Morfismos, Vol 14, No 2, 2010

VOLUMEN 14NÚMERO 2

JULIO A DICIEMBRE DE 2010ISSN: 1870-6525

MorfismosComunicaciones EstudiantilesDepartamento de Matematicas

Cinvestav

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• Isidoro Gitler • Jesus Gonzalez

Consejo Editorial

• Luis Ernesto Carrera • Beatris Adriana Escobedo• Samuel Gitler • Onesimo Hernandez-Lerma

• Ismael Hernandez Noriega • Hector Jasso Fuentes• Miguel Maldonado • Enrique Ramırez de Arellano

• Enrique Reyes

Editores Asociados

• Ricardo Berlanga • Emilio Lluis Puebla • Guillermo Pastor• Vıctor Perez Abreu • Carlos Prieto • Carlos Renterıa • Luis Verde

Apoyo Tecnico

• Irving Josue Flores Romero • Omar Hernandez Orozco• Roxana Martınez • Carlos Daniel Reyes Morales• Ivan Martın Suarez Barraza • Laura Valencia

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VOLUMEN 14NÚMERO 2

JULIO A DICIEMBRE DE 2010ISSN: 1870-6525

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Morfismos

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Morfismos

Contenido

A note on distributional equations in discounted risk processes

Gerardo Hernandez del Valle and Carlos G. Pacheco Gonzalez . . . . . . . . . . . 1

Cohomology groups of configuration spaces of pairs of points in real projectivespaces

Jesus Gonzalez and Peter Landweber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

An upper bound on the size of irreducible quadrangulations

Gloria Aguilar Cruz and Francisco Javier Zaragoza Martınez . . . . . . . . . . . 61

Morfismos, Vol. 14, No. 2, 2010, pp. 1–15

A note on distributional equations in discountedrisk processes !

Gerardo Hernandez del Valle Carlos G. Pacheco Gonzalez

Abstract

In this paper we give an account of the classical discounted riskprocesses and their limiting distributions. For the models consid-ered, we set the Markov chains embedded in the continuous-timeprocesses; we also set distributional equations for the limit dis-tributions. Additionally, we mention some applications regardingruin probabilities and optimal premium.

2010 Mathematics Subject Classification: 60K05.Keywords and phrases: Aggregate claim amount, discounted risk pro-cess, perpetuity, embedded Markov chains, distributional equations, ruinprobability, income rates.

1 Introduction

Let us introduce the model. We can think of costumers arriving ac-cording to a renewal process and at their arrival they bring a rewardaccording to a i.i.d. sequence Xi’s. Then the discounted reward at timet with discounted rate ! is

(1) Z(!) := Z(!)t =

Nt

i=1

Xie!!Ti , t " 0 ,

where N := {N(t), t " 0} is a renewal process with interarrival times"1, "2, . . .; which are independent identically distributed positive random

!Invited article. Research partially supported by the CONACYT-NSF(Mexico-USA) program.

1

2 Gerardo Hernandez del Valle and Carlos G. Pacheco Gonzalez

variables (i.i.d. positive r.v.s.). The process N is defined through

(2) Nt := max

!k :

k"

i=1

!i ! t

#.

Further, Ti :=$i

j=1 !j , i = 1, 2, . . ., which represents the arrival timesas mentioned before. Variables X1, X2, ... are i.i.d. positive r.v.s. and "is the continuous time interest rate. Throughout this paper we assumethat the interarrival times and claim sizes are independent, and we

denote by ! and X the generic random variables, such that !d= !1 and

Xd= X1. To avoid technical problems we assume that P (! = 0) < 1.We use model (1) in the context of insurance; variable ! represent

a generic arrival and X a generic claim size, thus Z(!)t is the present

value of the claims up to time t. One can see that process (1) is anextension of the well-known renewal reward processes (see for instance[2, 32]), and when " = 0 it resembles a particular instance of the so-called continuous–time random walk (one may find a summary on thistype of process in [29]). The renewal reward process is also called theaggregate claim amount in the insurance jargon (see [26]). Processesthat resemble (1) have been studied with other names; for example, theMarkov shot noise processes in [30, 27], and when the renewal processis Poisson it is called filtered Poisson processes in [16]. Process (1) haverenewal properties, feature that classifies it in a more general familycalled regenerative processes, as labeled in [2]); or it is a particularinstance of a semi-Markov process, see e.g. [21]. Specifically, process(1) has been studied previously in [5, 9, 17, 18, 31, 19].

Let Z! be the limit of Z(!)t when t " #, when it is well defined,

intuitively one expects the following distributional equation to hold,

(3) Z! d= e"!" (Z! +X).

This equation is derived from a recursive random equation; regardingrecursive equations the reader might find interesting the many ideas in[10].

In section 2, we give the details to derive previous equation using theso-called embedded Markov chains ; this technique has been well used inother papers, see e.g. [12]. Following this idea, in section 3, we do thesame for the so-called discounted risk process. Then, we mentioned howone can find the moments recursively (see e.g. [18]). In section 4 it is

A note on distributional equations in discounted risk processes 3

used the tools developed to give straight forward applications, namelyfor studying the ruin probability and a perpetual cash flow. In addition,in subsection 4.3 we study a model where rate income and severity aresensitive to the price per contract, so that the insurer should choose anoptimal premium.

2 An embedded Markov chain

The term embedded Markov chain (EMC) refers to the concept of hav-ing a discrete-time process embedded within a continuous-time process.An important feature of model (1) is that it renews/regenerates at thevery time of an arrival Ti; this helps to identify Markov chains (MCs)embedded in the process. The EMC helps to study the limit behaviorof process (1) by studying the corresponding stationary distributions.The type of MCs that arise here can be compared to the so-called per-petuities (see also [20]), and in turn they give rise to distributional equa-tions (DEs), also called stochastic or random equations (some relevantreferences about DEs are [1, 14, 30]). An important application of thedistributional equations is finding properties of stationary distributions,such as moments or even the distribution itself (see e.g. [12, 22]).

Notice first that, since the trajectories are increasing, limt!" Z(!)t

always exists by monotone convergence theorem, so that it convergencesin distribution. Also,

Proposition 2.1. If X has finite mean, then Z(!)t converges in distri-

bution as t ! " to a random variable Z# with finite mean.

Proof. By Fatou´s Lemma,

(4) E(Z#) #"!

i=1

E(Xi)E(e$!Ti).

Finally, we know that E(e$!Ti) = Ei(e$!" ) and that E(e$!" ) < 1, then

(5) E(Z#) # E(X)"!

i=1

Ei(e$!" ) < ".

Having E(X) < " will be necessary in next section, when askingfor the positive safety loading condition.


Proposition 2.2. Let Yn be the process Z(!) evaluated at the time of

the n arrival, that is, Yn := Z(!)Tn

. Then the following identity holds

(6) Yn+1d= Xne

!!"n + e!!"nYn, n = 0, 1, . . . ,

with Y0 := 0. Here Xn, !n and Yn are independent for each n.

Proof. Using the definition of the process

Z(!)Tn+1

=

NTn+1!

i=1

Xie!!Ti =

n+1!

i=1

Xie!!Ti

d= Xe!!" +

n+1!

i=2

Xie!!Ti d

= Xe!!" + e!!"n+1!

i=2

Xie!!

!ik=2 "k

d= Xe!!" + e!!"Z(!)

Tn.

Hence, equation (6) can be set.

Remark 2.3. It is said that process Y := {Y0, Y1, . . .} of previousresult is an embedded Markov chain of process Z(!). There are resultsregarding ergodic properties of stochastic processes through embeddedMarkov chains (see for instance [6]). Since the paths of Z(!) are piecewiseconstant between arrivals, we shall study the limit behavior of Z(!) bystudying the stationary distribution of Y .

Remark 2.4. Relation (6) is an identity of distributions which itselfprovides a method for approximating samples of Z" by running the MC.This is possible due to the fact that the stationary distribution of theMC is the limiting distribution of Zt as t ! ". An extensive study onstochastic equations of this type can be found in Vervaat [30].

The following result is easy to see from previous results, however thesame idea is used later in Theorem 3.1.

Proposition 2.5. Let

(7) Z" := limt#$

Z(!)t .

Then we have the following distributional equation

(8) Z" d= Xe!!" + e!!"Z".


Proof. Notice that (Xi, !i, Yi) ! (X, !, Z!) as i ! ". Since f(x, t, y) :=xe"!t + e"!ty is a continuous function, we can apply the continuousmapping theorem (see [4]) to obtain (8) from (6)

Remark 2.6. Often, the following class of distributional equation arisesin insurance applications (see for example [12, 14, 30]):

(9) Z#d= AZ# +B,

where A,B and Z# are random variables, and Z# may or may not beindependent of (A,B). The question is to find the distribution of Z#given the distribution of (A,B).

A typical application of the distributional equation (8) is using itfor computing the moments of Z!.

Corollary 2.7. Suppose that X has all its moments finite and that theLaplace transform of ! exists. Then, the k-moment of Z! satisfies thefollowing recursive formula

(10) E((Z!)k) =E!e"k!"

"

1# E (e"k!" )

k"1#

i=0

$k

i

%E(Xk"i)E((Z!)i),

for k=1,2,. . . .

Proof. This is done by a direct use of the Newton´s binomial theorem tothe distributional equation (9). First, expanding the binomial, takingexpectations and then solving for the k-moment.

The procedure described in Corollary 2.7 for finding moments throughdistributional equations is quite common in the literature, see for ex-ample [20, p. 465] or [22, 30]. Notice that using the distributionalequations one may also attempt to find the characteristic function orthe moment generating function.

3 The present value distribution

Using process (1), a popular model in insurance is the so-called totalsurplus (we also call it discounted risk process) given by

(11) Ut = " + r(t)# Z(!)t , t $ 0,


where r(t) :=! t0 !e

!!sds is the present value of the incomes received bythe company, which is determined with the premium rate !. Variable" is the initial capital of the company. In [31] model (11) is called therenewal risk process. Process U = {Ut, t ! 0} in (11) represents thepresent value of the total surplus (earnings or losses) of the companyup to time t.

It is known that Ut admits the representation

Ut := " +

" t

0e!!sdYs,

where Ys := ! +#N(s)

i=1 Xi, see e.g. [15]. At this point, it would beguarantied for the insurance company that limt"# Ut > 0 almost surely;from previous equation we notice that this condition is achieved if thepositive safety loading condition holds, that is if

!" #E(X)

#E(X)> 0,

with # := 1/E($) (see [13]). From now on we assume that our modelsatisfies this condition.

When t # $, Ut may converge to a random variable, which isinterpreted as the total earnings or losses of the perpetuity, that is tosay, the total outcome of the business; the interested reader can find amore extensive discussion of this in [24]. A natural question is to find theso-called present value distribution, which is defined as the distributionof

(12) U$ := limt"#

Ut.

Finding the present value distribution has been done for a general classof models based on the Poisson process; important references are [11,12, 14, 15]. Specially in [12] one finds a good account.

One can see that if Z(!)t converges in distribution as t # $, so does

Ut. To this end we have the following (see also [25]).

Theorem 3.1. Let Z be the process specified in (1). When limt"# Z(!)t

exists in distribution, we have the following distributional equation

(13) U$ d= %" e!!" (X + %" U$) ,


where ! = " + !" . Moreover, the distribution of U! coincides with the

stationary distribution of the following MC

(14) Wn+1 = !! e""#n+1 (Xn+1 + !!Wn) ,

with W0 " R.

Proof. First, we will exploit the renewal property to obtain an identityin distribution. Then, we appeal to the continuous mapping theorem toset equation (13), which itself gives rise to the MC (14).

