconcerning the transition probabilities are related to Nowak’s assump-tions: Nowak assumes that there is a Borel set C ∈ X and for everystationary strategy pair (π∞, ρ∞) a measure µ such that C is µ-smallwith respect to the Markov chain induced by this strategy pair. Weassume that C is only a µ-petite set with respect to a resolvent of thisMarkov chain; as against this, we demand that µ is independent of thecorresponding strategy pair. (For the definition of ”small sets” and”petite sets” see [10].)

The paper is organised as follows: In Section 2 the mathematicalmodel of Markov games is presented. Section 3 contains the assump-tions on the transition probabilities and on the stage costs, and also somepreliminary results. In Section 4 we study the expected average cost ofa fixed stationary strategy pair. We show that the Poisson equationhas a solution. In Section 5 we prove that the average cost optimal-ity equation has a solution and both players have optimal stationarystrategies.

2 The Mathematical Model

In this section we introduce the mathematical model of the stochasticgame considered in this paper.

Definition 2.1M = ((X,σX), (A,σA),A, (B,σB),B, p, k1, k2,E,F) is called a Markovgame if the elements of this tuple have the following meaning:

— (X,σX) is a standard Borel space, called the state space.

— A is a countable set and σA is the power set of A. A(x) ∈ Adenotes a finite set of actions of the first player for every x ∈ X.A is called the action space of the first player and A(x) is calledthe admissible action set of the first player at state x ∈ X.

AVERAGE OPTIMAL STRATEGIES IN MARKOV GAMES 17

— B is a countable set and σB is the power set of B. B(x) ∈ Bdenotes a finite set of actions of the second player for every x ∈ X.B is called the action space of the second player and B(x) is calledthe admissible action set of the second player at state x ∈ X.

— p is a transition probability from σX×A×B to σX, the transitionlaw.

— ki, i = 1, 2 , are σX×A×B-measurable functions, called stage costfunctions.

— Let Hn = (X×A×B)n×X for n ≥ 1, H0 = X. h ∈ Hn is calledthe history at time n.A transition probability πn from σHn to σA withπn(A(xn) | x0, a0, b0, . . . , xn) = 1 for all (x0, a0, b0, . . . , xn) ∈ Hn

is called a decision rule of the first player at time n.A transition probability ρn from σHn×A to σB withρn(B(xn) | x0, a0, b0, . . . , xn) = 1 for all (x0, a0, b0, . . . , xn) ∈ Hn

is called a decision rule of the second player at time n.A decision rule of the first [second] player is called Markoviff a transition probability πn from σHn to σA [ρn fromσHn to σB] exists such that πn(· | x0, a0, b0, . . . , xn) = πn(· | xn)[ ρn(· | x0, a0, b0, . . . , xn) = ρn(· | xn)] for all (x0, a0, b0, . . . , xn) ∈Hn ×A. (Notation: We identify πn as πn and ρn as ρn.)E and F denote non-empty sets of Markov decision rules.

A decision rule of the first [second] player is called determin-istic if a function en : Hn → A [fn : Hn → B] exists suchthat πn(en(hn) | hn) = 1 for all hn ∈ Hn [ρn(fn(hn) | hn) = 1 for all(hn) ∈ Hn ].

A sequence Π = (πn) or P = (ρn) of decision rules of the first orsecond player is called a strategy of that player.Strategies are called deterministic, or Markov iff all their decision ruleshave the corresponding property.A Markov strategy Π = (πn) or P = (ρn) is called stationary iffπ0 = π1 = π2 = . . . or ρ0 = ρ1 = ρ2 = . . .. (Notation: Π = π∞

or P = ρ∞.) We assume in this paper that the sets of all admissiblestrategies are E∞ and F∞. Hence, only Markov strategies are allowed.But by means of dynamic programming methods it is also possible toget corresponding results for Markov games with larger sets of admis-sible strategies. If E and F are the sets of all Markov decision rules

18 HEINZ-UWE KUENLE

(in the above sense) then we have a Markov game with perfect (orcomplete) information. In this case the action set of the second playermay depend also on the present action of the first player. If E is theset of all Markov decision rules but F is the set of all Markov deci-sion rules which do not depend on the present action of the first playerthen we have a usual Markov game with independent action choice. LetΩ := X×A×B×X×A×B× . . . and Ki,N (ω) :=

!Nj=0 k

i(xj , aj , bj)for ω = (x0, a0, b0, x1, . . . ) ∈ Ω, i = 1, 2, N ∈ N. By means of theIonescu-Tulcea Theorem (see, for instance, [11]), it follows that thereexists a suitable σ-algebra F in Ω and for every initial state x ∈ X andstrategy pair (Π, P ), Π = (πn), P = (ρn), a unique probability mea-sure Px,Π,P on F according to the transition probabilities πn, ρn and p.Furthermore, Ki,N is F-measurable for all i = 1, 2, N ∈ N. We set

V i,NΠP (x) =

"

ΩKi,N (ω)Px,Π,P (dω) (2.1)

and

ΦiΠP (x) = lim inf

N→∞

1

N + 1V i,NΠP (x) (2.2)

if the corresponding integrals exist.

Definition 2.2A strategy pair (Π∗, P ∗) is called a Nash equilibrium pair iff

Φ1Π∗P ∗ ≤ Φ1

ΠP ∗

Φ2Π∗P ∗ ≤ Φ2

Π∗P

for all strategy pairs (Π, P ).

In this paper we will consider especially zero-sum Markov games, thatmeans k1 = −k2. In this case we call a Nash equilibrium pair also anoptimal strategy pair. We set k := k1, V N

ΠP := V 1,NΠP , ΦΠP := Φ1

ΠP .

3 Assumptions and Preliminary Results

In this paper we use the same notation for a substochastic kernel andfor the ”expectation operator” with respect to this kernel, that means:If (Y,σY) and (Z,σZ) are standard Borel spaces, v : Y × Z → R a


σY×Z-measurable function, and q a substochastic kernel from (Y,σY)to (Z,σZ) then we put

qv(y) :=

!

Zq(dz | y)v(y, z) for all y ∈ Y

if this integral is well-defined.Furthermore, we define the operator T by

Tu = k + pu

for all σX-measurable u : X → R for which pu exists, that means,

Tu(x, a, b) = k(x, a, b) +

!

Xp(dξ | x, a, b)u(ξ)

for all x ∈ X, a ∈ A, b ∈ B.Let Π = (πn) ∈ E∞, P = (ρn) ∈ F∞. If V N

ΠP exists, then we get

V NΠP = π0ρ0k +

N"

j=1

π0ρ0p · · · pπjρjk.

For π ∈ E, ρ ∈ F we put (πρp)n := πρp(πρp)n−1 where (πρp)0 denotesthe identity. Let ϑ ∈ (0, 1). We set for every π ∈ E, ρ ∈ F, x ∈ X, andY ∈ σX

Qϑ,π,ρ(Y | x) := (1− ϑ)∞"

n=0

ϑn(πρp)nIY (x)

where IY is the characteristic function of the set Y .We remark that for a stationary strategy pair (π∞, ρ∞) the transi-

tion probability Qϑ,π,ρ is a resolvent of the corresponding Markov chain.

Assumption 3.1 There are: a nontrivial measure µ on σX; a set C ∈σX; a σX-measurable function W ≥ 1; and constants ϑ ∈ (0, 1), α ∈(0, 1), and β ∈ R, with the following properties:

(a)

Qϑ,π,ρ ≥ IC · µ

for all π ∈ E and ρ ∈ F,

20 HEINZ-UWE KUENLE

(b)

pW ≤ αW + ICβ,

(c)

supx∈X,a∈AAA(x),b∈BBB(x)

|k(x, a, b)|W (x)

< ∞.

Assumption 3.1 (a) means that C is a ”petite set”, (b) is called”geometric drift towards C ” (see Meyn and Tweedie [10]). We assumein this paper that Assumption 3.1 is satisfied.

Lemma 3.2 There are a σX-measurable function V with 1 ≤ W ≤V ≤ W + const, and a constant λ ∈ (0, 1) with

Qϑ,π,ρV ≤ λV + IC · µV (3.1)

and

ϑpV ≤ λV. (3.2)

Proof: Without loss of generality we assume β > 0. Let β′ := ϑ1−ϑβ,

W ′ := W + β′, and α′ := β′+αβ′+1 . Then it holds that α′ ∈ (α, 1) and

pW ′ = pW + β′

≤ αW + β′ + βIC

≤ α′W − (α′ − α)W + α′β′ + (1− α′)β′ + βIC

≤ α′W ′ − (α′ − α) + (1− α′)β′ + βIC

= α′W ′ + β′ + α− α′(β′ + 1) + βIC

= α′W ′ + βIC . (3.3)

Now let W ′′ := W ′ − β′IC = W + β′(1− IC). Then we get from (3.3 )

p(W ′′ + β′IC) = pW ′

≤ α′W ′ + βIC

= α′W ′′ + α′β′IC + βIC

= α′W ′′ + α′β′IC +1− ϑ

ϑβ′IC

= α′W ′′ +α′ϑ+ 1− ϑ

ϑβ′IC

≤ α′W ′′ +β′

ϑIC . (3.4)


We put α′′ := 1−ϑ1−α′ϑ . Then it holds that α′ = α′′+ϑ−1

α′′ϑ . For β′′ := α′′β′

it follows:

pW ′′ ≤ α′′ + ϑ− 1

α′′ϑW ′′ − β′′

α′′ pIC +β′′

α′′ϑIC .

Hence,

α′′ϑpW ′′ ≤ (α′′ + ϑ− 1)W ′′ − ϑβ′′pIC + β′′IC .

Then

(1− ϑ)W ′′ ≤ α′′W ′′ + β′′IC − ϑp(α′′W ′′ + β′′IC).

This implies

(1− ϑ)W ′′ ≤ α′′W ′′ + β′′IC − ϑπρp(α′′W ′′ + β′′IC)

for every π ∈ E, ρ ∈ F. Hence,

Qϑ,π,ρW′′ =

∞!

n=0

(1− ϑ)ϑn(πρp)nW ′′

≤∞!

n=0

ϑn(πρp)n(α′′W ′′ + β′′IC)

−∞!

n=1

ϑn(πρp)n(α′′W ′′ + β′′IC)

= α′′W ′′ + β′′IC . (3.5)

We choose ϑ′ ∈ (ϑ, 1) and set γ := max β′′

µ(X) ,β′

ϑ′−ϑ, λ′ := α′′+γ

1+γ , λ :=

maxλ′,ϑ′. It follows that α′′ < λ′ ≤ λ < 1 and λ′ − α′′ = (1 − λ′)γ.Hence,

(λ− α′′)W ′′ ≥ λ′ − α′′ ≥ (1− λ′)γ ≥ (1− λ)γ. (3.6)

We put V := W ′′ + γ. Obviously, V ≥ W ′′ ≥ 1 and V ≥ γ. Then itfollows

Qϑ,π,ρV = Qϑ,π,ρW′′ + γ

≤ α′′W ′′ + IC · β′′ + γ

≤ α′′W ′′ + IC · γµ(X) + γ

≤ α′′W ′′ + IC · µV + γ

≤ α′′W ′′ + IC · µV + (λ− α′′)W ′′ + λγ (see (3.6 ))

= λ(W ′′ + γ) + IC · µV= λV + IC · µV.

22 HEINZ-UWE KUENLE

Hence, (3.1 ) is proved.

From γ ≥ β′

ϑ′−ϑ it follows

ϑ′γ ≥ ϑγ + β′. (3.7)

Then

ϑpV = ϑpW ′′ + ϑγ

≤ α′ϑW ′′ + β′ + ϑγ (see (3.4 ))

≤ α′ϑW ′′ + ϑ′γ (see (3.7 ))

≤ ϑ′(W ′′ + γ)

= ϑ′V

≤ λV.

Hence, (3.2 ) is also proved.

4 Properties of Stationary Strategy Pairs

For a function u : X → R we put ∥u∥V := supx∈X|u(x)|V (x) . Furthermore,

we denote byV the set of all σX-measurable functions u with ∥u∥V < ∞.In the following we will assume that on V that metric is given which isinduced by the weighted supremum norm ∥ · ∥V . Then V is complete.

Lemma 4.1 ∥ supn∈N,π∈E,ρ∈F

(πρp)nV ∥V < ∞.

Proof: From Assumption 3.1(b) it follows that

(πρp)nW ≤ αnW +1

1− αβ.

By Lemma 3.2 we get

(πρp)nV ≤ (πρp)nW + const ≤ αnW + const′ ≤ αnV + const′.

The statement is implied by this.

Let Tw be the operator given by

Twu(x, a, b) := (1− ϑ)(ϑk(x, a, b) + w(x)) + ϑpu(x, a, b)

for all u ∈ V, x ∈ X, a ∈ A , b ∈ B. We note that Tw has essentiallythe same structure as the cost operator T used in stochastic dynamic


programming and stochastic game theory. This implies that some ofour proofs are very similar to known proofs. Therefore we will restrictourselves to a few remarks in these cases. (A very good exposition ofbasic ideas and recent developments in stochastic dynamic programmingcan be found in the books of Hernandez- Lerma and Lasserre [1], [2].)Obviously,

Twu = (1− ϑ)ϑT (u

1− ϑ) + (1− ϑ)w. (4.8)

Lemma 4.2 Let w ∈ V, π ∈ E, ρ ∈ F . Then the functional equation

u = πρTwu (4.9)

has a unique solution uw = Sπρw ∈ V and it holds:

Sπρw = limn→∞

(πρTw)nu = (1− ϑ)

∞!

n=0

ϑn(πρp)n(ϑπρk + w) (4.10)

for every u ∈ V.

Proof: We note that πρTwV ⊆ V. From (3.2 ) it follows that πρTw

is contracting on V with modulus λ. The rest of the proof follows byBanach’s Fixed Point Theorem.

We define a new operator Sγ,π,ρ by

Sγ,π,ρw := −(1− IC)γ + Sπρw − ICµw (4.11)

for π ∈ E, ρ ∈ F, w ∈ V where Sπρ is the operator defined by thefunctional equation (4.9 ). The following lemma gives some propertiesof this operator.

Lemma 4.3 (a) Sγ,π,ρV ⊆ V.

(b) Sγ,π,ρ is isotonic.

(c) Sγ,π,ρ is contracting.

Proof: (a) is obvious.

24 HEINZ-UWE KUENLE

(b) Using (4.10 ) we get

Sγ,π,ρw = −(1− IC)γ + (1− ϑ)∞!

n=0

ϑn(πρp)n(ϑπρk + w)− ICµw

= −(1− IC)γ + (1− ϑ)∞!

n=0

ϑn+1(πρp)nπρk

+(Qϑ,π,ρ − ICµ)w. (4.12)

The statement follows from Assumption 3.1 (a).

(c) By Lemma 3.2 and (4.12 ) we get for u, v ∈ V

|Sγ,π,ρu− Sγ,π,ρv| = |(Qϑ,π,ρ − ICµ)(u− v)|≤ (Qϑ,π,ρ − ICµ)V ∥u− v∥V≤ λV ∥u− v∥V . (4.13)

Lemma 4.4 The operator Sγ,π,ρ has in V a unique fixed point uγ,π,ρ.µuγ,π,ρ is continuous and non-increasing in γ.

Proof: The existence and uniqueness of the fixed point follows fromLemma 4.3 by Banach’s Fixed Point Theorem. From Sγ,π,ρv ≥ Sγ′,π,ρvfor γ ≤ γ′, and the isotonicity of Sγ,π,ρ it follows that uγ,π,ρ ≥ uγ′,π,ρ.Hence, µuγ,π,ρ ≥ µuγ′,π,ρ. Furthermore, for arbitrary γ, γ′

|uγ,π,ρ − uγ′,π,ρ| = |(1− IC)(γ′ − γ) + (Qϑ,π,ρ − ICµ)(uγ,π,ρ − uγ′,π,ρ)|

≤ |γ − γ′|V + λ∥uγ,π,ρ − uγ′,π,ρ∥V V

Hence,

∥uγ,π,ρ − uγ′,π,ρ∥V ≤ |γ − γ′|+ λ∥uγ,π,ρ − uγ′,π,ρ∥V

and

|µuγ,π,ρ − µuγ′,π,ρ| ≤ ∥uγ,π,ρ − uγ′,π,ρ∥V µV

≤ |γ − γ′|1− λ

µV .

Theorem 4.5 There exists a constant g and v ∈ V such that

g + v = πρk + πρpv. (4.14)

It holds:

g = Φπ∞ρ∞ .


Proof: From Lemma 4.4 it follows that there is a γ∗ with γ∗ = µuγ∗,π,ρ.Hence,

uγ∗,π,ρ = Sγ∗,π,ρuγ∗,π,ρ

= −(1− IC)γ∗ + Sπρuγ∗,π,ρ − ICµuγ∗,π,ρ

= Sπρuγ∗,π,ρ − γ∗. (4.15)

Let w∗ := uγ∗,π,ρ. If we put w = w∗ in (4.9 ), then we get

Sπρw∗ = (1− ϑ)(ϑπρk + w∗) + ϑπρpSπρw

∗.

It follows by (4.15 ) that

w∗ + γ∗ = (1− ϑ)(ϑπρk + w∗) + ϑπρp(w∗ + γ∗).

Therefore,

ϑw∗ + (1− ϑ)γ∗ = (1− ϑ)ϑπρk + ϑπρpw∗.

For g = γ∗

ϑ , v = w∗

1−ϑ we get (4.14 ). From (4.14 ) it follows

Ng =N−1!

n=0

(πρp)nπρk + (πρp)Nv − v.

If we consider Lemma 4.1 we get

g = limN→∞

1

N

N−1!

n=0

(πρp)nπρk = Φπ∞ρ∞ .

5 Existence of optimal stationary strategies

We give first a lemma which concerns a certain auxiliary one-stage game.The results of this lemma are well-known and can be derived, for in-stance, from the results in [12].

Lemma 5.1 Let u : X ×A ×B → R a σX×A×B-measurable function

with supx∈X,a∈AAA(x),b∈BBB(x)|u(x,a,b)|V (x) < ∞. Then it holds:

(a) infπ∈E supρ∈F πρu = supρ∈F infπ∈E πρu ∈ V.

(b) There are π∗ ∈ E, ρ∗ ∈ F with π∗ρu ≤ π∗ρ∗u ≤ πρ∗u for allπ ∈ E, ρ ∈ F.

26 HEINZ-UWE KUENLE

For a function v : X×A → R (v : X×B → R) we put Lv := infπ∈E πv(Uv := supρ∈F ρv). We can now prove the following lemma concerningan auxiliary functional equation.

Lemma 5.2 The functional equation

u = infπ∈E

supρ∈F

(1− ϑ)(ϑπρk + w) + ϑπρpu

= LUTwu

= (1− ϑ)ϑLUT (u

1− ϑ) + (1− ϑ)w (5.16)

has for every w ∈ V a unique solution u∗ =: Sw in V.

Proof: Let w ∈ V. Then it follows from Lemma 5.1 that LUTwV ⊆ V.Because πρTw is contracting on V, it holds for u, v ∈ V:

πρTwu ≤ πρTwv + λ∥u− v∥V V.

Since L and U are isotonic it follows:

LUTwu ≤ LUTwv + λ∥u− v∥V V.

Because u and v can be interchanged, we get that LUTw is alsocontracting. The statement follows by Banach’s Fixed Point Theorem.

In the following lemma Sπρ and S are the operators defined by thefunctional equations (4.9 ) and (5.16 ).

Lemma 5.3 For every w ∈ V there are π∗ ∈ E, ρ∗ ∈ F with

Sπ∗,ρw ≤ Sw ≤ Sπ,ρ∗w (5.17)

for all π ∈ E, ρ ∈ F . Furthermore,

Sw := infπ∈E

supρ∈F

Sπρw. (5.18)

Proof: It follows from Lemma 5.1 that there are π∗ ∈ E, ρ∗ ∈ F suchthat

π∗ρT (uw

1− ϑ) ≤ LUT (

uw1− ϑ

)

≤ πρ∗T (uw

1− ϑ) (5.19)


where uw = Sw. Hence,

π∗ρTwuw ≤ LUTwuw = uw ≤ πρ∗Twuw (5.20)

for all π ∈ E, ρ ∈ F. Assume that

(π∗ρTw)nuw ≤ uw ≤ (πρ∗Tw)

nuw (5.21)

for n ∈ N. Then it follows from (5.20 ) that

uw ≤ πρ∗Tw((πρ∗Tw)

nuw = (πρ∗Tw)n+1uw. (5.22)

Analogously,

uw ≥ (π∗ρTw)n+1uw. (5.23)

From (5.22 ) and (5.23 ) it follows by mathematical induction that(5.21 ) holds for all n ∈ N. For n → ∞ we get (5.17 ).(5.18 ) follows immediately from (5.17 ).

We define a new operator Sγ by

Sγw := −(1− IC)γ + Sw − ICµw

for π ∈ E, ρ ∈ F, w ∈ V, γ ∈ R. The following lemma gives someproperties of this operator.

Lemma 5.4 (a) SγV ⊆ V.

(b) Sγ is isotonic.

(c) Sγ is contracting with modulus λ.

(d) Sγ has in V a unique fixed point vγ . It holds limn→∞(Sγ)nu = vγfor every u ∈ V. Moreover, vγ is isotonic and continuous in γ.

Proof: (a) is obvious.

(b) From (4.11 ) and (5.18 ) it follows that

Sγw = infπ∈E

supρ∈F

Sγ,π,ρw.

By Lemma 4.3 we get the statement.

28 HEINZ-UWE KUENLE

(c) Let w′, w′′ ∈ V. By Lemma 5.3 it follows that there are π′′ ∈ E,ρ′ ∈ F, such that

Sw′ ≤ Sπ,ρ′w′

Sw′′ ≥ Sπ′′,ρw′′

for all π ∈ E, ρ ∈ F. Hence,

Sγw′ − Sγw

′′ = −(1− IC)γ + Sw′ − ICµw′

−(−(1− IC)γ + Sw′′ − ICµw′′)

≤ −(1− IC)γ + Sπ′′,ρ′w′ − ICµw

′

−(−(1− IC)γ + Sπ′′,ρ′w′′ − ICµw

′′)

= Sγ,π′′,ρ′w′ − Sγ,π′′,ρ′w

′′

≤ λV ∥w′ − w′′∥V

since Sγ,π′′,ρ′ is contracting (see Lemma 4.3). Because w′ and w′′ canbe interchanged, we get the statement.

(d) The existence of a unique fixed point vγ ∈ V andlimn→∞(Sγ)nu = vγ for every u ∈ V follows from Banach’s Fixed PointTheorem. For γ′ ≤ γ it holds

Sγw ≤ Sγ′w = Sγw + (1− IC)(γ − γ′) ≤ Sγw + (γ − γ′)V.

Assume that for n > 1

Sn−1γ vγ′ ≤ vγ′ ≤ Sn−1

γ vγ′ +γ − γ′

1− λV.

Then it follows

Snγ vγ′ ≤ Sγ′Sn−1

γ vγ′ ≤ Sγ′vγ′ = vγ′ ≤ Sγ′(Sn−1γ vγ′ +

γ − γ′

1− λV )

≤ Sγ(Sn−1γ vγ′ +

γ − γ′

1− λV ) + (γ − γ′)V

≤ Snγ vγ′ +

λ(γ − γ′)

1− λV + (γ − γ′)V (see (c))


= Snγ vγ′ +

γ − γ′

1− λV.

Hence, by mathematical induction we find that this inequality holds forall n ∈ N. For n → ∞ it follows

vγ ≤ vγ′ ≤ vγ +γ − γ′

1− λV.

The rest of the statement is implied by this.

Theorem 5.5 There are g = const and v ∈ V with

g + v = LUTv. (5.24)

It holds

g = infΠ∈E∞

supP∈F∞

ΦΠP .

Furthermore, there is an optimal stationary strategy pair.

Proof: From Lemma 5.4 it follows that µvγ is non-increasing in γ.Therefore, there is a γ∗ with γ∗ = µvγ∗ .

vγ∗ = Sγ∗vγ∗

= −(1− IC)γ∗ + Svγ∗ − ICµvγ∗

= Svγ∗ − γ∗. (5.25)

Let w∗ := vγ∗ . If we put w = w∗ in (5.16 ) then we get

Sw∗ = LU((1− ϑ)(ϑk + w∗) + ϑpSw∗).

It follows by (5.25 )

w∗ + γ∗ = LU((1− ϑ)(ϑk + w∗) + ϑp(w∗ + γ∗)).

Therefore,

ϑw∗ + (1− ϑ)γ∗ = LU((1− ϑ)ϑk + ϑpw∗).

For g = γ∗

ϑ , v = w∗

1−ϑ we get (5.24 ).From (5.24 ) and Lemma 5.1 it follows that there are π∗ ∈ E, ρ∗ ∈ F,

with

π∗ρnTvγ∗ − g ≤ vγ∗ ≤ πnρ∗Tvγ∗ + ε− g

30 HEINZ-UWE KUENLE

for all Π = (πn) ∈ E∞, P = (ρn) ∈ F∞. It follows

π∗ρ0Tπ∗ρ1T · · ·π∗ρNTvγ∗ − (N + 1)g

≤ vγ∗ ≤ π0ρ∗Tπ1ρ

∗T · · ·πNρ∗Tvγ∗ − (N + 1)g

For N → ∞ we get

ΦΠρ∗∞ ≤ g ≤ Φπ∗∞P

for all Π ∈ E∞, P ∈ F∞. This implies

g = infΠ∈E∞

supP∈F∞

ΦΠP

and the optimality of (π∗∞, ρ∗∞).

Heinz-Uwe KuenleBrandenburgische Technische Universitat CottbusInstitut fur MathematikPF 101344D-03013 CottbusGERMANYPhone: +49 (0355) 69 3151Fax: +49 (0355) 69 [email protected]

References

[1] Hernandez-Lerma, O.; Lasserre, J. B.: Discrete-Time MarkovControl Processes: Basic Optimality Criteria. Applications ofMathematics 30. Springer-Verlag, New York, 1996

[2] Hernandez-Lerma, O.; Lasserre, J. B.: Further Topics onDiscrete-Time Markov Control Processes. Applications of Math-ematics 42. Springer-Verlag, New York, 1999

[3] Kuenle, H.-U.: Stochastische Spiele und Entscheidungsmodelle.Teubner-Texte zur Mathematik 89. Teubner-Verlag, Leipzig,1986

[4] Kuenle, H.-U.: Stochastic games with complete information andaverage cost criterion. Advances in Dynamic Games and Applica-tions [edited by J.A. Filar, V. Gaitsgory, K. Mizukami] (Annals ofthe International Society of Dynamic Games, Vol. 5) Birkhauser,Boston, 2000, 325– 338


[5] Kuenle, H.-U.: Equilibrium strategies in stochastic games withadditive cost and transition structure. International Game The-ory Review, 1, 1999, 131–147

[6] Kuenle, H.-U.: On multichain Markov games. Annals of the In-ternational Society of Dynamic Games. Birkhauser, to appear

[7] Maitra, A.; Sudderth, W.: Borel stochastic games with limsuppayoff. Ann. Probab., 21, 1993, 861–885

[8] Maitra, A.; Sudderth, W.: Finitely additive and measurablestochastic games. Internat. J. Game Theory, 22, 1993, 201–223

[9] Maitra, A.; Sudderth, W.: Finitely additive stochastic gameswith Borel measurable payoffs. Internat. J. Game Theory, 27,1998, 257–267

[10] Meyn, S. P.; Tweedie, R. L.: Markov Chains and Stochastic Sta-bility. Communication and Control Engineering Series. Springer-Verlag, London, 1993

[11] Neveu, J.: Mathematical Foundations of the Calculus of Proba-bility. Holden-Day, San Francisco 1965

[12] Nowak, A. S.: Minimax selection theorems. J. Math. Anal. Appl.103, 1984, 106–116.

[13] Nowak, A. S.: Zero-sum average payoff stochastic games withgeneral state space. Games and Econ. Behavior 7, 1994, 221–232

[14] Nowak, A. S.: Optimal strategies in a class of zero-sum ergodicstochastic games.Math. Meth. Oper. Res. 50, 1999, 399–419

[15] Rieder, U.: Average optimality in Markov games with generalstate space. Proc. 3rd International Conf. on Approximation andOptimization, Puebla, 1995


Little cubes and homotopy theory ∗

Dai Tamaki

.

AbstractThis is a survey article on the theory of configuration spaces oflittle cubes, which is naturally related to the homotopy commu-tativity of double or more highly iterated loop spaces. The articlebegins with historical background and then surveys important ap-plications to homotopy theory.

1991 Mathematics Subject Classification: 55-02, 55P35Keywords and phrases: Configuration space, loop space, quasifibration

1 Introduction

A little n-cube is the image of an affine embedding

c : In → In

whose image has all edges parallel to the corresponding edges of thestandard n-cube In.

c

∗Invited article

33

34 DAI TAMAKI

The configuration space of j little n-cubes Cn(j) is the set of j-tuples(c1, · · · , cj) of little n-cubes whose images have disjoint interiors fromeach other. Cn(j) is topologized with the compact-open topology.

Since a little cube is determined by its image, it is safe to discuss onlittle cubes by drawing a picture like:

c1

c2

The reader might notice that he or she has seen this kind of picturein a textbook of algebraic topology.

It is an elementary theorem in homotopy theory that the n-th ho-motopy group of a pointed space X, πn(X), is an Abelian group, orequivalently the n-th loop space on X, ΩnX, is a homotopy commuta-tive H-space, if n ≥ 2.

In most textbooks on algebraic topology, this fact is explained asfollows: Suppose an element α ∈ πn(X) is represented by a continuousmap

f : (In, ∂In) −→ (X, ∗),

where ∗ denotes the base point of X. Pick up another element β ∈πn(X) represented by

g : (In, ∂In) −→ (X, ∗).

The product of α and β in πn(X), α+ β, is represented by the map

f + g : (In, ∂In) −→ (X, ∗)

defined by the following picture.

LITTLE CUBES AND HOMOTOPY THEORY 35

.....................................

......................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................

............................................................

.........................................................

...................................................................................................................................................................................................................................................................................................................

X

! ∗........................................................................................................................................................................................................................................................................................................................................................................

........................................................

.................................................................................................................................................................................................................................................................................................

f

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................

.....................

g

.................................................

...................................................................

.......................................................................................................................................................................................................................................................................................................................................................⑦

On the other hand, β + α is represented by the map

g + f : (In, ∂In) −→ (X, ∗)

defined by the following picture.

......................................

.........................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................

............................................................

.........................................................

...................................................................................................................................................................................................................................................................................................................

X

! ∗........................................................................................................................................................................................................................................................................................................................................................................

........................................................

.................................................................................................................................................................................................................................................................................................

g

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................f

....................................

..........................................

..................................................

.......................................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................⑦

If n ≥ 2, a homotopy connecting these two maps can be obtainedby rotating two (rectangular) cubes in In.

f g f

g

g

f

g f

36 DAI TAMAKI

Notice that domains of f + g and g+ f can be regarded as elementsof Cn(2) and the above picture can be interpreted as a path in Cn(2),since we need to keep two cubes separated while rotating.

In fact, Cn(2) is path-connected (more generally Cn(j) is known tobe (n− 2)-connected). The path-connectivity of Cn(2) implies the com-mutativity of πn(X) or the homotopy commutativity of ΩnX. This canbe considered as the first application of the topology of little cubes tohomotopy theory.