By using the definition of Tn, n = 1, 2, . . ., we have

UTn = " +

! Tn

0#e""sds!

NTn"

i=0

Xie""Ti

= " +

! Tn

0#e""sds!

#X1e

""#1 +n"

i=2

Xie""Ti

$

d= " +

! Tn

0#e""sds!

#Xe""# + e""#

n"1"

i=1

Xie""Ti

$

= " +

! Tn

0#e!!sds!

#Xe!!" + e!!"

#n!1"

i=1

Xie!!Ti ± " ±

! Tn!1

0#e!!sds

$$

= "+

! Tn

0#e""sds!

%Xe""# + e""#

%!UTn!1 + " +

! Tn!1

0#e""sds

&&.

Thus, when taking limits we have distributional equation (13).

Remark 3.2. Notice that the moments of U! may be computed usingequation (13) as in Corollary 2.7, however this approach does not givea formula as friendly as recurrence (10). It is more convenient to usethe fact that

(15) U! = !! Z!, with ! = " +#

$,

which yields

(16) E'(U!)k

(= (!!)k

k"

i=0

%k

i

&(!1)iE

'(Z!)k"i

(, k = 1, 2, . . . .


0 5 10 15 20 25 30 350.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

ï�� ï�� ï�� ï�� ï� 0 5 100.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Figure 1: Histograms of Z! and U!.

Corollary 2.7 and Remark 3.2 find the moments of Z! and U!, re-spectively. An interesting point is to find the actual limit distribution,i.e. finding the solutions (8) and (13). Next, we use the Markov chainsto perform numerical approximations.

Example 3.3. In figure 1 we show the approximations of Z! and U! fora model where the claim sizes and the interarrival times are exponential,both with parameters 1; and we have taken ! = 0.1, " = 5 and # = 0.3.We have run 106 times the corresponding Markov chains (6) and (14),and obtained numerically the histograms with partition 200.

4 Applications

Now we present some applications of the embedded Markov chains andthe distributional equations. First, we find a bound for the ruin prob-ability. Then, we discuss about the probability of ending negative inperpetual cash flow. Finally, we give an example to show that the in-come rate (i.e. #) may not be set too high or too low.

4.1 A bound for the ruin probability

Calculating the ruin probability has generated a great deal of interestin risk theory for discounted and non-discounted sums. Since the cele-brated works of Lundberg and Cramer, many articles and books havebeen published to address this problem; few references are [3, 13, 20, 26],and a concise summary can be found in [7].

Consider model (11). The ruin probability is defined as follows:Given the initial capital ", variable $! is the first time when Ut goesbelow 0 (when the company goes bankrupt or ruined). It is expressed


as P (!! < !) where

(17) !! = inf {s : Us < 0} .

Ruin probability has been studied extensively with model (1) whenthe interarrival times are exponential r.v.s and " = 0 (see e.g. [13]).Here we take " > 0 but now we assume that # is exponentially dis-tributed. Under these assumptions, Harrison [15] gives bounds for theruin probability in terms of the present value distribution.

Proposition 4.1.1. Using model (11) with # being an exponential ran-

dom variable, suppose that Z! d= limt"# Z(")

t is well defined. Then thefollowing upper bound for the ruin probability holds

(18) P (!! < !) " P (Z! > $ + %/")

P (Z! > %/").

Proof. By Corollary 2.4 in [15] we have that

(19) P (!! < !) " H(#$)

H(0),

where H is the distribution function of #$ + limt"# Ut. Recall thatlimt"# r(t) = %/".

Previous sections give grounds for finding numerically the bound forthe ruin probability. This is easily achieved by approximating P (Z! >z) using the embedded Markov chain of process Z("), which is explainedin Remark 2.4 and carried out in Example 3.3.

4.2 The probability of long-run negative dividends

We now turn to the study of the long time behavior of Ut as t $ !,which is the perpetual cash flow (such as in pension schemes); we maythink of this as the “total/final outcome of the business”. In this paperwe study the probability of ending up loosing at the infinite horizon:

(20) P (U! < 0).

We call quantity (20) the business-ruin probability. The name is moti-vated from the fact that U! may represent the total discounted dividendsof a company, and if it is negative, it means that the business did notwork.


Despite the aforementioned motivation, in the case of the insurancecompany, it is quite not realistic to take t ! ", because the insurerwould not continue operating in case of bankruptcy (i.e. when Ut < 0).

Compare to the classical concept of ruin probability, the business-ruin probability is a less stringent version. This, due to the fact thatprocess U can go below 0 but may end up positive as t ! ". Therefore,the event of going business-ruin implies that process U went negativeat some point, thus

(21) P (U! < 0) # P (!! < "),

where !! = inf {s : Us < 0}.A natural question is to find an income rate " that guaranties certain

level of total earnings. Moreover, we may find " that helps to achievelow probability of ending up loosing or a high probability of ending upearning. Notice that U! depends on the rate ", and we can write U!(")to emphasize this. The following definition gives criteria to choose anincome rate.

Definition 4.2.1. For # $ (0, 1), whenever it exists we call the quantity"" the #-percentile income rate if it is such that

(22) P (U!("") $ A) = #,

for some Borel set A. And we call "# , with $>0, the $-mean income rateif it is such that

(23) E(U!("#)) = $.

Generally, we would be interested in an income rate that allows usto either minimize the potential loss, maximize profit, or simply suchthat we reach a minimum level of profit. The previous definition takesinto account these ideas, see next remark.

Remark 4.2.2. For the #-percentile income rate, one needs to be morespecific. For instance, if we are interested on minimizing loss, a naturalchoice for A is (%", %], where % & 0 is a minimum level of tolerance.Likewise, we can set A = [%,") if we want to achieve certain level ofprofit. In any case, the calculation of "" requires the knowledge of thedistribution of U! (the present value distribution).

The calculation of "# is direct from equation (13). Equating theexpectation E(U!) to $, and solving for "# we have that


Proposition 4.2.3. The !-mean income rate is given by

(24) "! = #2(! ! $)

!1! E(e!"# )

"+ E(X)E(e!"# )

1! E(e!"# ).

Example 4.2.4. In Example 3.3, for " = 0.3, we have that P (U" <0) " 0.71. Thus the %!percentile premium rate is "$ = 0.3 with % " 0.71.This is an unfavorable scenario for the company, because with a highprobability the business will end up loosing.

4.3 A control problem for the insurance company

The income rate " is a quantity that depends on the price per contract.That is, the rate of income is a factor that can be determined by howcheap or expensive the actual price of the contract is.

Let p be the price per contract. Price p may be so expensive that noone would be able to a!ord it (and thus no income would be obtained);or, the price could be so cheap that even though many would buy it, theincome rate would not be enough to pay the potential losses. The valueof p would a!ect the income rate " and the frequency of arrivals, definedby & . Thus, the company does not want to set a very expensive or verycheap price per contract, rather it needs to find an optimal price.

Consider the distributional equation (13). If we take expectation ofboth sides of (13), and solve for E(U") we obtain

(25) E(U") =$ + %

" ! E(e!"# )(E(X) + $ + %" )

1! E(e!"# ).

Here, " and E(e!"# ) are functions of p. Then, an optimal price p canbe found by maximizing (25).

Now, we give a model that is specified by p.Suppose that & is an exponential r.v. with mean 1

&(p) , where

(26) '(p) =a

pb, for a > 0, b > 0.

Moreover, suppose that "(p) is given by

(27) "(p) = pce!dp, for c > 0, d # 0.

If p is small (cheap), more people would buy a contract, and thus morepotential loses might arrive in the future. If p is large (expensive),


0 5 10 15 20 25 30 35 40 45 504

6

8

10

12

14

16

18

20

22

24

Figure 2: E(U!) as function of p

less people would buy the insurance, thus the company would have lesslosses. These phenomena are reflected in (26) and (27).

Now we have that

E!e"!"

"=

!(p)

!(p) + ".

Furthermore, if X ! exp(µ), formula (25) becomes

(28) E(U!) =

## +

$(p)

"" !(p)

!(p) + "

#µ+ # +

$(p)

"

$$!(p) + "

".

Using Example 3.3 and setting a = 0.05, b = 0.1, c = 0.5 and d = 0.05 todefine functions (26) and (27), in Figure 4.3 we plot E(U!) as functionof p. We can see that E(U!) attains a maximum: the optimal price percontract. Related to this application, see [28] for control problems ininsurance.

AcknowledgementWe thank Jose Alfredo Lopez Mimbela for telling us about the ne-

cessity of including the safety loading condition. We also thank the


anonymous referee for pointing out numerous corrections and sugges-tions.

Gerardo Hernandez del ValleStatistics Department,Columbia University,1255 Amsterdam Ave. MailCode 4403, New York, N.Y.,[email protected]

Carlos G. Pacheco GonzalezDepartamento de Matematicas,CINVESTAV-IPN,A. Postal 14-740Mexico D.F. 07000, MEXICO,[email protected]

References

[1] Aldous D.J.; Bandyopadhyay A., A survey of max-type recursivedistributional equations, Annals of Applied Probability 15 No. 2(2005), 1047–1110.

[2] Asmussen S., Applied Probability and Queues, Springer, 2003.

[3] Asmussen S., Ruin Probabilities, World Scientific, 2000.

[4] Billinsley P., Convergence of Probability Measures, John Wiley &Sons, 1968.

[5] Boogaert P.; Haezendonck J.; Delbaen F., Limit theorems for thepresent value of the surplus of an insurance portfolio, Insurance:Mathematics and Economics 7 (1988), 131–138.

[6] Borovkov A. A., Ergodicity and Stability of Stochastic Processes,John Wiley & Sons, 1998.

[7] Cai J., Cramer-Lundberg asymptotics, Encyclopedia of ActuarialScience (2004), John Wiley & Sons.

[8] Cai J., Ruin probabilities and penalty functions with stochasticrates of interest, Stochastic Processes and their Application 112(2004), 53–78.

[9] Delbaen F.; Haezendonck J., Classical risk theory in an economicenvironment, Insurance: Mathematics and Economics 6 (1987),85–116.

[10] Diaconis P.; Freedman D., Iterated random functions, SIAM Rev41 (1999), 45–76.


[11] Dufresne D., The distribution of a perpetuity, with applications torisk theory and pension funding, Scandinavian Actuarial Journal9 (1990), 39–79.

[12] Gjessing H.K.; Paulsen J., Present value distribution with applica-tions to ruin theory and stochastic equations, Stochastic Processesand their Applications 71 (1997), 123–144.

[13] Grandell J., Aspects of Risk Theory, Springer, 1991.

[14] Goldie C.M., Implicit renewal theory and tails of solutions of ran-dom equations, Annals of Applied Probability 1 No. 1 (1991),126–166.

[15] Harrison J.M., Ruin problems with compounding assets, StochasticProcesses and their Applications 5 (1977), 67–79.

[16] Lefebvre M., Applied Stochastic Processes, Springer, 2007.

[17] Leveille G.; Garrido J., Moments of compound renewal sums withdiscounted claims, Insurance: Mathematics and Economics 28(2001), 217–231.

[18] Leveille G.; Garrido J., Recursive moments of compound renewalsums with discounted claims, Scandinavian Actuarial Journal 2(2001), 98–110.

[19] Leveille G.; Garrido J.; Wang Y. F., Moment generating functionof compound renewal sums with discounted claims, ScandinavianActuarial Journal 1 (2009), 1–20.

[20] Embrechts P.; Kluppelberg C.; Mikosch T., Modelling ExtremalEvents for Insurance and Finance, Springer, 1997.