It is natural to extend this idea to study higher homotopy commu-tativity of iterated loop spaces. Boardman and Vogt began the firstsystematic study in this direction [5, 6]. A few years later, Peter Mayestablished a concrete relationship between iterated loop spaces and lit-tle cubes by proving the recognition principle [40]. The study of littlecubes also led him to introduce the notion of operad, which turned outto be an important object naturally appearing in many fields of mathe-matics and mathematical physics. But we are not going to cover operadin this article. Those who are interested in operad are recommended totake a look at [36], for example.

The following is the organization of this article:

Section 2. Basic Definitions and Fundamental Facts: recalls ba-sic definitions and classical results used in later sections.

Section 3. Stable Splitting of Iterated Loop Spaces: reviews thedevelopment of homotopy theory stimulated by Snaith’s theoremof stable splitting of ΩnΣnX.

Section 4. Constructing Maps and Spaces: shows that various im-portant maps and spaces have been constructed by using littlecubes.

Section 5. Problems: is a collection of open problems related to littlecubes.

2 Basic Definitions and Fundamental Facts

Let us begin with a precise definition of little cube.


Definition 2.1.1 For a positive integer n, a little n-cube c is a mapIn −→ In which can be decomposed into the following form:

c = ℓ1 × · · ·× ℓn

where each ℓi : I −→ I is an orientation preserving Affine embed-ding. The space of little n-cubes is denoted by Cn(1) and topologizedwith compact-open topology. Thus Cn(1) is a subspace of Map(In, In).

The configuration space of j little n-cubes Cn(j) is defined as follows:

Cn(j) = (c1, · · · , cj) ∈ Cn(1)j |ci(IntIn) ∩ ck(IntIn) = ∅ if i = k

Cn(j) admits a natural action of the symmetric group of j-letters,Σj, by permuting the indices of little cubes.

Cn = Cn(j)j is called the little n-cube operad.We also need the case n = ∞ to study infinite loop spaces. Define

C∞(j) = colim−→n

Cn(j)

where colimit is taken over the inclusion maps induced by

In × 0 → In+1.

Obviously C∞(j) inherits the action of Σj.

Thanks to the following fact, we do not have to struggle with thecompact-open topology on Cn(j).

Lemma 2.1.2 Define a map

ξn : Cn(1) −→ I2n

by ξn(c) = (c(14 , · · · ,14), c(

34 , · · · ,

34)), then ξn is an embedding of Cn(1)

as an open subset of I2n.

As is stated in Introduction, the following is one of the most funda-mental properties of little cube.

Lemma 2.1.3 Cn(j) is (n−2)-connected. Hence C∞(j) is contractible.

The following map plays a central role in studying the relationshipbetween the configuration space of little cubes Cn(j) and an n-fold loopspace ΩnX.

38 DAI TAMAKI

Definition 2.1.4 For a pointed space X, define

θn,j : Cn(j)× (ΩnX)j −→ ΩnX

as follows: For (c1, · · · , cj) ∈ Cn(j) and ω1, · · · ,ωj ∈ ΩnX,

θn,j(c1, · · · , cj ;ω1, · · · ,ωj) : In −→ X

is defined to be ωi on each cube ci(In) and maps the outside of littlecubes to the basepoint. More precisely

θn,j(c1, · · · , cj ;ω1, · · · ,ωj)(t) =

=

!ωi(c

−1i (t)) if t ∈ ci(In) for some 0 ≤ i ≤ j,

∗ otherwise.

θn,j is obviously Σj-equivariant, hence we have an induced map onthe quotient, which is denoted by the same notation:

θn,j : Cn(j)×Σj (ΩnX)j −→ ΩnX

These maps θn,j satisfy the following compatibility conditions.

Lemma 2.1.5 Let ∗ denote the constant loop to the basepoint of ΩnX,then, for (c1, · · · , cj) ∈ Cn(j) and ω1, · · · ,ωi−1,ωi+1, · · · ,ωj ∈ ΩnX, wehave

θn,j(c1, · · · , cj ;ω1, · · · ,ωi−1, ∗,ωi+1, · · · ,ωj)

= θn,j−1(c1, · · · , ci−1, ci+1, · · · , cj ;ω1, · · · ,ωi−1,ωi+1, · · · ,ωj)

Definition 2.1.6 For a pointed space Y , an action of Cn is a collectionof maps

θj : Cn(j)×Σj Yj −→ Y

for j = 0, 1, · · · satisfying the same relations as in Lemma 2.1.5.

In other words, an n-fold loop space has an action of Cn. May provedthat, conversely, the existence of an action of Cn on a pointed space Ycan be used as a criterion for Y to be equivalent to an n-fold loop space.

Theorem 2.1.7 (Recognition Principle) Let X be a path-connectedspace with a nondegenerate basepoint. Then X has a weak homotopytype of an n-fold loop space if and only if it has an action of little n-cube operad.


The relations in Lemma 2.1.5 can be used to pierce the spacesCn(j)×Σj (Ω

nX)j together to get a single space Cn(ΩnX) as follows.

Definition 2.1.8 Let Y be a pointed space with basepoint ∗. Generatean equivalence relation ∼ on

∐j Cn(j)×Σj Y

j by relations

(c1, · · · , cj ; y1, · · · , yi−1, ∗, yi+1, · · · , yj)∼ (c1, · · · , ci−1, ci+1, · · · , cj ; y1, · · · , yi−1, yi+1, · · · , yj).

Define Cn(Y ) by

Cn(Y ) =

⎛

⎝∐

j

Cn(j)×Σj Yj

⎞

⎠ / ∼ .

Corollary 2.1.9 θn,j induces a well-defined continuous map

θn : Cn(ΩnX) −→ ΩnX.

Note that Cn(Y ) can be defined for any pointed space Y and wehave a natural map

σn : Y −→ Cn(Y )

induced by the inclusion of Cn(1)×Y . The reader can easily check thatθn σn ≃ id if Y = ΩnX. In other words, if Y is equivalent to an n-foldloop space ΩnX, Y is a retract of Cn(Y ).

Y Cn(Y )

Y.

σn

id

θn

Notice that the evaluation map on ΩnX

eval : Σn(ΩnX) −→ X

and the Freudenthal suspension

En : X −→ ΩnΣnX

40 DAI TAMAKI

make the following diagram commutative

ΩnX ΩnΣn(ΩnX)

ΩnX.

En

#

id

Ωneval

This similarity between the functors Cn and ΩnΣn can be explainedby the following important theorem of May.

Theorem 2.1.10 (Approximation Theorem) Let X denote a path-connected space with a nondegenerate basepoint. Then we have a naturalweak homotopy equivalence

Cn(X) ≃ ΩnΣnX.

Recognition Principle is important because it is the origin of thetheory of operad. But for practical applications in homotopy theory,Approximation Theorem is far more useful, because it gives us a com-binatorial model for iterated loop spaces and thus making them easierto handle.

Besides May’s model, some other combinatorial models for ΩnΣnXhave been discovered. The oldest is the reduced product of James[27] which is a model for ΩΣX. Milgram [41] constructed a model forΩnΣnX generalizing James’ construction. Segal [43] suggested to usethe configuration space of distinct points instead of little cubes. Theresulting space is equivalent to the little cube model. May’s approachwas improved by J.Caruso and S.Waner [11] to prove an approxima-tion theorem for nonconnected spaces. There are also simplicial modelsfound by Milnor [42] for n = 1, by Barratt and Eccles [2] for n = ∞and by Jeff Smith [44] for general n.

These are essentially equivalent to each other. Among these models,however, the “little cube model” (and its variants, like the one usingconfiguration space of distinct points) has been most popular. This ispartly because Victor Snaith used the little cube model to prove thestable splitting theorem, which is the subject of the next section.


3 Stable Splitting of Iterated Loop Spaces

3.1 Snaith’s Theorem

A stable splitting (in fact, a splitting after a single suspension) of ΩΣXwas proved by James in 1955 [27] using a combinatorial constructionof ΩΣX, called the James construction or reduced product. Importanttools in classical homotopy theory, like the (James)-Hopf invariant

H : ΩSn+1 −→ ΩS2n+1

and EHP fibration

Sn E−→ ΩSn+1 H−→ ΩS2n+1

have been constructed by using this splitting.The little cube model of ΩnΣnX can be regarded as a generalization

of the James construction. In fact, Snaith [45] proved a stable splittingtheorem for ΩnΣnX by using the little cube model.

Theorem 3.1.1 For a pointed space X with a nondegenerate basepointand a positive integer n, we have a weak stable homotopy equivalence

ΩnΣnX ≃S

∞∨

j=1

Cn(j)+ ∧Σj X∧j .(1)

This holds also for n = ∞.

Note that Cn(X) is naturally filtered by the number of cubes

FmCn(X) =

⎛

⎝∐

j≤m

Cn(j)×Σj Xj

⎞

⎠ / ∼

and the right hand side of the above splitting is the “subquotient” ofCn(X) with respect to this filtration.

Later, Cohen, May and Taylor generalized Snaith’s theorem to moregeneral settings [15]. For those who are curious about how to constructthese stable splittings, the appendix of [12] will be of interest.

It is also worthwhile to note that these splittings can be obtainedby a very simple argument in the category of spectra. This fact, provedby R.L.Cohen [24], depends on the existence of the Σj-equivariant halfsmash product of spectra constructed by Lewis, May and Steinberger[37].

42 DAI TAMAKI

3.2 Applications of Stable Splitting

After the stable splitting theorem was proved by Snaith, the case n = 2and X = S2k−1 became a subject of intense study in 70’s. Mahowald[38] found a new infinite family in the 2-primary components of thestable homotopy groups of spheres by using the summands of the stablesplitting of Ω2S9. An odd primary analogue was obtained by R.L.Cohen[23].

These facts suggest that the cohomology of the summands of theSnaith splitting of Ω2S2k+1 is equipped with an important module struc-ture over the Steenrod algebra. To be more precise, we need the follow-ing definition.

Definition 3.2.1 For any prime p and integer k ≥ 0, define a moduleM(k, p) over the mod p Steenrod algebra as follows:

M(k, p) =

!A2/(χ(Sq

i)|i > k) if p = 2,Ap/(χ(βϵP i)|i > k, ϵ = 0, 1) if p is odd,

where χ denotes the canonical anti-automorphism on Ap.

Theorem 3.2.2 For any prime p and integer k ≥ 0, there exists ap-local spectrum B(k, p) satisfying the following properties:

1. H∗(B(k, p);Z/pZ) ∼= M(k, p) as modules over the Steenrod alge-bra.

2. Let jk : B(k, p) −→ HZ/pZ be a generator of H0(B(k, p);Z/pZ).Then for any CW complex X, the induced map of generalized ho-mology theories

(jk)∗ : B(k, p)∗(X) −→ H∗(X;Z/pZ)

is surjective for q ≤ 2k + 1 if p = 2 and for q ≤ 2p(k + 1)− 1 if pis odd.

Furthermore such a spectrum is unique up to homotopy.

Definition 3.2.3 B(k, p) is called the Brown-Gitler spectrum [7].

Mahowald in [38] conjectured that the stable summands of Ω2S2k+1

localized at 2 realize the Brown-Gitler spectra. This fact was proved byBrown and Peterson [8].


Theorem 3.2.4 Localized at 2, we have the following homotopy equiv-alence

C2(j)+ ∧Σj (S2k−1)∧j ≃ Σj(2k−1)B([ j2 ]).

Odd primary analogs are proved by R.Cohen [23].

Theorem 3.2.5 Localized at an odd prime p, we have the followinghomotopy equivalence when j = mp+ r for p > r > 0:

C2(pj)+ ∧Σpj (S2k−1)∧pj ≃ Σj(2pk−2)B(m).

There is another way of describing the summands of the stable split-ting of ΩnSn+k closely related to the above theorems.

Consider the vector bundle

pn,j : Cn(j)×Σj Rj −→ Cn(j)/Σj .

It is easy to see that the Thom complex of pn,j , T (pn,j), is Cn(j)+∧Σj

Sj which is a stable summand of ΩnSn+1. More generally, we have

T (pn,j ⊕ · · ·⊕ pn,j︸︷︷︸k

) = Cn(j)+ ∧Σj Sjk.

It is worthwhile to note that the infinite families in the stable homo-topy groups found by Mahowald and R.Cohen comes from the trivialityof p2,j⊕p2,j proved by F.Cohen, Mahowald and Milgram [14]. The orderof pn,j in general was studied in [13].

Mahowald’s construction of infinite families in the stable homotopygroups of spheres is one of the most important applications of Ω2S2k+1

in stable homotopy theory. Note that the classical applications of theJames splitting of ΩS2k+1 live in the unstable world: For example, themod p Hopf invariant

Hp : ΩS2n+1 −→ ΩS2np+1

is defined to be the adjoint to the following composition

ΣΩS2n+1 ≃ Σ

⎛

⎝∨

j

(S2n)∧j

⎞

⎠ −→ Σ(S2n)∧p = S2np+1.

In order to get an unstable application of the splitting of Ω2S2k+1,for example a “secondary Hopf invariant”, we need to desuspend the

44 DAI TAMAKI

Snaith splitting. However, as stated in [12], Ω2Σ2X does not split infinitely many suspensions. This is one of the crucial differences betweenΩΣX and ΩnΣnX for n ≥ 2.

This is the difficulty in applying ΩnΣnX in unstable homotopy the-ory when n ≥ 2. But F.Cohen, May and Taylor [15] proved that eachpiece can be split off in finitely many suspensions.

Theorem 3.2.6 Fix a positive integer k, then there exists a positiveinteger L (depending on k) and a map

ΣLCn(X) −→ ΣLCn(k)+ ∧Σk X∧k

which is equivalent to the composition of the Snaith splitting and theprojection on the k-th component after suspending infinitely many times.

Furthermore F.Cohen proved that L can be taken to be 2k whenn = 2 [12]. This allowed him to construct a map

σn : W (n) −→ Ω2pW (n+ 1)

which can be considered to be the “secondary double-suspension”, whereW (n) is the homotopy theoretic fiber of the double suspension map

E2 : S2n−1 −→ Ω2S2n+1.

σn played an essential role in the work of Mahowald and Thomp-son [39, 48] where they determined the unstable v1-periodic homotopygroups of spheres.

B.Gray began a systematic study in this direction in [28]. He intro-duced the notion of EHP spectra which is an unstable way of studyingToda-Smith spectra V (n) in the fashion of Cohen-Moore-Neisendorfer[18, 19, 20]. In order his program to be accomplished, we need to findvarious unstable maps between loop spaces. As in the case of the clas-sical EHP sequence, those maps could be constructed by using unsta-ble splitting of loop spaces. In fact, Gray is very close to proving theexistence of an EHP spectrum for V (0). His construction would becompleted, if we could prove Ω2Σ2X localized at an odd prime splitsafter suspending twice, which is conjectured in [28]. However nothingis known about localization of ΩnΣnX.


4 Constructing Maps and Spaces

As we have seen in the previous section, little cube model played an im-portant role in the construction and applications of the stable splittingsof iterated loop spaces.

It is often helpful to have extra geometric information even thoughwhat we need is abstract homotopy-theoretic results.

As is suggested by Quillen’s work on closed model category, homo-topy theory (stable or unstable) is fairly formal. It can be axiomatizednicely. But we need concrete models for practical applications.

Little cubes can be used to construct important spaces and maps.The followings are some of the examples.

4.1 Fibrations

May proved the Approximation theorem (Theorem 3.1.1) by construct-ing a little cube model for the path-loop fibration:

ΩnΣnX −→ PΩn−1ΣnX −→ Ωn−1ΣnX.

Definition 4.1.1 For a pair of pointed space (X,A), define En(X,A)to be the subspace of Cn(X) consisting of elements (c1, · · · , cj ;x1, · · · , xj)satisfying the following properties: if xi1 , · · · , xiℓ ∈ A then

(prn−1(ci1), · · · , prn−1(ciℓ)) ∈ Cn−1(ℓ),

where prn−1 is the projection onto the last (n− 1)-coordinates.

Theorem 4.1.2 (May) 1. If X is a path-connected pointed spacewith nondegenerate basepoint, we have a quasifibration

Cn(X) −→ En(CX,X) −→ Cn−1(ΣX)(2)

and natural maps

α : Cn(X) −→ ΩnΣnX

α : En(CX,X) −→ PΩn−1ΣnX

making the following diagram commutative up to homotopy

Cn(X) En(CX,X) Cn−1(ΣX)

ΩnΣnX PΩn−1ΣnX Ωn−1ΣnX.

α

α

α

46 DAI TAMAKI

2. En(CX,X) is contractible. Therefore α is a weak homotopy equiv-alence.

Since path-loop fibration is a principal fibration with contractibletotal space, principal fibrations with fiber ΩnΣnX are pull-back of thepath-loop fibration. Thus it is natural to expect to obtain a little cubemodel of such a fibration by pulling back May’s quasifibration (2). Un-fortunately, however, taking a pull-back in general does not preservequasifibration.

The quasifibration (2) is not just a quasifibration. The base space isfiltered and the quasifibration satisfies a kind of local-triviality on eachsuccessive difference of the filtration.

Thanks to this additional structure, a pull-back of (2) has a chanceto be a quasifibration. In fact, S.-C. Wong proved the following [49]:

Definition 4.1.3 For a pointed space X, define W (k, n,ΣX) to be thehomotopy fiber of the Freudenthal suspension map:

Ωk−1En : Ωk−1ΣkX −→ Ωn+k−1Σn+kX.

Namely W (k, n,ΣX) is defined by the following pull-back diagram:

W (k, n,ΣX) PΩn+k−1Σn+kX

Ωk−1ΣkX Ωn+k−1Σn+kX.

Ωk−1En

Definition 4.1.4 For a pair of pointed spaces (X,A), define ξ(k, n;X,A)by the following pull-back diagram:

ξ(k, n;X,A) En+k−1(X,A)

Ck−1(X/A) Cn+k−1(X/A).

π(k,n)

πn+k

σn

Theorem 4.1.5 (Wong) For k ≥ 1, n ≥ 0 and a strong NDR pair(X,A),

π(k, n) : ξ(k, n;X,A) −→ Ck−1(X/A)

is a quasifibration with fiber Cn+k(A). Thus we have a weak homotopyequivalence

ξ(k, n;X,A) ≃ W (k, n;X/A).


Wong used ξ(k, n;X,A) to prove a stable splitting of W (k, n;X/A).Since the secondary-suspension σn was constructed by splitting off thep-adic piece of Ω2S2n+1 after the 2p-fold suspension, it would be in-teresting if we could desuspend Wong’s stable splitting to construct“tertiary suspension map”.

Wong’s idea can be also applied to the following case.Since Cn(X) has a structure of a monoid by concatenation and the

equivalenceCn(X) ≃ ΩnΣnX

is an equivalence of Hopf spaces, the p-th power map

p× : ΩnΣnX −→ ΩnΣnX

is equivalent to the p-th power map on Cn(X). Thus we have a diagram

? En(CX,X)

Cn(ΣX) Cn(ΣX).

πn

p×

in which all maps are defined in terms of little cubes.The homotopy fiber of p× : ΩnΣn+1X −→ ΩnΣn+1X is the mapping

space from the mod p Moore space Map∗(Pn+1(p),Σn+1X). Following

Wong’s idea K. Iwama recently proved the following [29].

Theorem 4.1.6 Let X be a pointed space with (X, ∗) a strong NDRpair. Define En(p)(CX,X) by the following pull-back diagram

En(p)(CX,X) En(CX,X)

Cn(ΣX) Cn(ΣX).

πn(p)

πn

p×

Then πn(p) is a quasifibration with fiber Cn+1(X). Therefore we have aweak homotopy equivalence

En(p)(CX,X) ≃ Map∗(Pn+1(p),Σn+1X),

where Pn+1(p) = Sn ∪p en+1.

48 DAI TAMAKI

It is not known if En(p)(CX,X) has a stable splitting. Althoughthe proof of the above theorem is parallel to that of Wong’s, the proof ofstable splitting by Wong cannot be applied in this case. The difficultycomes from the fact that the p-th power map on Cn(ΣX) does notpreserve the filtration.

4.2 Kahn-Priddy Transfer

As the Snaith splitting suggests, little cubes are useful for handlingstable maps in concrete ways. Another example of applications of lit-tle cube in stable homotopy theory is the construction of transfer byD.S.Kahn and S.B. Priddy.

Transfer is a map going to the wrong direction in the homology orcohomology, classically in group cohomology. It is a well-known factthat they can be realized as stable maps between spaces. One of theexamples is constructed by Kahn and Priddy by using the little cubemodel of Ω∞Σ∞X [30, 31].

Suppose we are given an N -fold covering space

p : E −→ B.

The symmetric group of N -letters acts on each fiber by deck transforma-tion and p can be considered to be a fiber bundle with fiber π1(B)/π1(E)and structure group ΣN . Since C∞(N) is a contractible space with afree ΣN -action, the projection

π : C∞(N) −→ C∞(N)/ΣN

is a universal ΣN -bundle. Thus we have the following pull-back diagram

E C∞(N)×ΣN (π1(B)/π1(E))N

B C∞(N)/ΣN .

ϕ

p

π

ϕ

DefineΦ : B −→ C∞(N)×ΣN EN

byΦ(p(x)) = (ϕ(x), xτ1, · · · , xτN )


where τ1, · · · , τN is a choice of coset representatives of π1(B)/π1(E).Composed with

C∞(N)×ΣN EN −→ C∞(E)

we have

B+Φ−→ C∞(E+) ≃ Ω∞Σ∞(E+).

The Kahn-Priddy transfer is the stable map adjoint to this map:

p! : Σ∞B+ −→ Σ∞E+.

Let p be a prime number. Kahn and Priddy applied this constructionto the covering

EΣp −→ BΣp

to get the famous Kahn-Priddy theorem which states

πS∗ (BΣp) −→ πS

∗ (S0)

is surjective on the p-primary components.

4.3 Strong Convergent Cobar Spectral Sequence

Little cubes can be used to construct a spectral sequence converging tothe homology of iterated loop spaces:

E2 ∼= Cotorh∗(Ωn−1ΣnX)(h∗, h∗) =⇒ h∗(ΩnΣnX).(3)

The following is a quick review of the construction by the author [46].Throughout this subsection h∗(−) denotes a multiplicative homologytheory.

In order to construct a spectral sequence (3), the most natural ideawould be to try to find a filtration on ΩnΣnX with which the E1-term ofthe resulting spectral sequence becomes the algebraic cobar constructionon the coalgebra h∗(Ωn−1ΣnX). Thanks to the stable splitting (1), itenough to define a filtration on each Cn(j), separately.

Suppose we have a filtration F−qCn(j) on each Cn(j). Define

F−sCn(X) =!

j

F−sCn(j)+ ∧Σj X∧j .

50 DAI TAMAKI

F−sCn(X) is a stable filtration for ΩnΣnX. The E1-term of the spec-tral sequence defined by this filtration is

E1−s,t = h−s+t(F−sCn(X), F−s−1Cn(X))

=!

j

h−s+t(F−sCn(j)+ ∧Σj X∧j , F−s−1Cn(j)+ ∧Σj X

∧j)

=!

j

h−s+t(F−sCn(j)/F−s−1Cn(j) ∧Σj X∧j).

On the other hand, the tensor algebra on h∗(Ωn−1ΣnX) has sum-mands of the following form:

h∗((Cn−1(j1)× · · ·× Cn−1(js))+ ∧Σj1×···×Σjs(ΣX)∧(j1+···+js)).

Thus what we need is filtrations on Cn(j) so that F−sCn(j) −F−s−1Cn(j) becomes s vertically aligned stacks of cubes. In order todefine such filtrations, we need auxiliary functions on Cn(j).

Letpr1 : Cn(1) −→ C1(1)

be the map induced by the projection onto the first coordinate. It isnot difficult to find a function

d : Cn(1)× Cn(1) −→ [0, 1]

with the following properties.

d(c, c′) = 0 ⇐⇒ pr1(c)(12) ∈ pr1(c

′)([0, 1]) or pr1(c′)(12) ∈ pr1(c)([0, 1])

d(c, c′) = 1 ⇐⇒ pr1(c)(12) = pr1(c

′)(12).

For c = (c1, · · · , cj) ∈ Cn(j), a stack in c is a subset of c1, · · · , cj.We say that a stack ci|i ∈ S is stable under gravity if and only ifd(ci1 , ci2) = 0 for any i1, i2 ∈ S.

With these terminologies we define a filtration on little cubes asfollows.

Definition 4.3.1 Let F0Cn(j) = F−1Cn(j) = Cn(j). For q > 0, c =(c1, · · · , cj) ∈ F−s−1Cn(j) if and only if, for any partition 1, · · · , j =S1

"· · ·

"Sq with S1 = ∅, · · · , Sq = ∅, at least one of the stacks corre-

sponding to S1, · · · , Sq is not stable under gravity.This filtration is called the gravity filtration. The spectral sequence

induced by this filtration is called the gravity spectral sequence.


On the contrary to our intension, it is not known whether the E1-term of the spectral sequence defined by this filtration is isomorphicto the cobar construction. However the following fact proved in [46] isenough to identify the E2-term with Cotor.

Theorem 4.3.2 There exists a stable filtration on En(CX,X) with thefollowing isomorphism of differential graded h∗-modules, if h∗(Ωn−1ΣnX)is flat over h∗:

E1(En) ∼= E1(Cn)⊗h∗ h∗(Cn−1(ΣX)),

where Er(En) is the spectral sequence induced by the filtration onEn(CX,X) and Er(Cn) is the spectral sequence induced by the gravityfiltration on Cn(X).

Furthermore (E1(En), d1) is acyclic.

Corollary 4.3.3 If h∗(Ωn−1ΣnX) is flat over h∗, we have the followingisomorphism

E2(Cn) ∼= Cotorh∗(Ωn−1ΣnX)(h∗, h∗).

For a pointed space X, if h∗(X) is a flat h∗-module, we have a spec-tral sequence, so-called the classical Eilenberg-Moore spectral sequencewith

E2 ∼= Cotorh∗(X)(h∗, h∗).

The E∞-term of this spectral sequence is not in general directly re-lated to h∗(ΩX). However, the gravity spectral sequence does calculateh∗(ΩnΣnX), since it is a direct sum of the spectral sequences definedby finite filtrations.

Theorem 4.3.4 The gravity spectral sequence converges to h∗(ΩnΣnX).

Although the gravity spectral sequence has this important property,it also has a disadvantage: the E1-term is mysterious. In order to usethe gravity spectral sequence for practical computations, it is importantto find good generators in the E2-term. In the case of the classicalEilenberg-Moore spectral sequence, we have the Dyer-Lashof operationswhich are defined in terms of the cobar construction. The followingfact helps us to find generators in the E2-term of the gravity spectralsequence.

Theorem 4.3.5 ([47]) If h∗(Ωn−1ΣnX) is flat over h∗, the gravityspectral sequence is isomorphic to the classical Eilenberg-Moore spectralsequence from the E2-term on.

52 DAI TAMAKI

4.4 Little Cubes with Overlappings Allowed

We have been considering little cubes disjoint from each other, so far. Ifwe remove this disjointness condition, we obtain loop spaces of differenttypes.

Definition 4.4.1 Define Dn(j) = Cn(1)j. This is the space of j littlen-cubes with overlappings allowed.

c1

c2

c3

c4

The symmetric group of j-letters acts on Dn(j) by renumbering ofcubes and the definition of Cn(X) can be applied without modificationto get a functor Dn(X).

Since the little cube operad Cn acts on Dn(X) in a natural way,Dn(X) is an n-fold loop space by the recognition principle of May. Infact, we can easily see that this is equivalent to a classical constructiondue to Dold and Thom, i.e. infinite symmetric product, and thus it hasa structure of an infinite loop space.

Definition 4.4.2 For a pointed space X and a nonnegative integer i,define

SPi(X) =

⎛

⎝i∐

j=0

Xj/Σj

⎞

⎠ / ∼

where the equivalence relation “∼” is generated by the relation removingbasepoint.

SPi(X) is called the i-th symmetric product of X. When i = ∞, itis called the infinite symmetric product.


Since there is no restriction on the movements of cubes, Dn(j) isΣj-equivariantly contractible. Thus we have

Proposition 4.4.3 For any pointed space X, Dn(X) is homotopy equiv-alent to SP∞(X).

The following is a classical theorem of Dold and Thom [25],

Theorem 4.4.4 For a path-connected pointed CW-complex X, we havethe following natural isomorphism

πn(SP∞(X)) ∼= !Hn(X)

As an immediate corollary, we have

Theorem 4.4.5 For a pointed connected CW-complex X of finite type,we have the following homotopy equivalence

SP∞(X) ≃∞"

n=0

K(πn(X), n).

Cn(X) and Dn(X) are two extreme cases: cubes are disjoint fromeach other in Cn(j), while any kinds of overlapping are allowed in Dn(j).π∗(C∞(X)) is the stable homotopy groups of X, while π∗(D∞(X)) isthe (reduced) integral homology groups of X.

Let us consider the following intermediate objects.

Definition 4.4.6 Let Din(j) be the subspace of Dn(j), consisting of j

little cube (c1, · · · , cj) satisfying the following property: each t ∈ In iscontained in the interior of the image of at most i cubes.

Din(j) inherits the action of Σj. For a pointed space X, define

Din(X) in the same way as Cn(X) is defined.

Now we have successive inclusions

Cn(X) = D0n(X) ⊂ D1

n(X) ⊂ · · · ⊂ D∞n (X) = Dn(X).

The commutativity of the following diagram

Cn(X) Dn(X)

X

⊂

"""

54 DAI TAMAKI

and the fact that the induced map on homotopy groups

π∗(X) −→ π∗(Dn(X)) ∼= π∗(SP∞(X)) ∼= !H∗(X)

coincides with the Hurewicz homomorphism implies that the Hurewiczhomomorphism factors through π∗(Di

n(X))

π∗(Din(X)) !H∗(X)

π∗(X).

⊂

F. Kato determined the homotopy type of Din(X) [33].

Theorem 4.4.7 For a path-connected pointed space X with (X, ∗) astrong NDR pair, we have the following weak homotopy equivalence

Din(X) ≃ ΩnSPiΣnX.

The idea of the proof is to extend May’s construction of the quasi-fibration (2).

Since

Di0(j) =

"∗ if j ≤ i∅ if j > i

we haveDi

0(X) = SPi(X).

Once we have a quasifibration

Din(X) −→ E −→ Di

n−1(ΣX)

with E contractible, we have the desired weak homotopy equivalence

Din(X) ≃ ΩDi

n−1(ΣX) ≃ · · · ≃ ΩnDi0(Σ

nX) = ΩnSPiΣnX.

Definition 4.4.8 For a pair of pointed spaces (X,A), define Ein(X,A)

to be the subspace of Cin(X) consisting of elements (c1, · · · , cj ;x1, · · · , xj)

satisfying the following properties: if xi1 , · · · , xiℓ ∈ A then

(prn−1(ci1), · · · , prn−1(ciℓ)) ∈ Cin−1(ℓ).


Proposition 4.4.9 For a pointed space X with (X, ∗) a strong NDRpair, the projection

Ein(CX,X) −→ Di

n−1(ΣX)

is a quasifibration with fiber Din(X).

Furthermore Ein(CX,X) is contractible if X is path-connected.

Sadok Kallel [32] independently studied an analogous constructionCi(Rn;X) by using (labelled) configuration space of points in Rn, in-stead of little cubes, and proved a homotopy equivalence

Ci(Rn;X) ≃ ΩnSPiΣnX

for connected CW complexes together with a “delooped version” of thishomotopy equivalence.

We should point out that most of the constructions using little n-cubes can be also done by using configuration space of points in Rn.One of the advantages of using configuration space of points rather thanlittle cubes is that Rn can be replaced with any space M . Kallel alsostudied Cd(M ;X) where M is a stably parallelizable smooth manifoldwith nonempty boundary.