[21] Korolyuk V.S.; Korolyuk V.V., Stochastic Models of Systems,Kluwer Academic Publishers, 1999.

[22] Pacheco-Gonzalez C. G.; Stoyanov J., A class of Markov chainswith Beta ergodic distributions, The Mathematical Scientist 33No. 2 (2008), 110–119.

[23] Paulsen J., Risk theory in a stochastic economic environment,Stochastic Processes and their Applications 46 (1993), 327–361.


[24] Paulsen J., Present value of some insurance portfolio, Scandina-vian Actuarial Journal 20 No. 3 (1997), 260–260.

[25] Paulsen J., Ruin models with investment income, Probability Sur-veys 5 (2008), 416–434.

[26] Rolski T.; Schmidli H.; Schmidt V.; Teugels J., Stochastic Pro-cesses for Insurance and Finance, John Wiley & Sons, 1999.

[27] Ross S.M., Stochastic Processes, John Wiley & Sons, 1996.

[28] Schmidli H., Stochastic Control in Insurance, Springer, 2008.

[29] Scalas E., Five years of continuous-time random walks in econo-physics, In: The Complex Networks of Economic Interactions,Lectures Notes in Economics and Mathematical Systems, 5672006, Springer Verlag Berlin, 3–16.

[30] Vervaat W., On stochastic di!erence equation and a representa-tion of non-negative infinitely divisible random variable, Advancesin Applied Probability 11 (1979), 750–783.

[31] Yuen K.C.; Wang G.; Wu R., On the renewal risk process withstochastic interest rate, Stochastic Processes and their Applica-tions 116 (2006), 1496–1510.

[32] Whitt W., Stochastic-Processes Limits, Springer, 2002.


Cohomology groups of configuration spaces ofpairs of points in real projective spaces

zelaznoGsuseJ 1 Peter Landweber

Abstract

The Stiefel manifold Vm+1,2 of 2-frames in Rm+1 is acted uponby the orthogonal group O(2). By restriction, there are corre-sponding actions of the dihedral group of order 8, D8, and ofthe rank-2 elementary 2-group Z2 !Z2. We use the Cartan-Lerayspectral sequences of these actions to compute the integral homol-ogy and cohomology groups of the configuration spaces B(Pm, 2)and F (Pm, 2) of (unordered and ordered) pairs of points on thereal projective space Pm.

2010 Mathematics Subject Classification: 55R80, 55T10, 55M30, 57R19,

57R40.

Keywords and phrases: 2-point configuration spaces, dihedral group oforder 8, twisted Poincare duality, torsion linking form.

1 Introduction

The integral cohomology rings of the configuration spaces F (Pm, 2) andB(Pm, 2) of two distinct points, ordered and unordered respectively, inthe m-dimensional real projective space Pm have recently been com-puted in [6]. The method in that paper relies on a rather technicalbookkeeping in the corresponding Bockstein spectral sequences. As aconsequence, a reader following the details in that work might miss partof the geometrical insight of the problem (in Definition 1.4 and subse-quent considerations). To help remedy such a situation, we o!er in thispaper an alternative approach to the additive structure.

1Partially supported by CONACYT Research Grant 102783.

17

18 Jesus Gonzalez and Peter Landweber

The basic results are presented in Theorems 1.1 and 1.2 below, wherethe notation !k" stands for the elementary abelian 2-group of rank k,Z2 # · · · # Z2 (k times), and where we write {k} as a shorthand for!k" # Z4.

Theorem 1.1. For n > 0,

H i(F (P2n, 2)) =

!"""""""""#

"""""""""$

Z, i = 0 or i = 4n$ 1;%i2 + 1

&, i even, 1 % i % 2n;%

i!12

&, i odd, 1 % i % 2n;%

2n+ 1$ i2

&, i even, 2n < i < 4n$ 1;%

2n$ i+12

&, i odd, 2n < i < 4n$ 1;

0, otherwise.

For n & 0,

H i(F (P2n+1, 2)) =

!"""""""""""#

"""""""""""$

Z, i = 0;%i2 + 1

&, i even, 1 % i % 2n;%

i!12

&, i odd, 1 % i % 2n;

Z# !n", i = 2n+ 1;%2n+ 1$ i

2

&, i even, 2n+ 1 < i % 4n+ 1;%

2n+ 1$ i!12

&, i odd, 2n+ 1 < i % 4n+ 1;

0, otherwise.

Theorem 1.2. Let 0 % b % 3. For n > 0,

H4a+b(B(P2n, 2)) =

!"""""""""""""""""""#

"""""""""""""""""""$

Z, 4a+ b = 0 or 4a+ b = 4n$ 1;

{2a}, b = 0 < a, 4a+ b % 2n;

!2a" , b = 1, 4a+ b % 2n;

!2a+ 2" , b = 2, 4a+ b % 2n;

!2a+ 1" , b = 3, 4a+ b % 2n;

{2n$ 2a}, b = 0, 2n < 4a+ b < 4n$ 1;

!2n$ 2a$ 1", b = 1, 2n < 4a+ b < 4n$ 1;

!2n$ 2a", b = 2, 2n < 4a+ b < 4n$ 1;

!2n$ 2a$ 2", b = 3, 2n < 4a+ b < 4n$ 1;

0, otherwise.

Cohomology of two points in projective spaces 19

For n ! 0,

H4a+b(B(P2n+1, 2)) =

!""""""""""""""""""#

""""""""""""""""""$

Z, 4a+ b = 0;

{2a}, b = 0 < a, 4a+ b < 2n+ 1;

!2a" , b = 1, 4a+ b < 2n+ 1;

!2a+ 2" , b = 2, 4a+ b < 2n+ 1;

!2a+ 1" , b = 3, 4a+ b < 2n+ 1;

Z# !n", 4a+ b = 2n+ 1;

{2n$ 2a}, b = 0, 2n+ 1 < 4a+ b % 4n+ 1;

!2n+ 1$ 2a", b = 1, 2n+ 1 < 4a+ b % 4n+ 1;

!2n$ 2a", b & {2, 3}, 2n+ 1 < 4a+ b % 4n+ 1;

0, otherwise.

As noted in [6], Theorems 1.1 and 1.2 can be coupled with the Uni-versal Coe!cient Theorem (UCT), expressing homology in terms of co-homology (e.g. [22, Theorem 56.1]), in order to give explicit descriptionsof the corresponding integral homology groups. Another immediateconsequence is that, together with Poincare duality (in its not neces-sarily orientable version, cf. [17, Theorem 3H.6] or [24, Theorem 4.51]),Theorems 1.1 and 1.2 give a corresponding explicit description of thew1-twisted homology and cohomology groups of F (Pm, 2) and B(Pm, 2).Details are given in Section 4—a second contribution not discussed in [6].

Remark 1.3. Note that, after inverting 2, both B(Pm, 2) and F (Pm, 2)are homology spheres. This assertion can be considered as a partialgeneralization of the fact that both F (P1, 2) and B(P1, 2) have the ho-motopy type of a circle; for B(P1, 2) this follows from Lemma 1.6 andExample 3.4 below, while the situation for F (P1, 2) comes from thefact that P1 is a Lie group—so that F (P1, 2) is in fact di"eomorphicto S1 " (S1 # {1}). In particular, any product of positive dimensionalclasses in either H!(F (P1, 2)) or H!(B(P1, 2)) is trivial. The trivial-product property also holds for both H!(F (P2, 2)) and H!(B(P2, 2)) inview of the P2-case in Theorems 1.1 and 1.2. For m ! 3, the multiplica-tive structure of H!(F (Pm, 2)) and H!(B(Pm, 2)) was first worked outin [5].

Definition 1.4. Recall that D8 can be expressed as the usual wreathproduct extension

(1) 1 $ Z2 " Z2 $ D8 $ Z2 $ 1.

Let !1, !2 % D8 generate the normal subgroup Z2"Z2, and let (the classof) ! % D8 generate the quotient group Z2 so that, via conjugation,


! switches !1 and !2. D8 acts freely on the Stiefel manifold Vn,2 oforthonormal 2-frames in Rn by setting

!(v1, v2) = (v2, v1), !1(v1, v2) = (!v1, v2), and !2(v1, v2) = (v1,!v2).

This describes a group inclusion D8 "" O(2) where the rotation !!1 isa generator for Z4 = D8 # SO(2).

Notation 1.5. Throughout the paper the letter G stands for eitherD8 or its subgroup Z2 $ Z2 in (1). Likewise, Em = Em,G denotes theorbit space of the G-action on Vm+1,2 indicated in Definition 1.4, and# : Vm+1,2 " Em,G represents the canonical projection. Our interest liesin the (kernel of the) morphism induced in cohomology by the map

(2) p = pm,G : Em " BG

that classifies the G-action on Vm+1,2.

Lemma 1.6 ([15, Proposition 2.6]). Em is a strong deformation retractof B(Pm, 2) if G = D8, and of F (Pm, 2) if G = Z2 $ Z2.

Thus, the cohomology properties of the configuration spaces we areinterested in—and of (2), for that matter—can be approached via theCartan-Leray spectral sequence (CLSS) of the G-action on Vm+1,2. Suchan analysis yields:

Proposition 1.7. Let m be even. The map p! : H i(BG) " H i(Em) is:

1. an isomorphism for i % m;

2. an epimorphism with nonzero kernel for m < i < 2m! 1;

3. the zero map for 2m! 1 % i.

Proposition 1.8. Let m be odd. The map p! : H i(BG) " H i(Em) is:

1. an isomorphism for i < m;

2. a monomorphism onto the torsion subgroup of H i(Em) for i = m;

3. an epimorphism with nonzero kernel for m < i % 2m! 1.

4. the zero map for 2m! 1 < i.


Kernels in the above two results are carefully described in [6]. Theapproach in this paper allows us to prove Propositions 1.7 and 1.8,except for item 3 in Proposition 1.8 if G = D8 and m ! 3 mod 4.

Since the ring H!(BG) is well known (see Theorem 2.3 and thecomments following Lemma 2.8), the multiplicative structure ofH!(Em)through dimensions at most m follows from the four results stated inthis section. Of course, the ring structure in larger dimensions dependson giving explicit generators for the ideal Ker(p!). In this direction wenote that the methods in this paper also yield:

Proposition 1.9. Let G = D8. Assume m "! 3 mod 4 and considerthe map in (2). In dimensions at most 2m # 1, every nonzero elementin Ker(p!) has order 2, i.e. 2 · Ker(p!) = 0 in those dimensions. Infact, every 4!-dimensional integral cohomology class in BD8 generatinga Z4-group maps under p! into a class which also generates a Z4-groupprovided ! < m/2—otherwise the class maps trivially for dimensionalreasons.

Remark 1.10. By Lemma 2.8 below, Ker(p!) is also killed by multipli-cation by 2 when G = Z2 $ Z2 (any m, any dimension). Our approachallows us to explicitly describe the (dimension-wise) 2-rank of Ker(p!)in the cases where we know this is an F2-vector space (i.e. when eitherG = Z2$Z2 or m "! 3 mod 4, see Examples 5.3 and 5.7). Unfortunatelythe methods used in the proofs of Propositions 1.7–1.9 break down forE4n+3,D8 , and Section 7 discusses a few such aspects focusing attentionon the case n = 0.