5 Problems

We conclude this article with open problems related to little cubes.

1. Find a little cube model of the fiber of the secondary suspension

σn : W (n) −→ Ω2pW (n+ 1).

2. Find a stable splitting of Map∗(Pn(p),ΣnX).

3. Prove that, localized away from 2, Ω2Σ2X splits into a wedge ofthe (localized) Snaith summands after suspending twice.

4. For each prime p, find a combinatorial model for ΩnΣnX localizedat or away from p.

5. Compute the homology of ΩnSPiΣnX.

56 DAI TAMAKI

AcknowledgmentThe author is very grateful to Jesus Gonzalez, CINVESTAV, for

inviting him to write this article for Morfismos, which gave him a goodchance to review the development of the theory of little cubes. He wouldalso like to thank the referees for informing him of Kallel’s work and forpointing out ambiguous assumptions in some theorems.

Dai TamakiDepartment of Mathematical SciencesFaculty of Science, Shinshu University3-1-1 Asahi, Matsumoto, [email protected]

References

[1] Kudo T. and Araki S., Topology of Hn-spaces and H-squaringoperations., Memoirs of the Faculty of Science, Kyusyu Univ.,Ser. A, 10 (1956), 85–120.

[2] Barratt M.G. and Eccles P.J., Γ+-structures – I: A free groupfunctor for stable homotopy theory., Topology, 13 (1974), 25–45.

[3] Barratt M.G. and Eccles P.J., Γ+-structures – II: A recognitionprinciple for infinite loop spaces., Topology, 13 (1974), 113–126.

[4] Barratt M.G. and Eccles P.J., Γ+-structures – III: The stablestructure of Ω∞Σ∞A., Topology, 13 (1974), 199–207.

[5] Boardman J.M. and Vogt R.M., Homotopy-everything H-spaces.,Bulletin of the American Mathematical Society, 74 (1968), 1117–1122.

[6] Boardman J.M. and Vogt R.M., Homotopy Invariant AlgebraicStructures on Topological Spaces., Lecture Notes in Mathematics,vol. 347, Springer-Verlag, 1973.

[7] Brown E.H. and Gitler S., A spectrum whose cohomology is acertain cyclic module over the Steenrod algbebra., Topology, 12(1973), 283–295.

[8] Brown E.H. and Peterson F.P., On the stable decomposition ofΩ2Sr+2., Transactions of the American Mathematical Society, 243(1978), 287–298.


[9] Caruso Jeffrey L., A simpler approximation to QX., Transactionsof the American Mathematical Society, 265 (1981), 163–167.

[10] Caruso J., Cohen F.R., May J.P. and Taylor L.R., James maps,Segal maps, and the Kahn-Priddy theorem., Transactions of theAmerican Mathematical Society, 281 (1984), 243–283.

[11] Caruso J. and Waner S., An approximation to ΩnΣnX., Transac-tions of the American Mathematical Society, 265 (1981), 147–162.

[12] Cohen Frederick R., The unstable decomposition of Ω2Σ2X andits applications., Mathematische Zeitschrift, 182 (1983), 553–568.

[13] Cohen F.R., Cohen R.L., Kuhn N.J. and Neisendorfer J.L., Bun-dles over configuration spaces., Pacific Journal of Mathematics,104 (1983), 47–54.

[14] Cohen F.R., Mahowald M.E. and Milgram M.J., The stable de-composition for the double loop space of a sphere., Proceedings ofSymposia in Pure Mathematics vol. 32 (1978), 225–228.

[15] Cohen F.R., May J.P. and Taylor L.R., Splitting of certain spacesCX., Mathematical Proceedings of the Cambridge PhilosophicalSociety, 84 (1978), 465–496.

[16] Cohen F.R., May J.P. and Taylor L.R., Splitting of some morespaces., Mathematical Proceedings of the Cambridge Philosophi-cal Society, 86 (1979), 227–236.

[17] Cohen F.R., May J.P. and Taylor L.R., James maps and En ringsspaces., Transactions of the American Mathematical Society, 281(1984), 285–295.

[18] Cohen F.R., Moore, J.C. and Neisendorfer J.L., Torsion in homo-topy groups., Annals of Mathematics, 109 (1979), 121–168.

[19] Cohen F.R., Moore, J.C. and Neisendorfer J.L., The double sus-pension and exponents of the homotopy groups of spheres., Annalsof Mathematics, 110 (1979), 549–565.

[20] Cohen F.R., Moore, J.C. and Neisendorfer J.L., Exponents in ho-motopy theory., In Algebraic Topology and Algebraic K-theory,Annals of Mathematical Studies No. 113, Princeton UniversityPress, 1987, 3–34.

58 DAI TAMAKI

[21] Cohen F.R., Lada T.J. and May J.P., The Homology of IteratedLoop Spaces., Lecture Notes in Mathematics, vol. 533, Springer-Verlag, 1976.

[22] Cohen R.L. The geometry of Ω2S3 and braid orientation., Inven-tiones Mathematicae, 54 (1979), 53–67.

[23] Cohen R.L. Odd primary infinite families in stable homotopy the-ory., Memoirs of the American Mathematical Society, No. 242,1981.

[24] Cohen R.L. Stable proofs of stable splittings., Mathematical Pro-ceedings of the Cambridge Philosophical Society, 88 (1980), 149–151.

[25] Dold A. and Thom R., Quasifaserungen und unendliche sym-metrische Produkte., Annals of Mathematics, 67 (1958), 239–281.

[26] Dyer E. and Lashof R.K., Homology of iterated loop spaces., Amer-ican Journal of Mathematics, 84 (1962), 35–88.

[27] James I., Reduced product spaces., Annals of Mathematics, 62(1955), 170–197.

[28] Gray B., EHP spectra and periodicity. I: geometric constructions.,Transactions of the American Mathematical Society, 340 (1993),595–616

[29] Iwama K., Constructing mapping spaces from Moore spaces vialittle cubes., Master’s thesis, Department of Mathematics, ShinshuUniversity, March 1998.

[30] Kahn D.S. and Priddy S.B., Applications of the transfer to stablehomotopy theory., Bulletin of the American Mathematical Society,78 (1972), 981–987.

[31] Kahn D.S. and Priddy S.B., The transfer and stable homotopy the-ory., Mathematical Proceedings of the Cambridge PhilosophicalSociety, 83 (1978), 103–111.

[32] Kallel S., An interpolation between homology and stable homo-topy., preprint.


[33] Kato F., On the space realizing the intermediate stages of theHurewicz homomorphism., Master’s thesis, Department of Math-ematics, Shinshu University, March 1996.

[34] Kuhn Nicholas J., The geometry of the James-Hopf maps., PacificJournal of Mathematics, 102 (1982), 397–412.

[35] Kuhn Nicholas J., The homology of the James-Hopf maps., IllinoisJournal of Mathematics, 27 (1983), 315–333.

[36] Loday J.-L., Stasheff J.D. and Voronov A.A. ed. Operads: Pro-ceedings of Renaissance Conferences., Contemporary Mathemat-ics 202, American Mathematical Society, 1997.

[37] Lewis L.G., May J.P. and Steinberger M. Equivariant StableHomotopy Theory., Lecture Notes in Mathematcis, vol. 1213,Springer-Verlag, 1986.

[38] Mahowald M.E., A new infinite family in 2πS∗ ., Topology, 16

(1977), 249–256.

[39] Mahowald M.E., The image of J and the EHP sequence., Annalsof Mathematics, 116 (1982), 65–112.

[40] May J.P., The Geometry of Iterated Loop Spaces., Lecture Notesin Mathematics, vol. 271, Springer-Verlag, 1972.

[41] Milgram R.J., Iterated loop spaces., Annals of Mathematics, 84(1966), 386–403.

[42] Milnor J.W., On the construction FK., Reprinted in J.F. Adams,“Algebraic Topology – A Student’s Guide”, London MahtematicalSociety Lecture Notes Series 4, Camridge University Press, 1972.

[43] Segal G.B., Configuration spaces and iterated loop spaces., Inven-tiones Mathematicae, 21 (1973), 213–221.

[44] Smith J.H., Simplicial group models for ΩnSnX., Israel Journalof Mathematics, 66 (1989), 330–350.

[45] Snaith V.P.A., Stable decomposition for ΩnSnX., Journal of theLondon Mathematical Society (2), 7 (1974), 577–583.

[46] Tamaki D. A dual Rothenberg-Steenrod spectral sequence., Topol-ogy, 33 (1994), 631–662.

60 DAI TAMAKI

[47] Tamaki D. Remarks on the cobar-type Eilenberg-Moore spectralsequences., preprint.

[48] Thompson R.D., The v1-periodic homotopy groups of an unstablesphere at odd primes., Transactions of the American MathematicalSociety, 319 (1990), 535–559.

[49] Wong S.-C., The fiber of the iterated Freudenthal suspension.,Mathematische Zeitschrift, 215 (1994), 377–414.


Hipergrupos y algebras de Bose-Mesner∗

Isaıas Lopez 1

ResumenEn este artıculo se prueba que toda algebra de Bose-Mesner esun hipergrupo con el producto usual y con el producto Hadamardde matrices. Ademas, presentamos los diferentes tipos de iso-morfismos entre algebras de Bose-Mesner y sus relaciones con loshipergrupos.

1991 Mathematics Subject Clasification: 05E45Keywords and phrases: algebras de Bose-Mesner, hipergrupos

1 Introduccion

Existen varios objetos matematicos cuya esencia es el de un esquemade asociacion y que se conocen con varios nombres, pero esencialmenteson el mismo concepto matematico; por ejemplo, algebras de adyacencia,algebras de Bose-Mesner, anillo centralizador, anillo de Hecke, anillo deSchur, algebra de caracteres, hipergrupos, grupos probabilısticos.

Los hipergrupos son una generalizacion de los grupos, en donde elproducto de dos elementos esta determinado por una funcion de dis-

arapodoterbos,aeraatsenenoicagitsevniahcumetsixE.noicubirtgeneralizar conceptos basicos de la teorıa de grupos, por ejemplo, enhipergrupos existe el teorema de Lagrange.

En el presente trabajo demostramos que toda algebra de Bose-Mesner,bajo cierta normalizacion, es un hipergrupo, tanto con el producto usualde matrices como con el producto de Hadamard (ver los Teoremas 3.3y 3.4). Esto permite dar una vision distinta de conceptos tales comoisomorfismos y dualidades en algebras de Bose-Mesner.

∗Este trabajo se realizo con el apoyo de CONACyT, a traves del proyecto No.29275E.

1Estudiante de doctorado del Departamento de Matematicas, CINVESTAV-IPN.Becario de CONACyT.

61

62 ISAIAS LOPEZ

2 Hipergrupos

La teorıa general de hipergrupos fue introducida por Dunkl [4],Jewett [9] y Spector [11] de manera independiente, y tratan a los hiper-grupos como un caso especial.

Una interpretacion fısica es la siguiente: Un hipergrupo conmuta-tivo finito es una coleccion de partıculas, digamos c0, c1, . . . , cn, enlas cuales esta permitido la interaccion por colision entre ellas. Cuandodos partıculas colisionan forman una tercera partıcula. Si colisionamosci con cj la probabilidad de que resulte la partıcula ck es nk

ij y estenumero es fijo. La partıcula c0 tiene la propiedad de ser absorbida encualquier colision; la llamamos un foton. Cada partıcula tiene una anti-partıcula la cual esta especificada por la siguiente regla: La colision dedos partıculas tiene una probabilidad positiva de resultar en un foton siy solo si las dos partıculas son anti-partıculas una de la otra. La inter-accion de las partıculas son independientes del orden, del tiempo y desu posicion en el espacio. La estructura del sistema esta completamentedeterminado por las probabilidades nk

ij las cuales son invariantes bajointercambio de todas las partıculas con sus anti-partıculas.

Definicion 1.1 Un hipergrupo generalizado es una pareja (H ⊂ A)donde H = c0, c1, . . . , cn y A es una algebra asociativa con identidadc0 y con involucion ⋆ sobre C, que satisface las siguientes condiciones:

(A1) H es una base de A.

(A2) H⋆ = H, que denotaremos por c⋆i = cσ(i).

(A3) La estructura constante nkij ∈ C definida por:

cicj =n!

k=0

nkijck

satisface lo siguientec⋆i = cj ⇐⇒ n0

ij > 0,

c⋆i = cj ⇐⇒ n0ij = 0.

En el resto de este trabajoH denotara a un hipergrupo generalizado.Si A es conmutativo, entonces H es conmutativo. Ademas, se dice

que H es Hermitiano si c⋆i = ci ∀i; real si nkij ∈ R ∀i, j, k; positivo

HIPERGRUPOS Y ALGEBRAS DE BOSE-MESNER 63

si nkij ! 0 ∀i, j, k. Por otra parte, decimos que H es normalizado si

satisface que

(A4)!

k

nkij = 1 ∀i, j.

Un hipergrupo generalizado que es real y normalizado se dice quees hipergrupo signado. Un hipergrupo generalizado que es positivo ynormalizado se llama simplemente un hipergrupo.

Definimos el peso de ci como

w(ci) = (n0iσ(i))

−1 > 0(1.1)

y el peso de H como

w(H) =n!

i=0

w(ci).(1.2)

Consideremos la siguiente forma alternativa de (A4):

(A4′) w(ci)−1w(cj)

−1 =!

k

nkijw(ck)

−1.

Un hipergrupo que es real y satisface (A4′) se dice que es un ensamble.Aunque la condicion (A4) parezca mas facil de probar que (A4′), estono siempre es ası, es por eso la siguiente proposicion.

Proposicion 1.2 Existe una correspondencia uno a uno entre ensam-bles e hipergrupos signados.

Demostracion: Si H = c0, c1, . . . , cn es un hipergrupo signado, te-nemos que H = w(c0)c0, w(c1)c1, . . . , w(cn)cn es un ensamble.

Reciprocamente, para H = c0, c1, . . . , cn un ensamble tenemosque H = w(c0)c0, w(c1)c1, . . . , w(cn)cn es un hipergrupo signado.

De aquı en adelante, a menos que se diga lo contrario, solo conside-raremos hipergrupos que son conmutativos.

Proposicion 1.3 La *-algebra es semisimple.

Demostracion: Se sigue del hecho de que la *-algebra no tiene ele-mentos idempotentes.

Para a ∈ A, ad(a) ∈ End(A) denotara el operador multiplicacionpor a.

64 ISAIAS LOPEZ

Lema 1.4 El conjunto ad(ci) es linealmente independiente en End(A).

Demostracion: Si!

i riad(ci) = 0, entonces multiplicando por c⋆jy considerando el coeficiente de c0 tenemos que rj = 0.

El algebra ad(A) ⊂ End(A) tiene dimension n + 1 y es conmuta-tiva y semisimple. Esto nos dice que End(A) es isomorfa al algebrade operadores diagonalizables en M(n + 1,C). Por lo tanto, podemosencontrar una base eo, e1, . . . , en de A en la cual los operadores ad(ci)son diagonales, esto es,

ciej = χj(ci)ej ∀i, j(1.3)

para alguna funcion χj tal que

ejek = δjkej ,(1.4)

donde δij es la delta de Kroenecker.Si F(H) denota el espacio de todas las funciones de H en los comple-

jos, entonces el conjunto de funciones χi es linealmente independienteen F(H) .

Definicion 1.5 Un caracter de H es cualquier χ ∈ F(H) que satisface

χ(ci)χ(cj) ="

k

nkijχ(ck) ∀i, j.(1.5)

Si la extension lineal de χ en A se denota tambien por χ, tendremosla siguiente formulacion equivalente.

χ(ci)χ(cj) = χ(cicj) ∀i, j.(1.6)

Al conjunto de todos los caracteres de H lo denotaremos por H.

Proposicion 1.6 H = χ0,χ1, . . . ,χn y

χi(c⋆j ) = χi(cj) ∀i, j.(1.7)


Demostracion: Tenemos que ciej = χj(ci)ej para toda i, j, de dondeχj(ci)χj(cs)ej = cicsej =

!knkisckej =

!knkijχj(ck)ej , es decir,

χj(ci)χj(cs) =!knkijχj(ck). De aquı se sigue la proposicion porque A es

isomorfa a la *-algebra Cn+1 y esta tiene exactamente n+ 1 caracteresque satisfacen las condiciones establecidas.

Claramente la funcion identicamente 1 es un caracter de H, el cualdenotaremos por χ0 .

Ahora deseamos ver que H es una base ortogonal de F(H). Paraesto, para todo f, g ∈ F(H), definimos f⋆(ci) = f(c⋆i ) e introducimos elproducto interno

⟨f, g⟩ = 1

w(H)

"

i

w(ci)f(ci)g(ci).(1.8)

Como H y e0, e1, . . . , en son bases para A podemos encontrar cons-tantes αk

j ∈ C tales que

ej ="

k

αkj ck ∀j.(1.9)

Multiplicando ambos lados por c⋆i y comparando los coeficientes de c0podemos observar que

αij = w(ci)χj(c

⋆i )α

0j ,(1.10)

de donde

ej = α0j

"

k

w(ck)χj(c⋆k)ck.(1.11)

Combinando esto con la ecuacion (1.4) y comparando los coeficientes dec0 obtenemos:

δij = α0j

"

k

w(ck)χi(ck)χj(c⋆k) = α0

jw(H)⟨χi,χj⟩.(1.12)

De aquı tenemos el siguiente lema:

66 ISAIAS LOPEZ

Lema 1.7 H = χ0,χ1, . . . ,χn es una base ortogonal de F(H) conrespecto al producto interno definido en (1.8).

Como F(H) es una *-algebra con unidad χ0 bajo la multiplicacionpunto a punto y conjugacion compleja, podemos escribir

χiχj =!

k

mkijχk con mk

ij ∈ C.(1.13)

Ademas, si χ⋆i = χj podemos definir el peso de χi como w(χi) =

(m0ij)

−1. Entonces

⟨χi,χi⟩ = ⟨χiχi,χ0⟩= ⟨χiχ

⋆i ,χ0⟩

= w(χi)−1⟨χ0,χ0⟩

= w(χi)−1.(1.14)

De aquı concluimos que w(χi) ∈ R y w(χi) > 0. Ademas, usando laecuacion (1.11) tenemos:

Proposicion 1.8

ei =w(χi)

w(H)

!

k

w(ck)χ⋆i (ck)ck ∀i.(1.15)

En particular,

e0 =1

w(H)

!

k

w(ck)ck.(1.16)

En conclusion, tenemos el siguiente resultado:

Teorema 1.9 Si H es un hipergrupo signado, tambien lo es H. Ademasw(H) = w(H).

2 Algebras de Bose-Mesner

El concepto de esquema de asociacion es importante en algebra com-binatoria. Dicho concepto aparece en el estudio de codigos, disenos,


graficas de distancia regular, invariantes de nudos y en muchos otrostemas. El estudio de los esquemas de asociacion lo inicio Delsarte en1973 (ver [3]). En el caso simetrico los esquemas de asociacion son esen-cialmente particiones de una grafica completa en subgraficas regularesque estan relacionadas entre sı de alguna manera especıfica. Para unanalisis extensivo de este tema se pueden consultar [1] y [2].

Definicion 2.1 Un esquema de asociacion de clase d es una familia de0,1-matrices Ai|i = 0, . . . , d de orden n que satisfacen lo siguiente:

(B1) A0 = I (I es la matriz identidad).

(B2)!d

i=0Ai = J (J es la matriz con todas sus entradas igual a uno).

(B3) Para todo i, tAi = Aσ(i), para algun σ(i) ∈ 0, 1, . . . , d(tA denota la traspuesta de A ).

(B4) Ai Aj = δijAi ( denota el producto de Hadamard de matrices,entrada por entrada).

(B5) AiAj = AjAi =d!

k=0pkijAk ( equivalentemente, pkij = pkji para todo

i, j, k ).

El algebra generada por Ai|i = 0, 1, . . . , d sobre C es una algebraconmutativa con el producto usual y con el producto de Hadamard dematrices y la familia Ai|i = 0, 1, . . . , d es una base para esta algebra,la cual es conocida como el algebra de Bose-Mesner del esquema deasociacion de clase d. Esta algebra tiene la propiedad de ser cerradabajo trasposicion compleja.

En otro contexto, a este tipo de algebras tambien se les llama algebrasDoble-Frobenius (ver [10]).

En el caso en que tAi = Ai para todo i, tenemos una algebra deBose-Mesner simetrica.

Sea

ni = poiσ(i),(2.1)

en donde ni es el numero de elementos en la diagonal de la matrizAiAσ(i). El entero positivo ni se llama la valencia de Ai, y es claro que

68 ISAIAS LOPEZ

n0 = 1,

ni = nσ(i),

n = n0 + n1 + · · ·+ nd.

La siguiente proposicion es importante para relacionar las algebrasde Bose-Mesner con los hipergrupos. Su demostracion se puede ver en[1].

Proposicion 2.2

(i) pk0j = δjk,

(ii) p0ij = niδiσ(i),

(iii) pkij = pσ(k)σ(i)σ(j),

(iv)d!

j=0pkij = ni,

(v) nkpkij = njpjσ(i)k = nipikσ(j),

(vi)d!

k=0pkijp

luk =

d!s=0

psuiplsj .

Como las matrices A0, A1, . . . , Ad son normales y conmutan porparejas, ellas son diagonalizables simultaneamente por una matriz uni-taria. Ası, podemos encontrar una descomposicion de Cn como sumadirecta de d + 1 eigenespacios de dimension fj , 0 ≤ j ≤ d (ver [5] y[7]). Ademas, como J pertenece al algebra, n es un eigenvalor de multi-plicidad 1 y entonces podemos suponer que f0 = 1. Los fi se llaman lasmultiplicidades de los esquemas. Sea E0, E1, . . . , Ed la base de idem-potentes ortogonales con el producto usual de matrices del algebra deBose-Mesner, cada una de ellas corresponde a los proyectores de Cn encada diferente eigenespacio, ademas, fi = tr(Ei), tenemos lo siguiente:

(C1) E0 =1nJ ,

(C2) E0 + E1 + · · ·+ Ed = I,

(C3) EiEj = δijEi,


(C4) tEi = Eσ(i) para algun σ(i) ∈ 0, 1, . . . , d,

(C5) Ei Ej = Ej Ei =d!

k=0qkijEk.

Los coeficientes qkij son llamados los parametros de Krein del esque-ma de asociacion.

Como E0, E1, . . . , Ed es una base para el algebra de Bose-Mesner,tenemos

Ai =d"

j=0

Pj(i)Ej .(2.2)

Analogamente

Ei =1

n

d"

j=0

Qj(i)Ak.(2.3)

Sea

P = (Pj(i)),(2.4)

cuya entrada (i, j) de la matriz es Pj(i), y sea

Q = (Qj(i)),(2.5)

cuya entrada (i, j) de la matriz es Qj(i). Las matrices P y Q se llaman laprimera y segunda matriz del esquema de asociacion, respectivamente.Ellas satisfacen lo siguiente:

PQ = QP = nI.(2.6)

Usando las ecuaciones (2.2) y (2.3) junto con las propiedades (A4)y (C3) obtenemos:

AiEj =Pj(i)Ej ,(2.7)

Ei Aj =1

nQj(i)Aj .(2.8)

70 ISAIAS LOPEZ

La ecuacion (2.7) muestra que cada vector columna de Ej es unvector propio de Ai asociado con el valor propio Pj(i). Analogamente,la ecuacion (2.8) nos permite decir que cada vector columna de Aj es unvector propio de Ei asociado al valor propio 1

nQj(i) respecto al productoHadamard.

Los parametros de Krein tienen las siguientes propiedades, similaresa los de la Proposicion 2.2. La demostracion de esta proposicion sepuede consultar en [1].

Proposicion 2.3

(i) qk0j = δjk,

(ii) q0ij = fiδiσ(j),

(iii) qkij = qσ(k)σ(i)σ(j),

(iv)d!

j=0qkij = fi,

(v) fkqkij = fjqjσ(i)k = fiqikσ(j),

(vi)d!

k=0qkijq

luk =

d!s=0

qsuiqlsj.

Finalmente, las entradas de la primera y segunda eigenmatrices P yQ satisfacen las siguientes relaciones:

Proposicion 2.4

(i) Pj(i)Pk(i) =d!

l=0pljkPl(i),

(ii) Qj(i)Qk(i) =d!

l=0qljkQl(i).

3 Hipergrupos y algebras de Bose-Mesner

Consideremos una algebra de Bose-Mesner A = A0, A1, . . . , Ad dedimension d+ 1 sobre C. Recordemos que A es una algebra asociativa,conmutativa con respecto al producto usual de matrices, cuya identidad


es I. Asimismo, A es una algebra asociativa, conmutativa con respectoal producto de Hadamard de matrices y con identidad J.

Lema 3.1 A es un hipergrupo generalizado con el producto usual dematrices y base A0, A1, . . . , Ad.

Demostracion: Sea A0, A1, . . . , Ad un esquema de asociacion yA = ⟨A0, A1, . . . , Ad⟩ el algebra de Bose-Mesner generada por dichoesquema. En particular, A es una algebra asociativa, conmutativa, conidentidad A0 y con involucion el mapeo traspuesta conjugada. Ademas,claramente se tiene (A1) y (A2), de modo que solo resta probar (A3).Para esto, observemos que de la Proposicion 2.2(ii):

P 0ij =

!0 si j = σ(i),

ni si j = σ(i).

De aquı se sigue (A3).Por el lema 3.1 el peso de Ai es:

w(Ai) ="p0iσ(i)

#−1> 0.(3.1)

Proposicion 3.2 A es un ensamble con el producto usual de matrices.

Demostracion: Es suficiente probar el axioma (A4’). Usando la Proposi-cion 2.2 tenemos:

d$

k=0

pkijw (Ak)−1 =

d$

k=0

pkijp0kσ(k)

=d$

k=0

pkijnkδkσ(k) =d$

k=0

pkij

%d$

r=0

ptkr

&

=d$

k=0

d$

r=0

pkijptkr =

d$

r=0

d$

k=0

pkijptkr

=d$

r=0

d$

k=0

pkijptkr =

d$

s=0

d$

r=0

psriptsj

=d$

s=0

ptsj

%d$

r=0

psri

&=

d$

s=0

ptsjni

=ninj = p0iσ(i)p0jσ(j)

=w(Ai)−1w(Aj)

−1.

72 ISAIAS LOPEZ

De la correspondencia uno a uno entre hipergrupos signados y en-sambles se deduce el siguiente resultado.

Teorema 3.3 !A = ⟨w(A0)A0, w(A1)A1, . . . , w(Ad)Ad⟩ es un hipergruposignado bajo el producto usual de matrices.

Finalmente, de la Proposicion 2.3 se deduce el siguiente resultado,similar al teorema anterior.

Teorema 3.4 A genera un hipergrupo signado bajo el producto Hadamardde matrices y base E0, E1, . . . , Ed.

4 Isomorfismos en algebras de Bose-Mesner

Algunos de los ejemplos clasicos de algebras de Bose-Mesner sonlos obtenidos a traves de graficas fuertemente regulares y de distanciaregular [2]. La clasificacion de graficas fuertemente regulares y de dis-tancia regular se realiza por medio del algebra de Bose-Mesner asociada.Aunque la clasificacion de graficas de distancia regular es un poco mascompleja, ambas ocupan el concepto de BM-isomorfismo, que se defineen esta seccion. Otra de las aplicaciones de la teorıa de esquemas deasociacion es la clasificacion de modelos spin (invariantes de nudos) (ver[6]), en donde juega un papel importante el concepto de dualidad entrealgebras de Bose-Mesner.

Sean A y B dos algebras de Bose-Mesner y ψ un isomorfismo deespacios vectoriales de A en B.

Definicion 4.1 Se dice que ψ es un BM-isomorfismo de A en B sisatisface:

ψ(AB) = ψ(A)ψ(B) y ψ(A B) = ψ(A) ψ(B)

para toda A,B ∈ A.

Un ejemplo de un BM-isomorfismo se obtiene a traves de una matrizde permutacion. Es decir, si P es una matriz de permutacion, entoncesψ(A) = P−1AP define un BM-isomorfismo. De hecho, este isomorfismoes conocido como isomorfismo combinatorial.

Si A tiene como base usual Ai | i = 1, . . . , d y su base de idem-potentes ortogonales es Ei | i = 1, . . . , d, tenemos que ψ(Ai) | i =


1, . . . , d y ψ(Ei) | i = 1, . . . , d son la base usual y la base de idem-potentes ortogonales de B, respectivamente.

Usando la ecuacion (2.7) obtenemos la siguiente relacion

ψ(AiEj) = ψ(Pj(i)Ej) = Pj(i)ψ(Ej) = ψ(Ai)ψ(Ej),

de la cual se deduce que A y B tienen el mismo conjunto de valorespropios y como ambas son diagonalizables simultaneamente por unamatriz unitaria obtenemos el siguiente resultado.

Teorema 4.2 Si ψ es un BM-isomorfismo de A en B, entonces existeuna matriz unitaria U tal que ψ(A) = U∗AU para toda A ∈ A.

La pregunta obvia, ya que toda matriz de permutacion es unitaria, essi la matriz unitaria que define el BM-isomorfismo del teorema anteriores una matriz de permutacion. Parece ser que la respuesta es negativa,como lo sugiere F. Jaeger [8], aunque hasta ahora no se cuenta ni conuna demostracion ni con un contraejemplo.

Definicion 4.3 Decimos que ψ es una dualidad de A en B si satisfaceque

ψ(AB) = ψ(A) ψ(B) y ψ(A B) =1

nψ(A)ψ(B)

para toda A,B ∈ A.

Es claro que si ψ es una dualidad de A en B, entonces 1nψ

−1 es unadualidad de B en A. Denotaremos por (A,B) a una pareja de algebrasde Bose-Mesner en donde existe una dualidad, y en este caso diremosque (A,B) forman una pareja dual de algebras de Bose-Mesner. En elcaso de que la dualidad este definida de A en sı misma, ψ es llamadauna dualidad fuerte si satisface que ψ2 = nτ , donde τ denota al mapeotrasposicion. A esta algebra se le llama una algebra de Bose-Mesnerautodual.

Es facil ver que la composicion entre una dualidad y un BM-isomor–fismo es una dualidad. Ademas, la composicion entre dualidades, bajocierta normalizacion, es un BM-isomorfismo. Por otra parte, salvo lacomposicion por un BM-isomorfismo, las dualidades entre algebras deBose-Mesner son unicas (ver [6]).

Proposicion 4.4 [6]. Sea (A,B) una pareja dual de algebras de Bose-Mesner. Entonces se cumple lo siguiente:

74 ISAIAS LOPEZ

i) Los numeros de interseccion de A son iguales a los parametros deKrein de B y viceversa.

ii) La primera eigenmatriz de A es igual a la segunda eigenmatriz deB y viceversa.

iii) Toda dualidad conmuta con el mapeo trasposicion.

De las ecuaciones (2.7) y (1.3) tenemos que el conjunto de caracteresde A es χ0,χ1, . . . ,χn, donde χj esta definida por

AiEj = Pj(i)Ej = χj(Ai)Ej .

En este caso, a A la llamaremos la algebra primal de caracteres delalgebra de Bose-Mesner A. De manera analoga, viendo a A como unhipergrupo con la base de idempotentes, los caracteres estan determina-dos por la ecuacion (2.8) y en este caso diremos que A es la algebra dualde caracteres de A. Observese que el concepto de caracter es consistentecon la Proposicion 2.4.