The spectral sequence methods in this paper are similar in spiritto those in [3] and [9]. In the latter reference, Feichtner and Zieglerdescribe the integral cohomology rings of ordered configuration spaceson spheres by means of a full analysis of the Serre spectral sequence(SSS) associated to the Fadell-Neuwirth fibration " : F (Sk, n) % Sk

given by "(x1, . . . , xn) = xn (a similar study is carried out in [10],but in the context of ordered orbit configuration spaces). One of themain achievements of the present paper is a successful calculation ofcohomology groups of unordered configuration spaces (on real projectivespaces), where no Fadell-Neuwirth fibrations are available—instead werely on Lemma 1.6 and the CLSS2 of the G-action on Vm+1,2. Also worth

2Our CLSS calculations can also be done in terms of the SSS of the fibrationVm+1,2

!! Em,Gp! BG.


stressing is the fact that we succeed in computing cohomology groupswith integer coe!cients, whereas the Leray spectral sequence (and its"k-invariant version) for the inclusion F (X, k) !! Xk has proved to bee#ectively computable mainly when field coe!cients are used ([11, 28]).

A major obstacle we have to confront (not present in [9]) comes fromthe fact that the spectral sequences we encounter often have non-simplesystems of local coe!cients. This is also the situation in [3], wherethe two-hyperplane case of Grunbaum’s mass partition problem ([14])is studied from the Fadell-Husseini index theory viewpoint [7]. Indeed,Blagojevic and Ziegler deal with twisted coe!cients in their main SSS,namely the one associated to the Borel fibration

(3) Sm " Sm ! ED8 "D8 (Sm " Sm)

p! BD8

where the D8-action on Sm " Sm is the obvious extension of that inDefinition 1.4. Now, the main goal in [3] is to describe the kernel ofthe map induced by p in integral cohomology—the so-called Fadell-Husseini (Z-)index of D8 acting on Sm"Sm, IndexD8(S

m"Sm). SinceD8 acts freely on Vm+1,2, IndexD8(S

m " Sm) is contained in the kernelof the map induced in integral cohomology by the map p : Em ! BD8

in Proposition 1.9 (whether or not m # 3 mod 4). In particular, thework in [3] can be used to identify explicit elements in Ker(p!) and, asobserved in Remark 1.10, our approach allows us to assess, for m $#3 mod 4 (in Examples 5.3 and 5.7), how much of the actual kernel isstill lacking description: [3] gives just a bit less than half the expectedelements in Ker(p!).

2 Preliminary cohomology facts

As shown in [1] (see also [15] for a straightforward approach), themod 2 cohomology of D8 is a polynomial ring on three generatorsx, x1, x2 % H!(BD8;F2), the first two of dimension 1, and the last oneof dimension 2, subject to the single relation x2 = x · x1. The classes xiare the restrictions of the universal Stiefel-Whitney classes wi (i = 1, 2)under the map corresponding to the group inclusion D8 & O(2) in Def-inition 1.4. On the other hand, the class x is not characterized by therelation x2 = x ·x1, but by the requirement that, for all m, x pulls back,under the map pm,D8 in (2), to the map u : B(Pm, 2) ! P" classifyingthe obvios double cover F (Pm, 2) ! B(Pm, 2)—see [15, Proposition 3.5].In particular:


Lemma 2.1. For i ! 0, H i(BD8;F2) = "i+ 1#.

Corollary 2.2. For any m,

H i(B(Pm, 2);F2) =

!"#

"$

"i+ 1#, 0 $ i $ m% 1;

"2m% i#, m $ i $ 2m% 1;

0, otherwise.

Proof. The assertion for i ! 2m follows from Lemma 1.6 and dimen-sional considerations. Poincare duality implies that the assertion form $ i $ 2m%1 follows from that for 0 $ i $ m%1. Since Vm+1,2 is (m%2)-connected, the assertion for 0 $ i $ m% 1 follows from Lemma 2.1,using the fact (a consequence of [15, Proposition 3.6 and (3.8)]) that,

in the mod 2 SSS for the fibration Vm+1,2!& Em,D8

p& BD8, the twoindecomposable elements in H!(Vm+1,2;F2) transgress to nontrivial el-ements.

Let Z" denote the Z[D8]-module whose underlying group is free ona generator ! on which each of ", "1, "2 ' D8 acts via multiplicationby %1 (in particular, elements in D8 ( SO(2) act trivially). Corollar-ies 2.4 and 2.5 below are direct consequences of the following description,proved in [16] (see also [3, Theorem 4.5]), of the ring H!(BD8) and ofthe H!(BD8)-module H!(BD8;Z"):

Theorem 2.3 (Handel [16]). H!(BD8) is generated by classes µ2, #2,$3, and %4 subject to the relations 2µ2 = 2#2 = 2$3 = 4%4 = 0, #22 =µ2#2, and $2

3 = µ2%4. H!(BD8;Z") is the free H!(BD8)-module onclasses !1 and !2 subject to the relations 2!1 = 4!2 = 0, $3!1 = µ2!2,and %4!1 = $3!2. Subscripts in the notation of these six generatorsindicate their cohomology dimensions.

The notation a2, b2, c3, and d4 was used in [16] instead of the currentµ2, #2, $3, and %4. The change is made in order to avoid confusion withthe generic notation di for di!erentials in the several spectral sequencesconsidered in this paper.

Corollary 2.4. For a ! 0 and 0 $ b $ 3,

H4a+b(BD8) =

!""""""#

""""""$

Z, (a, b) = (0, 0);

{2a}, b = 0 < a;

"2a#, b = 1;

"2a+ 2#, b = 2;

"2a+ 1#, b = 3.


Corollary 2.5. For a ! 0 and 0 " b " 3,

H4a+b(BD8;Z!) =

!""""#

""""$

#2a$, b = 0;

#2a+ 1$, b = 1;

{2a}, b = 2;

#2a+ 2$, b = 3.

We show that, up to a certain symmetry condition (exemplified inTable 1 at the end of Section 4), the groups explicitly described by Corol-laries 2.4 and 2.5 delineate the additive structure of the graded groupH!(B(Pm, 2)). The corresponding situation for H!(F (Pm, 2)) uses thefollowing well-known analogues of Lemma 2.1 and Corollaries 2.2, 2.4and 2.5:

Lemma 2.6. For i ! 0, H i(P" % P";F2) = #i+ 1$.

Lemma 2.7. For any m,

H i(F (Pm, 2);F2) =

!"#

"$

#i+ 1$, 0 " i " m& 1;

#2m& i$, m " i " 2m& 1;

0, otherwise.

Lemma 2.8. For i ! 0,

H i(P" % P") =

!"#

"$

Z, i = 0;%i2 + 1

&, i even , i > 0;%

i#12

&, otherwise.

H i(P" % P";Z!) =

'%i2

&, i even;%

i+12

&, i odd.

Here Z! is regarded as a (Z2 % Z2)-module via the restricted structurecoming from the inclusion Z2 % Z2 !' D8.

Here are some brief comments on the proofs of Lemmas 2.6–2.8. Ofcourse, the ring structure H!(P"%P";F2) = F2[x1, y1] is standard (asin Theorem 2.3, subscripts for the cohomology classes in this paragraphindicate dimension). On the other hand, it is easily shown (see forinstance [17, Example 3E.5 on pages 306–307]) that H!(P" % P") is


the polynomial ring over the integers on three classes x2, y2, and z3subject to the four relations

(4) 2x2 = 0, 2y2 = 0, 2z3 = 0, and z23 = x2y2(x2 + y2).

These two facts yield Lemma 2.6 and the first equality in Lemma 2.8.Lemma 2.7 can be proved with the argument given for Corollary 2.2—replacing D8 by its subgroup Z2 ! Z2 in (1). Finally, both equalities inLemma 2.8 can be obtained as immediate consequences of the Kunnethexact sequence (for the second equality, note that Z! arises as the tensorsquare of the standard twisted coe!cients for a single factor P!).

Remark 2.9. For future reference we recall (again from Hatcher’s book)that the mod 2 reduction map H"(P! ! P!) " H"(P! ! P!;F2), amonomorphism in positive dimensions, is characterized by x2 #" x21,y2 #" y21, and z3 #" x1y1(x1 + y1).

3 Orientability properties of some quotients ofVn,2

Proofs in this section will be postponed until all relevant results havebeen presented. Recall that all Stiefel manifolds Vn,2 are orientable(actually parallelizable, cf. [26]). Even if some of the elements of a givensubgroup H of O(2) fail to act on Vn,2 in an orientation-preserving way,we could still use the possible orientability of the quotients Vn,2/H asan indication of the extent to which H, as a whole, is compatible withthe orientability of the several Vn,2. For example, while every element ofSO(2) gives an orientation-preserving di"eomorphism on each Vn,2, it iswell known that the Grassmannian Vn,2/O(2) of unoriented 2-planes inRn is orientable if and only if n is even (see for instance [23, Example 47on page 162]). We show that a similar—but shifted—result holds whenO(2) is replaced by D8.

Notation 3.1. For a subgroup H of O(2), we will use the shorthandVn,H to denote the quotient Vn,2/H. For instance Vm+1,G = Em,G, thespace in Notation 1.5.

Proposition 3.2. For n > 2, Vn,D8 is orientable if and only if n isodd. Consequently, for m > 1, the top dimensional cohomology groupof B(Pm, 2) is

H2m#1(B(Pm, 2)) =

!Z, for even m;

Z2, for odd m.


Remark 3.3. Proposition 3.2 holds (with the same proof) if D8 isreplaced by its subgroup Z2!Z2, and B(Pm, 2) is replaced by F (Pm, 2).It is interesting to compare both versions of Proposition 3.2 with the factthat, for m > 1, B(Pm, 2) is non-orientable, while F (Pm, 2) is orientableonly for odd m ([18, Lemma 2.6]).

Example 3.4. The cases with n = 2 and m = 1 in Proposition 3.2are special (compare to [18, Proposition 2.5]): Since the quotient ofV2,2 = S1 " S1 by the action of D8 # SO(2) is di!eomorphic to thedisjoint union of two copies of S1/Z4, we see that V2,D8

$= S1.

If we take the same orientation for both circles in V2,2 = S1 " S1,it is clear that the automorphism H1(V2,2) % H1(V2,2) induced by anelement r & D8 is represented by the matrix ( 0 1

1 0 ) if r & SO(2), but bythe matrix

!0 !1!1 0

"if r '& SO(2). For larger values of n, the method of

proof of Proposition 3.2 allows us to describe the action of D8 on theintegral cohomology ring of Vn,2. The answer is given in terms of thegenerators !, !1, !2 & D8 introduced in Definition 1.4.

Theorem 3.5. The three automorphisms !!, !!1, !!2 : H

q(Vn,2) % Hq(Vn,2)

agree. For n > 2, this common morphism is the identity except when nis even and q & {n( 2, 2n( 3}, in which case the common morphism ismultiplication by (1.

Theorem 3.5 should be read keeping in mind the well-known coho-mology ring H"(Vn,2). We recall its simple description after provingProposition 3.2. For the time being it su"ces to recall, for the purposesof Proposition 3.6 below, that Hn!1(Vn,2) = Z2 for odd n, n ) 3.

We use our approach to Theorem 3.5 in order to describe the integralcohomology ring of the oriented Grassmannian Vn,SO(2) for odd n, n ) 3.Although the result might be well known (Vn,SO(2) is a complex quadricof complex dimension n( 2), we include the details (an easy step fromthe constructions in this section) since we have not been able to find anexplicit reference in the literature.

Proposition 3.6. Assume n is odd, n = 2a + 1 with a ) 1. Let#z & H2(Vn,SO(2)) stand for the Euler class of the smooth principal S1-bundle

(5) S1 % Vn,2 % Vn,SO(2)


There is a class !x ! Hn!1(Vn,SO(2)) mapping under the projection in (5)to the nontrivial element in Hn!1(Vn,2). Furthermore, as a ring

H"(Vn,SO(2)) = Z[!x, !z ]/In

where In is the ideal generated by

(6) !x 2, !x !z a, and !z a " 2!x.

It should be noted that the second generator of In is superfluous. Weinclude it in the description since it will become clear, from the proofof Proposition 3.6, that the first two terms in (6) correspond to the twofamilies of di!erentials in the SSS of the fibration classifying (5), whilethe last term corresponds to the family of nontrivial extensions in theresulting E#-term.

Remark 3.7. It is illuminating to compare Proposition 3.6 with H. F.Lai’s computation of the cohomology ring H"(Vn,SO(2)) for even n, n #4. According to [20, Theorem 2], H"(V2a,SO(2)) = Z[!, !z ]/I2a where I2ais the ideal generated by

(7) !2 " "!!z a!1 and !z a " 2!!z.

Here " = 0 for a even, and " = 1 for a odd, while the generator! ! H2a!2(V2a,SO(2)) is the Poincare dual of the homology class rep-resented by the canonical (realification) embedding CPa!1 #$ V2a,SO(2)

(Lai also proves that ("1)a!1!!z a!1 is the top dimensional cohomologyclass in V2a,SO(2) corresponding to the canonical orientation of this man-ifold). The first fact to observe in Lai’s description of H"(V2a,SO(2)) isthat the two dimensionally forced relations !!z a = 0 and !z 2a!1 = 0 canbe algebraically deduced from the relations implied by (7). A similarsituation holds for H"(V2a+1,SO(2)), where the first two relations in (6),as well as the corresponding algebraically implied relation !z 2a = 0, areforced by dimensional considerations. But it is more interesting to com-pare Lai’s result with Proposition 3.6 through the canonical inclusions$n : Vn,SO(2) #$ Vn+1,SO(2) (n # 3). In fact, the relations given by thelast element both in (6) and (7) readily give

(8) $"2a(!x) = !!z and $"2a+1(!) = !x

for a # 2. Note that the second equality in (8) can be proved, for alla # 1, with the following alternative argument: From [20, Theorem 2],


2!! !z a " V2a+2,SO(2) is the Euler class of the canonical normal bundleof V2a+2,SO(2) and, therefore, maps trivially under "!2a+1. The secondequality in (8) then follows from the relation implied by the last ele-ment in (6). Needless to say, the usual cohomology ring H!(BSO(2)) isrecovered as the inverse limit of the maps "!n (of course BSO(2) # CP").

Proof of Proposition 3.2 from Theorem 3.5. Since the action of everyelement in D8 $ SO(2) preserves orientation in Vn,2, and since two ele-ments in D8!SO(2) must “di!er” by an orientation-preserving elementin D8, the first assertion in Proposition 3.2 will follow once we arguethat (say) # is orientation-preserving precisely when n is odd. But sucha fact is given by Theorem 3.5 in view of the UCT. The second assertionin Proposition 3.2 then follows from Lemma 1.6, [17, Corollary 3.28],and the UCT (recall dim(Vn,2) = 2n! 3).

We now start working toward the proof of Theorem 3.5, recallingin particular the cohomology ring H!(Vn,2). Let n > 2 and think ofVn,2 as the sphere bundle of the tangent bundle of Sn#1. The (inte-

gral cohomology) SSS for the fibration Sn#2 !% Vn,2"% Sn#1 (where

$(v1, v2) = v1 and "(w) = (e1, (0, w)) with e1 = (1, 0, . . . , 0)) starts as

(9) Ep,q2 =

"Z, (p, q) " {(0, 0), (n! 1, 0), (0, n! 2), (n! 1, n! 2)};0, otherwise;

and the only possibly nonzero di!erential is multiplication by the Eulercharacteristic of Sn#1 (see for instance [21, pages 153–154]). At anyrate, the only possibilities for a nonzero cohomology group Hq(Vn,2) areZ2 or Z. In the former case, any automorphism must be the identity.So the real task is to determine the action of the three elements inTheorem 3.5 on a cohomology group Hq(Vn,2) = Z.

Proof of Theorem 3.5. The fact that #! = #!1 = #!2 follows by observ-ing that the product of any two of the elements #, #1, and #2 lies inthe path connected group SO(2), and therefore determines an automor-phism Vn,2 % Vn,2 which is homotopic to the identity.

The analysis of the second assertion of Theorem 3.5 depends on theparity of n.

Case with n even, n > 2. The SSS (9) collapses, giving that H!(Vn,2)is an exterior algebra (over Z) on a pair of generators xn#2 and xn#1

(indices denote dimensions). The spectral sequence also gives that xn#2


maps under !! to the generator in Sn"2, whereas xn"1 is the imageunder "! of the generator in Sn"1. Now, the (obviously) commutativediagram

Sn"2

! !! !

"antipodal map

"2 "

Sn"2

Vn,2Vn,2

Sn"1

####$#

%%%%& #

implies that #!2 (and therefore #!1 and #!) is the identity on Hn"1(Vn,2),and that #!2 (and therefore #!1 and #!) act by multiplication by !1 onHn"2(Vn,2). The multiplicative structure then implies that the lastassertion holds also on H2n"3(Vn,2).

Case with n odd, n > 2. The description in (9) of the start ofthe SSS implies that the only nonzero cohomology groups of Vn,2 areHn"1(Vn,2) = Z2 and H i(Vn,2) = Z for i = 0, 2n ! 3. Thus, we onlyneed to make sure that

(10) #! : H2n"3(Vn,2) " H2n"3(Vn,2) is the identity morphism.

Choose generators x # Hn"1(Vn,2), y # H2n"3(Vn,2), and z # H2(CP#),and let Vn,SO(2) " CP# classify the circle fibration (5). Thus, the E2-term of the SSS for the fibration

(11) Vn,2 " Vn,SO(2) " CP#

takes the simple form

Z Z Z Z Z Z Z Z Z Z

• • • • • • • • • •

Z Z Z Z Z Z Z Z Z Z

1 z z2 z3 za!1 za za+1 zn!2 zn!1 zn . . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

x

y

where n = 2a + 1, and a bullet represents a copy of Z2. The proof ofProposition 3.6 below gives two rounds of di!erentials, both originating


on the top horizontal line; the element 2y is a cycle in the first round ofdi!erentials, but determines the second round of di!erentials by

(12) d2n!2(2y) = zn!1.

The key ingredient comes from the observation that ! and the involution" : Vn,SO(2) ! Vn,SO(2) that reverses orientation of an oriented 2-planefit into the pull-back diagram

(13)

Vn,2

! !

"!

"

c

"

Vn,2

Vn,SO(2)Vn,SO(2)

CP" CP""! !

where c stands for conjugation. [Indeed, thinking of Vn,SO(2) ! CP"

as an inclusion, " is the restriction of c, and ! becomes the equivalenceinduced on (selected) fibers.] Of course c#(z) = "z in H2(CP"), sothat

(14) c#(zn!1) = zn!1

(recall n is odd). Thus, in terms of the map of spectral sequencesdetermined by (13), conditions (12) and (14) force the relation !#(2y) =2y. This gives (10).

The proof of (10) we just gave (for odd n) can be simplified byworking over the rationals (see Remark 3.8 in the next paragraph). Wehave chosen the spectral sequence analysis of (11) since it leads us toProposition 3.6.

Remark 3.8. It is well known that whenever a finite group H actsfreely on a space X, with Y = X/H, the rational cohomology of Y mapsisomorphically onto theH-invariant elements in the rational cohomologyof X (see for instance [17, Proposition 3G.1]). We apply this fact to the8-fold covering projection # : Vn,2 ! Vn,D8 . Since the only nontrivialgroups Hq(Vn,2;Q) are Q for q = 0, 2n" 3 (this is where we use that nis odd), we get that the rational cohomology of Vn,D8 is Q in dimension0, vanishes in positive dimensions below 2n" 3, and is either Q or 0 in


the top dimension 2n! 3. But Vn,D8 is a manifold of odd dimension, soits Euler characteristic is zero; this forces the top rational cohomologyto be Q. Thus, every element in D8 acts as the identity on the toprational (and therefore integral) cohomology group of Vn,2. This givesin particular (10), the real content of Theorem 3.5 for an odd n.

As in the notation introduced right after (10), let z " H2(CP!) bea generator so that the element !z " H2(Vn,SO(2)) in Proposition 3.6 isthe image of z under the projection map in (11).

Proof of Proposition 3.6. The E2-term of the SSS for (11) has been in-dicated in the proof of Theorem 3.5. In that picture, the horizontalx-line consists of permanent cycles; indeed, there is no nontrivial tar-get in a Z group for a di!erential originating at a Z2 group. Sincedim(Vn,SO(2)) = 2n ! 4, the term xza must be killed by a di!erential,and the only way this can happen is by means of dn"1(y) = xza. Bymultiplicativity, this settles a whole family of di!erentials killing o! theelements xzi with i # a. Note that this still leaves groups 2 · Z in they-line (rather, the 2y-line). Just as before, dimensionality forces thedi!erential (12), and multiplicativity determines a corresponding fam-ily of di!erentials. What remains in the SSS after these two roundsof di!erentials—depicted below—consists of permanent cycles, so thespectral sequence collapses from this point on.

Z Z Z Z Z Z Z Z

• • • • •

1 z z2 z3 za!1 za za+1 zn!2. . .. . .

. . .x

Finally, we note that all possible extensions are nontrivial. Indeed,orientability of Vn,SO(2) gives H2n"4(Vn,SO(2)) = Z, which implies anontrivial extension involving xza"1 and zn"2. Since multiplication byz is monic in total dimensions less that 2n ! 4 of the E!-term, the5-Lemma (applied recursively) shows that the same assertion is truein H#(Vn,SO(2)). This forces the corresponding nontrivial extensions indegrees lower than 2n ! 4: an element of order 2 in low dimensionswould produce, after multiplication by z, a corresponding element oforder 2 in the top dimension. The proposition follows.


Lai’s description of the ring H!(V2a,SO(2)) given in Remark 3.7 canbe used to understand the full patter of di!erentials and extensions inthe SSS of (11) for n = 2a . Due to space limitations, details are notgiven here—but they are discussed in Remark 3.10 of the preliminaryversion [13] of this paper.

We close this section with an argument that explains, in a geometricway, the switch in parity of n when comparing the orientability proper-ties of Vn,O(2) to those of Vn,D8 . Let ! stand for the projection map inthe smooth fiber bundle (5). The tangent bundle Tn,2 to Vn.2 decom-poses as the Whitney sum

Tn,2!= !!(Tn,SO(2))" "

where Tn,SO(2) is the tangent bundle to Vn,SO(2), and " is the 1–dimensional bundle of tangents to the fibers—a trivial bundle since wehave the nowhere vanishing vector field obtained by di!erentiating thefree action of S1 on Vn.2. Note that # : Vn,2 # Vn,2 reverses orienta-tion on all fibers and so reverses a given orientation of ". Hence, #preserves a chosen orientation of Tn,2 precisely when the involution $in (13) reverses a chosen orientation of Tn,SO(2). But, as explained inthe proof of Proposition 3.2, Vn,D8 is orientable precisely when # isorientation-preserving. Likewise, Vn,O(2) is orientable precisely when $is orientation-preserving.

4 Torsion linking form and Theorems 1.1 and1.2

In this short section we outline an argument, based on the classicaltorsion linking form, that allows us to compute the cohomology groupsdescribed by Theorems 1.1 and 1.2 in all but three critical dimensions.The totality of dimensions (together with the proofs of Propositions 1.7–1.9) is considered in the next three sections—the first two of whichrepresent the bulk of spectral sequence computations in this paper.