El siguiente corolario es consecuencia inmediata de la Proposicion4.4.

Corolario 4.5 Sea (A,B) una pareja dual de algebras de Bose-Mesner.Entonces el algebra primal de caracteres de A es igual al algebra dual decaracteres de B y viceversa. En particular, si A es una algebra de Bose-Mesner autodual se tiene que las algebras primal y dual de caracterescoinciden.

5 Conclusiones

En esencia, lo que hemos probado es que bajo cierta normalizacionde los elementos de la base de una algebra de Bose-Mesner, esta tiene elmismo comportamiento algebraico operacional de un hipergrupo. Masaun, las algebras de Bose-Mesner estan contenidas en un conjunto masgrande, el conjunto de los hipergrupos. Por otro lado, viendo a unaalgebra de Bose-Mesner como un hipergrupo, los conceptos de isomor-fismos estan determinados por sus algebras de caracteres asociadas.


AgradecimientosAgradezco a los evaluadores las sugerencias y correcciones realizadas

a este trabajo. Asimismo, al Dr. Onesimo Hernandez-Lerma por larevision ortografica y sugerencias. Este trabajo se realizo bajo la super-vision del Dr. Isidoro Gitler.

Isaıas LopezDepartamento de MatematicasCINVESTAV-IPNA.P. 14− 740Mexico, D.F., C. P. [email protected]

Referencias

[1] E. Bannai and T Ito. Algebraic Combinatorics I. Benjamin-Cummings, Menlo Park, C. A, 1989.

[2] A. E. Brouwer, A. M. Cohen, A. Neumaier. Distance-RegularGraphs. Springer-Verlag, Berlin, Heidedelberg, New York 1989.

[3] P. Delsarte. An algebraic approach to the association achemes ofcoding theory, Phillips Research Reports Supplements 10 (1973).

[4] C.F Dunkl. The measure algebra of a locally compact hipergroup.Trans. Amer. Math. Soc. 179,(1973), No. 32(1), 331-348.

[5] F. R. Gantmacher. The Theory of Matrices, Chelsea PublishingCompany, New York, N. Y. 1960.

[6] I. Gitler and I. Lopez. Spin models, association schemes and ∆−Ytransformations, Morfismos, Vol. 3, No. 2 (1999), 31-55.

[7] R. Horn and C. R. Johnson. Matrix Analysis, Cambridge Univer-sity Press, 1991.

[8] F. Jaeger. On Spin models, triply regular association schemes, andduality, Journal of Algebraic Combinatorics 4 (1995), 103-144.

76 ISAIAS LOPEZ

[9] R. I. Jewett. Spaces with an abstract convolution of measures. Adv.Math. 18(1975), 1-101.

[10] M. Koppinen. On algebras with two multiplications, includingHopf algebras and Bose-Mesner algebras. J. Alg. 182 (1996), 256-273.

[11] R. Spector. Measures invariantes sur les hipergroupes. Trans.Amer. Math. Soc. 239(1978), 147-165.

[12] N. J. Wildberger. Lagrange’s theorem and integrality for finitecommutative hipergroups with applications to strongly regulargraphs. J. Alg. 182(1996), 1-37.

[13] N. J. Wildberger. Finite commutative hipergroups and aplicationfrom group theory to conformal field theory. Contemporany Math-ematics, Vol 183, Amer. Math. Soc., Providence(1995), 413-434.

Morfismos, Vol. 4, No. 2, 2000, pp.77–90

Sincronizacion de parejasde automatas celulares

J. Guillermo Sanchez Saint-Martin 1

Resumen

Consideramos un sistema compuesto por dos automatas celularesunidimensionales definidos en ZZN

k a partir de la misma regla afınR : ZZ3

k → ZZk. El automata gobernante evoluciona de formaautonoma. El automata gobernado copia algunas de las entradasdel gobernante despues de cada iteracion. La pareja de automatassincroniza si eventualmente el automata gobernado evolucionaidenticamente al automata gobernante, esto para toda pareja decondiciones iniciales. En este caso, la diferencia entre gobernantey gobernado sigue un comportamiento lineal en ZZN

k , de modo queel estudio de la sincronizacion equivale a realizar el estudio de lanilpotencia de una matriz en GL(N,ZZk), la cual depende solo delas posiciones relativas de la entradas del gobernante que serancopiadas por el gobernado. En general estudiamos la relacionentre las propiedades aritmeticas de k y la sincronizacion de losautomatas celulares. Tambien presentamos una herramienta parademostrar que no hay sincronizacion de parejas de automatas celu-lares en ciertos alfabetos.

2000 Mathematics Subject Clasification: 37B15Keywords and phrases: Automatas celulares, Nilpotencia de matrices.

1 Introduccion

Los automatas celulares (AC) fueron introducidos por J. Von Neumann-cudorperotuasocigoloibsametsisraledomarap,04sonasoledselanfia

tivos.1Estudiante de doctorado, Ciencias Aplicadas, IICO-UASLP, Becario de

CONACYT

77

78 GUILLERMO SANCHEZ

En la actualidad los automatas celulares son aplicados a diversasramas de la ciencia como lo son la fısica, la biologıa, la computacion, laquımica, etc.

La evolucion de un AC es sencilla de generar, y proporciona modelossimples para una gran variedad de fenomenos [1]. Un AC es una idea-lizacion matematica de sistemas fısicos en los cuales el espacio fase y eltiempo son discretos. El estado de un automata celular esta completa-mente determinado por los valores de todas sus celdas (celulas). El ACevoluciona de acuerdo a una regla local que depende de sus valores enuna vecindad.

Por otro lado, la sincronizacion de sistemas acoplados ha desper-tado gran interes en estudios recientes [2], en particular cuando los sis-temas que se acoplan son caoticos. Sistemas que muestran este com-portamiento son temporalmente caoticos, pero espacialmente ordena-dos o coherentes. La coherencia presente es de un tipo particular:las dinamicas son las mismas o casi las mismas por perıodos largosde tiempo para todos los sistemas acoplados o en gran parte de ellos.

Para el acoplamiento unidireccional de dos sistemas dinamicos senecesita un sistema que evoluciona de manera autonoma, llamado go-bernante y un sistema que evoluciona con cierto grado de libertad perotambien es influido por medio del acoplamiento por el sistema gober-nante, razon por la cual se le denomina sistema gobernado.

Los automatas celulares (AC) son sistemas dinamicos discretos queevolucionan a pasos discretos en el tiempo. El espacio fase de un ACde tamano N es el conjunto ZZN

k de todas las secuencias de N celdasque toman valores en ZZ/kZZ, y la evolucion esta definida por la repetidaiteracion de la ley de evolucion F : ZZN

k → ZZNk . En este trabajo estudia-

mos algunos aspectos relativos al comportamiento de AC’s afines derango 1 acoplados unidireccionalmente. En particular nos enfocamos alfenomeno de la sincronizacion de automatas acoplados unidireccional-mente.

En estos sistemas resulta que la sincronizacion esta sujeta a condi-ciones aritmeticas en los parametros que definen al sistema. Estas rela-ciones aritmeticas son punto central de este trabajo.

2 Sincronizacion de automatas celulares

Un Automata Celular (AC) es un sistema dinamico discreto definidoa partir de:

SINCRONIZACION DE AUTOMATAS 79

1. Un alfabeto A (conjunto finito de sımbolos)

2. Una regla local f : A2r+1 → A, donde el entero r es llamado rangodel automata.

La ley de evolucion F : AN → AN es tal que

F (a0a1...an−1) = b0b1...bn−1,

donde bi = f(ai−r...ai...ai+r) y la suma de ındices dentro del parentesisdebe entenderse modulo IN. Para este caso tomamos A = ZZk, k ∈ IN,por lo tanto el espacio fase es ZZN

k , al cual dotamos con las operacionesusuales coordenada a coordenada en ZZk.

Un automata celular es afın si su regla local es de la forma

f(ai−r...ai...ai+r) =!r

j=−r βjai+j + c, con βj ∈ ZZk,donde a (ai−r...ai...ai+r) le llamamos vecindad de ai de tamano r.

2.1 Condiciones de frontera

Para generar un crecimiento con un numero constante de celdas en cadatiempo, es necesario especificar condicones de frontera, en nuestro casoconsideramos el estado del AC con sus celdas arregladas en forma deanillo, de tal manera que la ultima celda es a su vez la celda anteriora la primera celda, a este arreglo le llamamos condiciones de fronteraperiodicas.

2.2 Sincronizacion en AC

Recordaremos como se acoplan unidireccionalmente dos sistemas dinami-cos. El sistema gobernante esta relacionado con el gobernado por mediode una funcion, la cual considera parte de la evolucion del sistema go-bernante en el gobernado.

En el caso del AC, la informacion de cuales coordenadas son copia-das en el AC gobernado esta dada por una secuencia de acoplamientoκ = (κ0,κ1, · · · ,κN−1) de 0’s y 1’s. Las coordenadas de κ que son igualesa 1 son llamadas coordenadas de acoplamiento.

Si los estados al tiempo t del gobernante y el gobernado son xt y yt,respectivamente, entonces el estado al tiempo t+ 1 del gobernante esxt+1 = F (xt), y para el gobernado tenemos yt+1 = κ ∗F (xt)+ (1− κ) ∗F (yt) donde u∗v es el producto coordenada a coordenada de los vectoresu y v. Aquı 1 representa un vector formado de 1’s y la sincronizacion se


da cuando la discrepancia al tiempo t entre el gobernado y el gobernantees nula. Esto es

zt = xt − yt = (1− κ) ∗[F (xt−1)− F (yt−1)

]= 0.

3 Nilpotencia de matrices y sincronizacion

El caso en el cual se considera una regla local afın f que toma en cuentasolo a los primeros vecinos, es conocido como caso afın de rango 1 [4].En este caso la ley de evolucion puede ser representada por una matriz,esto es:

f(x) = Mfx+ C,

con Mf =

⎛

⎜⎜⎜⎜⎜⎜⎝

β2 β3 0 0 0 · · · β1β1 β2 β3 0 0 · · · 00 β1 β2 β3 0 · · · 0...

. . ....

β3 0 0 0 · · · β1 β2

⎞

⎟⎟⎟⎟⎟⎟⎠(1)

y C =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ccc...ccc

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

La discrepancia esta dada por:

zt = xt − yt = F (xt−1)−[κ ∗ F (xt−1) + (1− κ) ∗ F (yt−1)

]

= Mfxt−1+C−κ(Mfxt−1)−C−(1−κ)∗[Mfyt−1 + C

]

= (1− κ) ∗Mfxt−1 − (1− κ) ∗Mfyt−1

= (1− κ) ∗Mf (xt−1 − yt−1)

= (1− κ) ∗Mfzt−1.

Esto es, la discrepancia al tiempo t depende solo de la discrepancia altiempo t− 1, y por lo tanto la sincronizacion se da si existe un T ∈ INtal que Mκ,f

T = 0 para todo z ∈ ZZNk , donde Mκ,f ≡ (1− κ) ∗Mf .

La matriz Mκ,f tiene la forma:


Mκ,f =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

β2 β3 0 . . . 0 0 0 · · · 0 0 0 · · · 0 β1

β1 β2 β3 . . . 0 0 0 · · · 0 0 0 · · · 0 0. . .

. . .. . .

0 · · · β1 β2 β3 0 0 0 · · · 0 0 · · · 0 00 · · · 0 β1 β2 0 0 · · · 0 0 0 · · · 0 00 · · · 0 0 · · · 0 0 · · · 0 0 0 · · · 0 00 0 · · · 0 0 0 β2 β3 0 0 0 · · · 0 00 0 · · · 0 0 0 β1 β2 β3 0 0 0 · · · 00 0 · · · 0 0 0 0 β1 β2 0 0 0 · · · 00 · · · 0 0 · · · 0 0 0 0 0 0 · · · 0 00 0 · · · 0 0 0 0 0 0 0 β2 β3 · · · 00 0 · · · 0 0 0 0 0 0 0 β1 β2 β3 · · ·

. . .. . .

. . .

β3 0 0 · · · 0 0 0 0 0 0 0 · · · β1 β2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Como los bloques de la matrizMκ,f son independientes, para calcular lanilpotencia de esta matriz basta calcular la nilpotencia de sus bloques.Utilizando el teorema de Hamilton-Cayley [5] podemos concluir que unamatriz de rango ℓ es nilpotente si su polinomio caracterıstico es de laforma

Pℓ(λ) = λℓ.

Definicion: Si una submatriz de Mk,f de tamano ℓ×ℓ es nilpotente,entonces ℓ es una longitud de sincronizacion para el sistema acopladodefinido por Mf y k.

Los polinomios de las submatrices de Mk,f obedecen la siguienteformula recursiva:

Pℓ(λ) = (λ− β2)Pℓ−1(λ)− β1β3Pℓ−2(λ).donde Pℓ(λ) es el polinomio caracterıstico de una submatriz de tamanoℓ× ℓ.

En conclusion, la sincronizacion de parejas de automatas celularesafines de rango 1 se reduce al estudio de la nilpotencia de matrices deltipo Mκ,f . Como una matriz por bloques es nilpotente si sus bloques loson, entonces basta estudiar la nilpotencia de bloques del tipo

⎛

⎜⎜⎜⎜⎜⎜⎝

β2 β3 0 · · · 0β1 β2 β3 · · · 0...

. . ....

0 · · · β1 β2 β30 0 · · · β1 β2

⎞

⎟⎟⎟⎟⎟⎟⎠.


4 Clases de equivalencia y no sincronizacion

Una vez fijo el alfabeto ZZk, sobre el cual definimos nuestros automatas,las matrices que consideraremos estan completamente determinadas portripletas (β1,β2,β3) ∈ (ZZk)3.

Definicion: Dos tripletas son equivalentes si y solo si los polinomiosPℓ que definen son iguales.

Definicion: La tripleta (β1,β2,β3) es similar a la tripleta (β1, β2, β3)(estan en la misma clase), si β1 = β1 y β2 = β2.

Dado que la nilpotencia solo depende del tipo de polinomio carac-terıstico, podemos reducir el estudio de todas las tripletas al estudio deuna tripleta representante por clase.

La tripleta (µ,β2, ν) para las cuales µν = 0 sera representada por(η,β2, 1) donde µν = η.

Definicion: Las tripletas triviales son aquellas (η, 0, 0) y (0, 0, η).Y son tales que Pℓ(λ) = λℓ ∀ ℓ.

Por lo anterior, para los anillos ZZk en lugar de estudiar las k3 posi-bilidades para las tripletas (β1,β2,β3), solo se analizaran k2 − k.

Mas aun, se clasificaran las tripletas (β1,β2,β3) de acuerdo al valordel producto β1β3 y al valor de β2 dependiendo de si es igual a cero ono, de manera que tenemos 2(k − 1) clases para el anillo ZZk.

Denotaremos las clases de la forma siguiente: para el caso en queβ2=0 tenemos la clase (η, 0, 1), y si β2 = 0 tenemos (η, 0, 1).

4.1 Reduccion de ZZk[λ]

Para reducir el anillo de polinomios ZZk[λ] a un anillo finito utilizaremosel siguiente hecho:

Sea φi : ZZk[λ] → ZZk la transformacion definida por

(a0 + a1λ+ ...+ anλn)φi = (a0 + a1i+ ...+ ani

n) mod k

la cual es un homomorfismo de ZZk[λ] en ZZk.Consideremos ahora los ideales Ji = p : pφi = 0 para i = 0, ..., k−

1. Entonces J = ∩k−1i=1 Ji es tambien un ideal y por lo tanto el cociente

ZZk[λ]/J es un anillo [6].

Al polinomio de grado menor en cada clase le llamaremos represen-tante canonico, el cual a lo mas tiene grado k − 1. Denotaremos al


representante canonico de la clase [Pℓ(λ)] donde Pℓ(λ) es el polinomiode alguna tripleta, por

Pℓ(λ) = aℓk−1λk−1 + aℓk−2λ

k−2 + ...+ aℓ0.