For a space X let THi(X;A) (respectively, TH i(X;A)) denote thetorsion subgroup of the ith homology (respectively, cohomology) groupofX with (possibly twisted) coe"cients A. As usual, omission of A fromthe notation indicates that a simple system of Z-coe"cients is used. Weare interested in the twisted coe"cients !Z arising from the orientation


character of a closed m-manifold X = M for, in such a case, there arenon-singular pairings

(15) TH i(M)! THj(M ; !Z) " Q/Z

(for i+ j = m+1), the so-called torsion linking forms, constructed fromthe UCT and Poincare duality. Although (15) seems to be best knownfor an orientable M (see for instance [27, pages 16–17 and 58–59]), theconstruction works just as well in a non-orientable setting. We brieflyrecall the details (in cohomological terms) for completeness.

Start by observing that for a finitely generated abelian group H =F #T with F free abelian and T a finite group, the group Ext1(H,Z) $=Ext1(T,Z) is canonically isomorphic to Hom(T,Q/Z), the Pontryagindual of T (verify this by using the exact sequence 0 " Z " Q " Q/Z "0, and noting that Q is injective while Hom(T,Q) = 0). In particular,the canonical isomorphism TH i(M) $= Ext1(THi!1(M),Z) coming fromthe UCT yields a non-singular pairing TH i(M) ! THi!1(M) " Q/Z.The form in (15) then follows by using Poincare duality (in its not neces-sarily orientable version, see [17, Theorem 3H.6] or [24, Theorem 4.51]).As explained by Barden in [2, Section 0.7] (in the orientable case), theresulting pairing can be interpreted geometrically as the classical torsionlinking number ([19, 25, 29]).

Recall the group G and orbit space Em in Notation 1.5. We nextindicate how the isomorphisms

(16) TH i(M) $= THj(M ; !Z), i+ j = 2m,

coming from (15) for M = Em can be used for computing most of theintegral cohomology groups of F (Pm, 2) and B(Pm, 2).

Since Vm+1,2 is (m%2)-connected3, the map in (2) is (m%1)-connected.Therefore it induces an isomorphism (respectively, monomorphism) incohomology with any—possibly twisted, in view of [30, Theorem 6.4.3"]—coe!cients in dimensions i & m%2 (respectively, i = m%1). Togetherwith Corollary 2.4 and Lemmas 1.6 and 2.8, this leads to the explicitdescription of the groups in Theorems 1.1 and 1.2 in dimensions atmost m%2. The corresponding groups in dimensions at least m+2 canthen be obtained from the isomorphisms (16) and the full description in

3Low-dimensional cases with m ! 3 are given special attention in Example 5.1,Remark 5.4, and (32) in the following sections.


Section 2 of the twisted and untwisted cohomology groups of BG. Notethat the last step requires knowing that, when Em is non-orientable (asdetermined in Proposition 3.2 and Remark 3.3), the twisted coe!cients!Z agree with those Z! used in Theorem 2.3. But such a requirement isa direct consequence of Theorem 3.5. Since the torsion-free subgroupsof H!(Em) are easily identifiable from a quick glance at the E2-termof the CLSS for the G-action on Vm+1,2, only the torsion subgroups inTheorems 1.1 and 1.2 in dimensions

(17) m! 1, m, and m+ 1

are lacking description in this argument.

A deeper analysis of the CLSS of the G-action on Vm+1,2 (workedout in Sections 5 and 6 for G = D8, and discussed briefly in Section 8 forG = Z2"Z2) will give us (among other things) a detailed description ofthe three missing cases in (17) except for the (m+1)-dimensional groupwhen G = D8 and m # 3 mod 4. Note that this apparently singularcase cannot be handled directly with the torsion linking form argumentin the previous paragraph because the connectivity of Vm+1,2 only givesthe injectivity, but not the surjectivity, of the first map in the composite

(18) Hm"1(BD8;Z!)p!!$ Hm"1(B(Pm, 2);Z!) %= Hm+1(B(Pm, 2)).

To overcome the problem, in Section 6 we perform a direct calcula-tion in the first two pages of the Bockstein spectral sequence (BSS)of B(P4a+3, 2) to prove that (18) is indeed an isomorphism for m #3 mod 4—therefore completing the proof of Theorems 1.1 and 1.2.

& = 2 3 4 5 6 7 8 9 10 11 12 13 14

H!(E2,D8 ) #2$

H!(E4,D8 ) #2$ #1$ {2} #1$ #2$

H!(E6,D8 ) #2$ #1$ {2} #2$ #4$ #2$ {2} #1$ #2$

H!(E8,D8 ) #2$ #1$ {2} #2$ #4$ #3$ {4} #3$ #4$ #2$ {2} #1$ #2$

Table 1: H!(Em,D8)%= H!(B(Pm, 2)) for m = 2, 4, 6, and 8

The isomorphisms in (16) yield a (twisted, in the non-orientablecase) symmetry for the torsion groups ofH!(Em). This is illustrated (for


G = D8 and in the orientable case) in Table 1 following the conventionsset in the very first paragraph of the paper.

5 Case of B(Pm, 2) for m !" 3 mod 4

This section and the next one contain a careful study of the CLSS ofthe D8-action on Vm+1,2 described in Definition 1.4; the corresponding(much simpler) analysis for the restricted (Z2 # Z2)-action is outlinedin Section 8. The CLSS approach will yield, in addition, direct proofsof Propositions 1.7–1.9. The reader is assumed to be familiar with theproperties of the CLSS of a regular covering space, complete details ofwhich first appeared in [4].

We start with the less involved situation of an even m and, as awarm-up, we consider first the case m = 2.

Example 5.1. Lemmas 1.6 and 2.1, Corollary 2.4, and Theorem 3.5imply that, in total dimensions at most dim(V3,D8) = 3, the (integralcohomology) CLSS for the D8-action on V3,2 starts as

Z1 2 3

1

2

3

$2% $1%

$1% $2%

Z

The only possible nontrivial di!erential in this range is d 0,23 : E 0,2

2 &E 3,0

2 , which must be an isomorphism in view of the second assertionin Proposition 3.2. This yields the P2-case in Theorem 1.2 and Propo-sitions 1.9 and 1.7 (with G = D8 in the latter one). As indicated inTable 1, the symmetry isomorphisms are invisible in the current situ-ation. It is worth noticing that the d3-di!erential originating at node(1, 2) must be injective. This observation will be the basis in our argu-ment for the general situation, where 2-rank considerations will be thecatalyst. Here and in what follows, by the 2-rank (or simply rank) of afinite abelian 2-group H we mean the rank (F2-dimension) of H ' F2.

Proof of Proposition 1.7 for G=D8, and of Proposition 1.9, both foreven m ( 4. The assertion in Proposition 1.7 for


• i ! 2m follows from Lemma 1.6 and the fact that dim(Vm+1,2) =2m" 1, and for

• i = 2m " 1 follows from the fact that H2m!1(BD8) is a tor-sion group (Corollary 2.4) while H2m!1(B(Pm, 2)) = Z (Proposi-tion 3.2).

We work with the (integral cohomology) CLSS for the D8-action onVm+1,2 in order to prove Proposition 1.9 and the assertions in Proposi-tion 1.7 for i < 2m" 1.

In view of Theorem 3.5, the spectral sequence has a simple systemof coe!cients and, from the description of H"(Vm+1,2) in the proof ofTheorem 3.5, it is concentrated in the three horizontal lines with q =0,m, 2m " 1. We can focus on the lines with q = 0,m in view of therange under current consideration. At the start of the CLSS there is acopy of

• H"(BD8) (described by Corollary 2.4) at the line with q = 0;

• H"(BD8,F2) (described by Lemma 2.1) at the line with q = m.

Note that the assertion in Proposition 1.7 for i < m is an obviousconsequence of the above description of the E2-term of the CLSS. Thecase i = m will follow once we show that the “first” potentially nontrivialdi"erential d0,m

m+1 : E0,m2 # Em+1,0

2 is injective. More generally, we showin the paragraph following (22) below that all di"erentials

(19) dm!!!1,mm+1 : Em!!!1,m

2 # E2m!!,02 with 0 < ! < m are injective.

From this, the assertion in Proposition 1.7 for m < i < 2m " 1 followsat once.

The information we need about di"erentials is forced by the “size”of their domains and codomains. For instance, since H2m!1(B(Pm, 2))is torsion-free, all of E2m!1,0

2 = H2m!1(BD8) = $m" 1% must be killedby di"erentials. But the only possibly nontrivial di"erential landingin E2m!1,0

2 is the one in (19) with ! = 1. The resulting surjective

dm!2,mm+1 map must be an isomorphism since its domain, Em!2,m

2 =Hm!2(BD8;F2) = $m" 1%, is isomorphic to its codomain.

The extra input we need in order to deal with the rest of the di"er-entials in (19) comes from the short exact sequences

(20) 0 # Coker(2i) # H i(B(Pm, 2);F2) # Ker(2i+1) # 0


obtained from the Bockstein long exact sequence

· · ·! H i(B(Pm, 2);F2)!i! H i(B(Pm, 2))

2i! H i(B(Pm, 2))"i! H i!1(B(Pm, 2);F2)! · · · .

From the E2-term of the spectral sequence we easily see that

H1(B(Pm, 2)) = 0

and thatH i(B(Pm, 2))

is a finite 2-torsion group for 1 < i < 2m " 1; let ri denote its 2-rank.Then Ker(2i) #= Coker(2i) #= $ri%, so that (20), Corollary 2.2, and aneasy induction (grounded by the fact that Ker(22m!1) = 0, in view ofthe second assertion in Proposition 3.2) yield

(21) r2m!# =

!a+ 1, ! = 2a;

a, ! = 2a+ 1;

for 2 & ! & m" 1. Under these conditions, the !-th di!erential in (19)takes the form

(22) $m" !%=Hm!#!1(BD8;F2)' H2m!#(BD8)

where

H2m!#(BD8)=

"#$

#%

&m" #

2

', ! ( 0 mod 4;(

m" #!22

), ! ( 2 mod 4;(

m" #+12

), otherwise.

But the cokernel of this map, which is a subgroup of H2m!#(B(Pm, 2)),must have 2-rank at most r2m!#. An easy counting argument (using theright exactness of the tensor product) shows that this is possible onlywith an injective di!erential (22) which, in the case of ! ( 0 mod 4,yields an injective map even after tensoring4 with Z2.

Note that, in total dimensions at most 2m " 2, the Em+2-term ofthe spectral sequence is concentrated on the base line (q = 0). Thus,for 2 & ! & m " 1, H2m!#(B(Pm, 2)) is the cokernel of the di!eren-tial (22)—which yields the surjectivity asserted in Proposition 1.7 in the

4This amounts to the fact that twice the generator of the Z4-summand in (22) isnot in the image of (22)—compare to the proof of Proposition 5.2.


range m < i < 2m! 1. Furthermore the kernel of p! : H2m"!(BD8) "H2m"!(B(Pm, 2)) is the elementary abelian 2-group specified on the lefthand side of (22). In fact, the observation in the second half of the finalassertion in the previous paragraph proves Proposition 1.9. !

As indicated in the last paragraph of the previous proof, for 2 #! # m ! 1 the CLSS analysis identifies the group H2m"!(B(Pm, 2)) asthe cokernel of (22). Thus, the following algebraic calculation of thesegroups not only gives us an alternative approach to that using the non-singularity of the torsion linking form, but it also allows us to recover(for m even and G = D8) the three missing cases in (17)—thereforecompleting the proof of the Peven-case of Theorem 1.2.

Proposition 5.2. For 2 # ! # m ! 1, the cokernel of the di!eren-tial (22) is isomorphic to

H2m"!(B(Pm, 2)) =

!"#

"$

%!2

&, ! $ 0 mod 4;'

!2 + 1

(, ! $ 2 mod 4;'

!"12

(, otherwise.

Proof. Cases with ! %$ 0 mod 4 follow from a simple count, so we onlyo!er an argument for ! $ 0 mod 4. Consider the diagram with exactrows

0 ! &m! !' ! %m! !

2

& ! H2m"!(B(Pm, 2)) ! 0

0 ! &m! !' ! 'm! !

2 + 1( ! '

!2 + 1

( ! 0

"

!""

!"where the top horizontal monomorphism is (22), and where the middlegroup on the bottom is included in the top one as the elements annihi-lated by multiplication by 2. The lower right group is & !2+1' by a simplecounting. The snake lemma shows that the right-hand-side vertical mapis injective with cokernel Z2; the resulting extension is nontrivial in viewof (21).

Example 5.3. For m even, [3, Theorem 1.4 (D)] identifies three explicitelements in the kernel of p! : H i(BD8) " H i(B(Pm, 2)): one for each ofi = m+2, i = m+3, and i = m+4. In particular, this produces at mostfour basis elements in the ideal Ker(p!) in dimensions at most m + 4.


However we have just seen that, for m+ 1 ! i ! 2m" 1, the kernel ofp! : H i(BD8) # H i(B(Pm, 2)) is an F2-vector space of dimension i"m.This means that through dimensions at most m + 4 (and with m > 4)there are at least six more basis elements remaining to be identified inKer(p!).

We next turn to the case when m is odd (a hypothesis in forcethroughout the rest of the section) assuming, from Lemma 5.5 on, thatm $ 1 mod 4.

Remark 5.4. Since the P1-case in Proposition 1.9 and Theorems 1.2and 1.8 is elementary (in view of Remark 1.3 and Corollary 2.4), we willimplicitly assume m %= 1.

The CLSS of the D8-action on Vm+1,2 now has a few extra complica-tions that turn the analysis of di!erentials into a harder task. To beginwith, we find a twisted system of local coe"cients (Theorem 3.5). As aZ[D8]-module, Hq(Vm+1,2) is:

• Z for q = 0,m;• Z! for q = m" 1, 2m" 1;• the zero module otherwise.

Thus, in total dimensions at most 2m " 2 the CLSS is concentratedon the three horizontal lines with q = 0,m " 1,m. [This is in fact thecase in total dimensions at most 2m " 1, since H0(BD8;Z!) = 0; thisobservation is not relevant for the actual group H2m"1(B(Pm, 2)) =Z2—given in the second assertion in Proposition 3.2—, but it will berelevant for the claimed surjectivity of the map p! : H2m"1(BD8) #H2m"1(B(Pm, 2)).] In more detail, at the start of the CLSS we havea copy of H!(BD8) at q = 0,m, and a copy of H!(BD8;Z!) at q =m " 1. It is the extra horizontal line at q = m " 1 (not present foran even m) that leads to potential d2-di!erentials—from the (q = m)-line to the (q = m" 1)-line. Sorting these di!erentials out is the maindi"culty (which we have been able to overcome only for m $ 1 mod4). Throughout the remainder of the section we work in terms of thisspectral sequence, making free use of the description of its E2-termcoming from Corollaries 2.4 and 2.5, as well as of its H!(BD8)-modulestructure. Note that the latter property implies that much of the globalstructure of the spectral sequence is dictated by di!erentials on the threeelements


• xm ! E0,m2 = H0(BD8;Hm(Vm+1,2)) = H0(BD8;Z) = Z;

• !1 ! E1,m!12 = H1(BD8;Hm!1(Vm+1,2)) = H1(BD8;Z!) = Z2;

• !2 ! E2,m!12 = H2(BD8;Hm!1(Vm+1,2)) = H2(BD8;Z!) = Z4;

each of which is a generator of the indicated group (notation is inspiredby that in Theorem 2.3 and in the proof of Theorem 3.5—for even n).

Lemma 5.5. For m " 1 mod 4 and m # 5, the nontrivial d2-di!erentials are given by d4i,m

2 ("i4xm) = 2"i4!2 for i # 0.

Proof. The only potentially nontrivial d2-di!erentials originate at the(q = m)-line and, in view of the module structure, all we need to showis that

(23) d2 : E0,m2 $ E2,m!1

2 has d2(xm) = 2!2

(here and in what follows we omit superscripts of di!erentials).

Let m = 4a + 1. Since H2m!1(B(Pm, 2)) = %1&, most of the ele-ments in E2m!1,0

2 = %4a& must be wiped out by di!erentials. The only

di!erentials landing in a E2m!1,0r (that originate at a nonzero group)

are

(24) dm : Em!1,m!1m $ E2m!1,0

m and dm+1 : Em!2,mm+1 $ E2m!1,0

m+1 .

But Em!1,m!12 = %2a& and Em!2,m

2 = %2a' 1&, so that rank considera-tions imply

(25) Em!2,m2 = Em!2,m

m+1 ,

with the two di!erentials in (24) injective. In particular we get that

(26) H2m!1(B(Pm, 2)) = %1& comes from E2m!1,0" = %1&.

Furthermore, (25) and the H#(BD8)-module structure in the spectralsequence imply that the di!erential in (23) cannot be surjective.

It remains to show that the di!erential in (23) is nonzero. We shallobtain a contradiction by assuming that d2(xm) = 0, so that everyelement in the (q = m)-line is a d2-cycle. Since H2m(B(Pm, 2)) = 0, allof E2m,0

2 = %4a + 2& must be wiped out by di!erentials, and under thecurrent hypothesis the only possible such di!erentials would be

dm : Em,m!1m = Em,m!1

2 = %2a+ 1& $ E2m,0m = E2m,0

2


anddm+1 : E

m!1,mm+1 = Em!1,m

2 = !2a" # Z4 $ E2m,0m+1

—indeed, E0,2m!12 = H0(BD8;Z!) = 0. Thus, the former di!erential

would have to be injective while the latter one would have to be surjec-tive with a Z2 kernel. But there are no further di!erentials that couldkill the resulting Em!1,m

m+2 = !1", in contradiction to (26).

Remark 5.6. In the preceding proof we made crucial use of theH"(BD8)-module structure in the spectral sequence in order to handled2-di!erentials. We show next that, just as in the proof of Proposi-tion 1.7 for G = D8, many of the properties of all higher di!erentials inthe case m % 1 mod 4 follow from the “size” of the resulting E3-term.

Proof of Theorem 1.8 for G = D8, and of Proposition 1.9, both form%1 mod 4. The d2 di!erentials in Lemma 5.5 replace, by a Z2-group,every instance of a Z4-group in the (q = m&1) and (q = m)-lines of theE2-term. This describes the E3-term, the starting stage of the CLSSin the following considerations (note that the E3-term agrees with theEm-term). With this information the idea of the proof is formally thesame as that in the case of an even m, namely: a little input from theBockstein long exact sequence for B(Pm, 2) forces the injectivity of allrelevant higher di!erentials (we give the explicit details for the reader’sbenefit).

Let m = 4a + 1 (recall we are assuming a ' 1). The crux of thematter is showing that the di!erentials

(27) dm : Em!",m!13 $ E2m!",0

3 with ! = 0, 1, 2, . . . ,m

and

(28) dm+1 : Em!"!1,m3 $ E2m!",0

m+1 with ! = 0, 1, 2, . . . ,m& 1

are injective and never hit twice the generator of a Z4-group. Thisassertion has already been shown for ! = 1 in the paragraph contain-ing (24). Likewise, the assertion for ! = 0 follows from (26) with thesame counting argument as the one used in the final paragraph of theproof of Lemma 5.5. Furthermore the case ! = m in (27) is obvious sinceE0,m!1

3 = H0(BD8;Z!) = 0. However, since E0,m3 = H0(BD8) = Z and

Em+1,03 = Hm+1(BD8) = !2a+ 2", the injectivity assertion needs to be

suitably interpreted for ! = m& 1 in (28); indeed, we will prove that

(29) dm+1 : E0,m3 $ Em+1,0

m+1


yields an injective map after tensoring with Z2.From the E3-term of the spectral sequence we easily see that

Hm(B(Pm, 2))

is the direct sum of a copy of Z and a finite 2-torsion group, whileH i(B(Pm, 2)) is a finite 2-torsion group for i != 0,m. We consider theanalogue of (20), the short exact sequences

(30) 0 " Coker(2i) " H i(B(Pm, 2);F2) " Ker(2i+1) " 0,

working here and below in the range m + 1 # i # 2m $ 2. Let ridenote the 2-rank of (the torsion subgroup of) H i(B(Pm, 2)), so thatKer(2i) %= Coker(2i) %= &ri'. Then Corollary 2.2, (30), and an easyinduction (grounded by the fact that Ker(22m!1) = &1', which in turncomes from the second assertion in Proposition 3.2) yield that

(31) r2m!! is the integral part of !+12 for 2 # ! # m$ 1.

Now, in the range of (31), Lemma 5.5 and Corollaries 2.4 and 2.5give

Em!!,m!13 =

!"2a+ 1$ !

2

#, ! even;"

2a$ !!12

#, ! odd;

Em!!!1,m3 =

$%&

%'

Z , ! = m$ 1;"2a+ 1$ !

2

#, ! even, ! < m$ 1;"

2a$ !+12

#, ! odd;

E2m!!,03 =

$%&

%'

"4a+ 2$ !

2

#, ! ( 0 mod 4;(

4a+ 1$ !2

)! ( 2 mod 4;"

4a$ !!12

#, otherwise;

and since E2m!!,0m+2 has 2-rank at most r2m!! (indeed, E

2m!!,0m+2 = E2m!!,0

"which is a subgroup of H2m!!(B(Pm, 2))), an easy counting argument(using, as in the case of an even m, the right exactness of the ten-sor product) gives that the di!erentials in (27) and (28) must yield aninjective map after tensoring with Z2. In particular they

(a) must be injective on the nose, except for the case discussed in (29);

(b) cannot hit twice the generator of a Z4-summand.


The already observed equalities E0,2m!12 = H0(BD8;Z!) = 0 to-

gether with (a) above imply that, in total dimensions t with t ! 2m" 1and t #= m, the Em+2-term of the spectral sequence is concentrated onthe base line (q = 0), while at higher lines (q > 0) the spectral sequenceonly has a Z-group—at node (0,m). This situation yields Theorem 1.8,while (b) above yields Proposition 1.9. !

A direct calculation (left to the reader) using the proved behaviorof the di!erentials in (27) and (28)—and using (twice) the analogue ofProposition 5.2 when ! $ 2 mod 4—gives

H2m!"(B(Pm, 2)) =

!"#

"$

%"2

&, ! $ 0 mod 4;'

"2 " 1

(, ! $ 2 mod 4;%

"+12

&, otherwise;

for 2 ! ! ! m " 1. Thus, as the reader can easily check using Corol-laries 2.4 and 2.5, instead of the symmetry isomorphisms exemplifiedin Table 1, the cohomology groups of B(Pm, 2) are now formed (aspredicted by the isomorphisms (16) of the previous section) by a combi-nation of H"(BD8) and H"(BD8;Z!)—in the lower and upper halves,respectively. Once again, the CLSS analysis not only o!ers an alter-native to the (torsion linking form) arguments in the previous section,but it allows us to recover, under the present hypotheses, the torsionsubgroup in the three missing dimensions in (17).