Es facil verificar que la suma de representantes canonicos es tambienun representante canonico. Por otro lado el representante canonico deun producto de clases se puede calcular a partir de los representantesde cada clase como sigue: sea P (λ) = ak−1λk−1 + ak−2λk−2 + ... + a0el representante de la clase [P ] y Q(λ) = bk−1λk−1 + bk−2λk−2 + ...+ b0el representante de la clase [Q], entonces el representante de la clase[F ] = [P ][Q] es f(λ) = ck−1λk−1 + ck−2λk−2 + ... + c0 donde los ci =!

s+t=τ(i) asbt mod k, donde τ : ZZk → ZZk es una funcion que se puededeterminar y depende en general de k.

Para el caso de alfabetos cuyas cardinalidades son numeros primosτ esta dada por

τ(i) =

"p− 1 si s+ t ≡ 0 mod p− 1 ,j si s+ t ≡ j mod p− 1.

La relacion de recurrencia entre polinomios caracterısticos en ZZk[λ],induce una relacion de recurrencia entre los representantes canonicos delas clases del anillo reducido ZZk

k[λ] a traves de una relacion lineal entrelos coeficientes:

(aℓ−1i , aℓi) → (aℓ+1

i ), para i = k − 1, ..., 0.

Ya que el objetivo es encontrar todos los polinomios para todas laslongitudes de sincronizacion utilizando esta recurrencia, y dado que estadepende de una matriz de tamano 2k×k que no se puede iterar, entoncesnos valdremos de un desplazamiento para generar una recurrencia linealdel tipo:

(aℓ−1i , aℓi) → (aℓi , a

ℓ+1i ), para i = k − 1, ..., 0.

Esta depende de la siguiente matriz de orden 2k×2k, que denotaremospor Hk y que sı puede ser iterada para encontrar los coeficientes de lospolinomios:


⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 · · · 0 0 1 0 · · · 0 00 0 · · · 0 0 0 1 · · · 0 0

. . . · · · . . .

0 0 · · · 0 0 0 0 · · · 1 00 0 · · · 0 0 0 0 · · · 0 1

−β1β3 0 0 · · · 0 −β2 1 0 · · · 00 −β1β3 0 · · · 0 0 −β2 1 · · · 0

. . . · · · . . .

0 0 · · · −β1β3 0 1 0 · · · −β2 10 0 · · · 0 −β1β3 0 0 · · · 0 −β2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

5 Resultados computacionales

De la relacion lineal entre los coeficientes de los representantes canonicospodemos observar los siguientes 2 hechos:

1. La matriz es eventualmente periodica, es decir existe un k < l,tal que Mk = M l, esto por la finitud del anillo de matricesM2k×2k[ZZk].

2. Si Pℓ(λ) = λℓ en ZZk[λ], entonces Pℓ ∈ [Pℓ] tiene la forma λχ(ℓ),donde χ(ℓ) puede ser determinado.

El Hecho 1 implica que solo hay un numero finito de casos queanalizar. Si para una tripleta dada ninguno de los representantes canonicosgenerados son del tipo λχ(ℓ), entonces no puede haber longitudes de sin-cronizacion.

Consideremos la funcion n #→ Hn, donde H es la matriz de tamano2k × 2k definida en la ecuacion anterior y donde las potencias sonevaluadas en M2k×2k[ZZk]. Esta funcion es eventualmente periodica, esdecir, existen T y P ∈ IN tales que HT = HT+P en ZZk.

En el caso en que k es primo, la funcion n #→ Hn es estrictamenteperiodica es decir, existe P ∈ IN tal que H = HP . En este caso decimosque la matriz H tiene perıodo P .

Para realizar el algoritmo computacional utilizamos el paquete scilabdel sistema operativo UNIX que fue disenado en IRIA [3]. Este algo-ritmo nos indica para cada k, las tripletas y las longitudes para las cualesel representante canonico de Pℓ es de la forma λχ(ℓ), iterando 100,000


polinomios. Se verifico que la mitad de las k2 − k matrices H posiblesen ZZp, para p primo mayor que 2, tienen perıodo p2− 1 y la otra mitadtiene perıodo 1

2p(p2 − 1). Aun no podemos determinar a cuales de las

matriz Hp le corresponde cada perıodo.

6 Resultados

Para las tripletas de la clase (η, 0, 1) no podemos aplicar el algoritmoantes mencionado. Para mostrar la sincronizacion enunciamos el sigui-ente teorema.

Teorema 6.0.1 Sea k ∈ N primo o potencia de primo, y sea C ∈ ZZk

tal que (C, k) = 1.Consideremos la familia Pℓ(λ) de polinomios en ZZk[λ], definidos por

la recurrencia Pℓ(λ) = λPℓ−1(λ) + CPℓ−2(λ).No existe ℓ ∈ N tal que Pℓ(λ) = λℓ.

Demostracion:Al calcular los polinomios por induccion obtenemos

P2m(λ) = λ2m +Xm(λ) + Cm

yP2m+1(λ) = λ2m+1 + 2mCλ2m−1 + Ym(λ) + (m+ 1)Cmλ

donde Xm(λ) y Ym(λ) son polinomios de grado 2m − 2 sin terminoindependiente ni lineal.

Parte 1: P2m(λ) = λ2m, ya que Cm no es congruente con 0 mod k,pues (C,k)=1.

Parte 2: P2m+1(λ) = λ2m+1.Prueba de Parte 2: Si P2m+1(λ) = λ2m+1, entonces 2mC ≡ 0 y

(m + 1)Cm ≡ 0 modulo k, y como (C, k) = 1 esto implica que 2m ≡ 0y m+ 1 ≡ 0 modulo k.

Probaremos que esto no es posible.Si k = pn, entonces 2m = βpn y m+ 1 = αpn(con p primo).Caso 1: Si p = 2 y n ≥ 2, entonces m = β2n−1 y m + 1 = α2n lo

cual es una contradiccion, pues hace a m par e impar a la vez.Caso 2: Si p > 2 y n≥1, entonces 2m = βpn y m + 1 = αpn.

La primera de estas igualdades implica que m = β′pn, por lo tanto(α− β)pn = 1. Esto es imposible pues 2 no es divisible por p, de modo


que 2 es divisible forzosamente por β, lo cual harıa que m y m+1 fueransimultaneamente multiplos de k.

Del teorema 6.0.1 se deriva el siguiente teorema:

Teorema 6.0.2 La pareja de AC con evolucionf(x) = Mx+ Cf(y) = κMx+ (1− κ) ∗My + Cdonde M ∈ MN [ZZp] tiene la forma de la ecuacion (1), con (β1,β2,β3) ∈

(η, 0, 1) y con η = 0, no sincronizan para κ = 0.

Teorema 6.0.3 Si conocemos las tripletas y sus longitudes de sincronizacionen alfabetos de la forma [ZZm × ZZn], para (m,n) = 1, podemos predecirlas tripletas y las longitudes de sincronizacion en [ZZmn].

Demostracion: Sea ϕ:ZZmn → ZZm × ZZn, tal que (m,n) = 1. Seaϕ : Mℓ[ZZmn] → Mℓ[ZZm × ZZn] el isomorfismo inducido por ϕ. ϕ es talque ϕ(A)(i, j) = ϕ(A(i, j)).

Es facil ver que siM ∈Mℓ[ZZmn] es nilpotente, ϕ[M] ∈Mℓ[ZZm × ZZn]tambien es nilpotente para (m,n) = 1 y viceversa.

Ejemplo Podemos predecir que la longitud de sincronizacion parala tripleta (5, 0, 1) en ZZ3

10 es ℓ = 2k − 1, pues sabemos que la tripleta(1, 0, 1) en ZZ3

2 tiene esta longitud de sincronizacion.

6.1 Resultados particulares

Las longitudes de sincronizacion para el sistema (ecuacion 1), en el casoen que M ∈ MN [ZZk] y k no es primo, dependen fuertemente de lafactorizacion de k. Del Tma. 6.0.3 se deduce facilmente lo que pasacuando k = pq con p y q primos diferentes.

A continuacion describimos el comportamiento de las longitudes desincronizacion, para algunos casos en que k es potencia de primo.

La formula recursiva para las tripletas de la clase (0, 2, 0) en ZZ34 es

de la siguiente forma:Pℓ(λ) = (2− λ)ℓ

Haciendo un calculo sencillo podemos observar que si ℓ es par sus poli-nomios caracterısticos tienen la forma Pℓ(λ) = λℓ, por lo tanto todo pares longitud de sincronizacion, lo mismo pasa para las tripletas de la clase(0, 4, 0) en ZZ3

8. Procediendo de manera similar podemos demostrar queℓ = 4m, m = 1, 2, 3, ... son longitudes de sincronizacion para las tripletasde las clases (0, 2, 0) y (0, 6, 0) en ZZ3

8.


Para la clase (2, 0, 1), en ZZ34 la formula de recurrencia es:

Pℓ+2(λ) = λPℓ+1(λ)− 2Pℓ(λ)

y sustituyendo Pℓ+1(λ) = λPℓ(λ)− 2Pℓ−1(λ) obtenemos

Pℓ+2(λ) = λ(λPℓ − 2Pℓ−1)− 2Pℓ

Pℓ+2(λ) = λ2Pℓ(λ)− 2λPℓ−1(λ))− 2Pℓ(λ),

y sustituyendo λPℓ−1(λ) = Pℓ(λ) + 2Pℓ−2(λ), para ℓ ≥ 2, obtenemos

Pℓ+2(λ) = λ2Pℓ(λ)− 2Pℓ(λ)− 4Pℓ−2(λ)− 2Pℓ(λ).

Por lo anterior Pℓ+2(λ) = λ2Pℓ(λ) ∀ℓ ≥ 2, de modo que

Pℓ+2(λ) =

!P2ℓ(λ) = λ2(ℓ−1)P2(λ) si ℓ es par,P2ℓ+1(λ) = λ2(ℓ1)P3(λ) si ℓ es impar.

Dado que P0(λ) = 1 y P1(λ) = λ, entonces P2(λ) = λ2 − 2, P3(λ) =λ3. Por lo tanto, cuando ℓ es impar tenemos P2ℓ+1(λ) = λ2ℓ+1, por loque todos los impares > 2 son longitudes de sincronizacion.

Para ℓ par tenemos

P2ℓ(λ) = λ2(ℓ)−1(λ2 − 2),

de modo que no hay sincronizacion para estas longitudes.

En ZZ38, para la clase (4, 0, 1), podemos realizar un calculo similar y

concluir que hay sincronizacion para ℓ impar.

Para ZZ39 las tripletas de la clase (0, 3, 0) tienen formula recursiva de

la forma:Pℓ(λ) = (λ+ 3)ℓ.

Es facil ver que ℓ = 3m, es longitud de sincronizacion para m =1, 2, 3, ... Para la clase (0, 6, 0), las longitudes de sincronizacion son lasmismas. Por otro lado, para las tripletas de la clase (3, 0, 1), la formulade recurrencia es:

Pℓ(λ) = λPℓ−1(λ)− 3Pℓ−2(λ),

de modo que P0(λ) = 1, P1(λ) = λ, P2(λ) = λ2 − 3, y P3(λ) = λ3 − 6λ.Probaremos por induccion que


P4+3m(λ) = λ4+3m

P5+3m(λ) = λ5+3m − 3λ3+3m

P6+3m(λ) = λ6+3m − 6λ4+3m

para todo m = 0, 1, 2, ...Si m = 0, se verifica directamente que P4(λ) = λ4, P5(λ) = λ5−3λ3,

y P6(λ) = λ6 − 6λ4.Suponemos que la hipotesis se cumple para un cierto m, entonces

P4+3(m+1)(λ) = P7+3m(λ) = λP6+3m(λ)− 3P5+3m(λ)

=λ(λ6+3m − 6λ4+3m) - 3(λ5+3m − 3λ3+3m)=λ7+3m − 6λ5+3m - 3λ5+3m + 9λ3+3m.

Como las operaciones se realizan mod 9, tenemos que

P7+3m(λ) = λ7+3m.

Procediendo de manera similar tenemos que

P8+3m(λ) = λ8+3m − 3λ6+3m,

yP9+3m(λ) = λ9+3m − 6λ7+3m.

Por lo tanto, ℓ = 4 + 3m es longitud de sincronizacion m = 0, 1, ...Finalmente, para la clase (6, 0, 1) podemos realizar calculos similaresdemostrando que hay sincronizacion para las mismas longitudes.

7 Comentarios finales

En este trabajo se analizo la sincronizacion de parejas de automatascelulares afines de rango 1, en los anillos ZZk.

Despues de realizar el estudio computacional de los anillos ZZ4, ZZ8 yZZ9, encontramos para que tripletas existen longitudes de sincronizacion,ası como tambien que no hay otras tripletas que sincronicen en estosanillos.

Demostramos, ayudados por la computadora, que no hay sincroniza-cion, exceptuando las tripletas triviales, en los campos ZZp para p de 3a 29.


Para los alfabetos ZZk, k = pq con (p, q) = 1 y por el Teorema 6.0.3podemos conocer las tripletas y sus longitudes de sincronizacion a partirde los resultados obtenidos en ZZp y ZZq.

En todos los campos ZZp, p > 2 que analizamos, la matriz Hp derecurrencia entre los representantes canonicos es tal que Hm

p = 1, dondeen la mitad de las matrices posibles m = (p2 − 1) y en la otra mitadm = 1

2p(p2 − 1).

A partir de los resultados anteriores podemos conjeturar que la formade Hp es responsable de:

• Perıodos de la forma (p2 − 1) y 12p(p

2 − 1)).

• Ausencia de representantes canonicos de la forma λχ(ℓ).

De aquı se deducirıa que para los alfabetos ZZp con p primo > 2, lasunicas tripletas con longitudes de sincronizacion son las triviales. Estose demostro para las tripletas del tipo (µ, 0, ν), µν = 0.

AgradecimientosAgradezco a los Dres. Valentin Afraimovich, Gelasio Salazar y Jesus

Urias por sus ensenanzas, las cuales fueron de gran ayuda para la rea-lizacion de este trabajo.

Un agradecimiento especial al Dr. Edgardo Ugalde por sus invalua-bles consejos durante la realizacion de este trabajo.

J. Guillermo Sanchez Saint-MartinIICO–UASLP,Av. Karakorum 1470, Lomas 4a.San Luis Potosı, S.L.P.78210 [email protected]

Referencias

[1] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod.Phys. 55, 601 (1983).

[2] V. S. Afraimovich, N. N. Verichev, and M. I. Rabinovich, Stochas-tic synchronization of oscillations in dissipative systems, Izv.Vyssh. Uchebn. Zaved. Radiofiz 29, 1050 (1986) [Sov. Radiophys.29, 795 (1986)].

[3] htpp://www.rocq.iria.fr/scilab/scilab.html


[4] J. Urias, G. Salazar y E. Ugalde, Synchronization of cellular Au-tomaton Pairs, CHAOS 8, 814 (1998).

[5] A. I. Maltsev, Fundamentos de algebra lineal, Editorial Mir Moscu.(1972).

[6] J. B. Fraleigh, Algebra abstracta, Addison-Wesley iberoamericana.(1987).

MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matema-ticas del CINVESTAV, se termino de imprimir en el mes de septiembre de 2000en el taller de reproduccion del mismo departamento localizado en Av. IPN2508, Col. San Pedro Zacatenco, Mexico, D.F. 07300. El tiraje en papel opalinaimportada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pastatintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contenido

Algebraic K-theory and the η-invariant

Jose Luis Cisneros Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Average optimal strategies in Markov games under a geometric drift condition

elneuKewU-znieH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Little cubes and homotopy theory

Dai Tamaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

rensM-esoBedsarbeglaysopurgrepiH

Isaıas Lopez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

seralulecsatamotuaedsajerapednoicazinorcniS

J. Guillermo Sanchez Saint-Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Morfismos, Vol 4, No 2, 2000

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