Example 5.7. For m $ 1 mod 4, [3, Theorem 1.4 (D)] identifies twoexplicit elements in the kernel of p" : H i(BD8) % H i(B(Pm, 2)): one foreach of i = m + 1 and i = m + 3. In particular, this produces at mostthree basis elements in the ideal Ker(p") in dimensions at most m+ 3.However it follows from the previous spectral sequence analysis that, form+ 1 ! i ! 2m" 1, the kernel of p" : H i(BD8) % H i(B(Pm, 2)) is anF2-vector space of dimension i "m + ("1)i. This means that throughdimensions at most m+3 (and with m & 5) there are at least four morebasis elements remaining to be identified in Ker(p").

6 Case of B(P4a+3, 2)

We now discuss some aspects of the spectral sequence of the previoussection in the unresolved case m $ 3 mod 4. Although we are unableto describe the pattern of di!erentials for such m, we show that enoughinformation can be collected to not only resolve the three missing cases


in (17), but also to conclude the proof of Theorem 1.8 for G = D8.Unless explicitly stated otherwise, the hypothesis m ! 3 mod 4 will bein force throughout the section.

Remark 6.1. The main problem that has prevented us from fully un-derstanding the spectral sequence of this section comes from the ap-parent fact that the algebraic input coming from the H!(BD8)-modulestructure in the CLSS—the crucial property used in the proof of Lemma5.5—does not give us enough information in order to determine thepattern of d2-di!erentials. New geometric insights seem to be neededinstead. Although it might be tempting to conjecture the validity ofLemma 5.5 for m ! 3 mod 4, we have not found concrete evidence sup-porting such a possibility. In fact, a careful analysis of the possible be-haviors of the spectral sequence for m = 3 (performed in Section 7) doesnot give even a more aesthetically pleasant reason for leaning towardthe possibility of having a valid Lemma 5.5 in the current congruence.A second problem arose in [13] when we noted that, even if the patternof d2-di!erentials were known for m ! 3 mod 4, there would seem to bea slight indeterminacy either in a few higher di!erentials (if Lemma 5.5holds for m ! 3 mod 4), or in a few possible extensions among the Ep,q

"groups (if Lemma 5.5 actually fails for m ! 3 mod 4). Even though wecannot resolve the current d2-related ambiguity, in [13, Example 6.4]we note that, at least for m = 3, it is possible to overcome the abovementioned problems about higher di!erentials or possible extensions bymaking use of the explicit description of H4(B(P3, 2))—given later inthe section (considerations previous to Remark 6.3) in regard to theclaimed surjectivity of (18); see also [12], where advantage is taken ofthe fact that P3 is a group. The explicit possibilities in the case of P3

are discussed in Section 7.

In the first result of this section, Theorem 1.8 for G = D8 andm ! 3 mod 4, we show that, despite the previous comments, the spec-tral sequence approach can still be used to compute H!(B(P4a+3, 2))just beyond the middle dimension (i.e., just before the first problematicd2-di!erential plays a decisive role). In particular, this computes thecorresponding groups in the first two of the three missing cases in (17).

Proposition 6.2. Let m = 4a+3. The map H i(BD8) " H i(B(Pm, 2))induced by (2) is:

1. an isomorphism for i < m;


2. a monomorphism onto the torsion subgroup of H i(B(Pm, 2)) =!2a+ 1" # Z for i = m;

3. the zero map for 2m$ 1 < i.

Proof. The argument parallels that used in the analysis of the CLSSwhen m % 1 mod 4. Here is the chart of the current E2-term throughtotal dimensions at most m+ 1:

Z1 2

m! 1

m

!2"

!1" Z4

Z

· · · m! 1 m m+ 1

! •

The star at node (m$1, 0) stands for !2a+2"; the bullet at node (m, 0)stands for !2a+1"; the solid box at node (m+1, 0) stands for {2a+2}.In this range there are only three possibly nonzero di!erentials:

• a d2 from node (0,m) to node (2,m$ 1);

• a dm from node (1,m$ 1) to node (m+ 1, 0);

• a dm+1 from node (0,m) to node (m+ 1, 0).

Whatever these d2 and dm+1 are, there will be a resulting E0,m" = Z.

On the other hand, the argument about 2-ranks in (20) and in (30),leading respectively to (21) and (31), now yields that the torsion 2-group Hm+1(B(Pm, 2)) has 2-rank 2a+ 1. Since Em+1,0

" is a subgroupof Hm+1(B(Pm, 2)), this forces the two di!erentials dm and dm+1 aboveto be nonzero, each one with cokernel of 2-rank one less than the 2-rankof its codomain. In fact, dm must have cokernel isomorphic to {2a+1},whereas the cokernel of dm+1 is either {2a} or !2a+1" (Remark 6.3, andespecially [13, Example 6.4], expand on these possibilities). What mat-ters here is the forced injectivity of dm, which implies E1,m!1

" = 0 and,therefore, the second assertion of the proposition—the first assertion isobvious from the CLSS, while the third one is elementary.

We now start work on the only groups in Theorem 1.2 not yet com-puted, namely Hm+1(B(Pm, 2)) for m = 4a + 3. As indicated in the


previous proof, these are torsion 2-groups of 2-rank 2a + 1. Further-more, (18) and Corollary 2.5 show that each such group contains a copyof {2a}, a 2-group of the same 2-rank as that of Hm+1(B(Pm, 2)). Inshowing that the two groups actually agree (thus completing the proof ofTheorem 1.2), a key fact comes from Fred Cohen’s observation (recalledin the paragraph previous to Remark 1.3) that there are no elements oforder 8. For instance,

when m = 3 the two groups must agree since(32)

both are cyclic (i.e., have 2-rank 1).

In order to deal with the situation for positive values of a, Cohen’sobservation is coupled with a few computations in the first two pages ofthe Bockstein spectral sequence (BSS) for B(Pm, 2): we will show thatthere is only one copy of Z4 (the one coming from the subgroup {2a})in the decomposition of Hm+1(B(Pm, 2)) as a sum of cyclic 2-groups—forcing Hm+1(B(Pm, 2)) = {2a}.

Remark 6.3. Before undertaking the BSS calculations (in Proposi-tion 6.4 below), we pause to observe that, unlike the Bockstein inputin all the previous CLSS-related proofs, the use of the BSS does notseem to give quite enough information in order to understand the pat-tern of d2-di!erentials in the current CLSS. Much of the problem liesin being able to decide the actual cokernel of the dm+1-di!erential inthe previous proof and, consequently, understand how the Z4-group inHm+1(B(Pm, 2)) arises in the current CLSS; either entirely at the q = 0line (as in all cases of the previous—and the next—section), or as anontrivial extension in the E! chart. The final section of the paperdiscusses in detail these possibilities in the case m = 3—which shouldbe compared to the much simpler situation in Example 5.1.

Recall from [8, 15] that the mod 2 cohomology ring of B(Pm, 2) ispolynomial on three classes x, x1, and x2, of respective dimensions 1, 1,and 2, subject to the three relations

(I) x2 = xx1;

(II)!

0"i"m2

"m! i

i

#xm#2i1 xi2 = 0;

(III)!

0"i"m+12

"m+ 1! i

i

#xm+1#2i1 xi2 = 0.


Further, the action of Sq1 is determined by (I) and

(33) Sq1x2 = x1x2.

[The following observations—proved in [8, 15], but not needed in thispaper—might help the reader to assimilate the facts just described: Thethree generators x, x1, and x2 are in fact the images under the mappm,D8 in (2) of the corresponding classes at the beginning of Section 2.In turn, the latter generators x1 and x2 come from the Stiefel-Whitneyclasses w1 and w2 in BO(2) under the classifying map for the inclu-sion D8 ! O(2). In these terms, (33) corresponds to the (simplified inBO(2)) Wu formula Sq1(w2) = w1w2. Finally, the two relations (II) and(III) correspond to the fact that the two dual Stiefel-Whitney classeswm and wm+1 in BO(2) generate the kernel of the map induced by theGrassmann inclusion Gm+1,2 ! BO(2).]

Let R stand for the subring generated by x1 and x2, so that there isan additive splitting

(34) H!(B(Pm, 2);F2) = R" x ·R

which is compatible with the action of Sq1 (note that multiplication byx determines an additive isomorphism R #= x ·R).

Proposition 6.4. Let m = 4a+3. With respect to the di!erential Sq1 :

• Hm+1(R; Sq1) = Z2.

• Hm+1(x ·R; Sq1) = 0.

Before proving this result, let us indicate how it can be used toshow that (18) is an isomorphism for m = 4a + 3. As explained in theparagraph containing (32), we must have

(35) 2 ·H4a+4(B(P4a+3, 2)) = $r% with r & 1

and we need to show that r = 1 is in fact the case. Consider theBockstein exact couple

H!(B(P 4a+3, 2)) !2H!(B(P 4a+3, 2))

"""""# !

H!(B(P 4a+3, 2);F2).$$$$$%

"


In the (unravelled) derived exact couple

· · · ! 2 ·H4a+4(B(P 4a+3, 2))2! 2 ·H4a+4(B(P 4a+3, 2)) !

! H4a+4(H!(B(P 4a+3, 2);F2); Sq1) ! 2 ·H4a+5(B(P 4a+3, 2)) ! · · ·

we have 2 ·H4a+5(B(P 4a+3, 2)) = 0 since H4a+5(B(P 4a+3, 2)) = "2a+1#—argued in Section 4 by means of the (twisted) torsion linking form.Together with (35), this implies that the map

(36) "r# = 2 ·H4a+4(B(P 4a+3, 2)) ! H4a+4(H!(B(P 4a+3, 2);F2); Sq1)

in the above exact sequence is an isomorphism. Proposition 6.4 and (34)then imply the required conclusion r = 1.

Proof of Proposition 6.4. Note that every binomial coe!cient in (II)with i $% 0 mod 4 is congruent to zero mod 2. Therefore relation (II)can be rewritten as

(37) x4a+31 =

a/2!

j=1

"a& j

j

#x4(a"2j)+31 x4j2 .

Likewise, every binomial coe!cient in (III) with i % 3 mod 4 is con-gruent to zero mod 2. Then, taking into account (37), relation (III)becomes

x2a+22 = x4a+4

1 +!

i"!

"4a+ 4& i

i

#x4a+4#2i1 xi

2(38)

=

a/2!

j=1

"a& j

j

#x4(a#2j)+41 x4j

2 +!

i"!

"4a+ 4& i

i

#x4a+4#2i1 xi

2

where " is the set of integers i with 1 ' i ' 2a + 1 and i $% 3 mod 4.Using (37) and (38) it is a simple matter to write down a basis for Rand x ·R in dimensions 4a+ 3, 4a+ 4, and 4a+ 5. The information issummarized (under the assumption a > 0, which is no real restrictionin view of (32)) in the following chart, where elements in a column forma basis in the indicated dimension, and where crossed out terms canbe expressed as linear combination of the other ones in view of (37)and (38).


4a+ 3

x4a+31!!!

x4a+11 x2

x4a!11 x2

2

x4a!31 x3

2...

x31x

2a2

x1x2a+12

" 0

" 0

" 0

"

"

"

"

4a+ 4

x4a+41!!!

!!!

x4a+21 x2

x4a1 x2

2

x4a!21 x3

2...

x41x

2a2

x21x

2a+12

x2a+22

4a+ 5

x4a+51!!!

x4a+31 x2###

###

x4a+11 x2

2

x4a!11 x3

2

x4a!31 x4

2...

x31x

2a+12

x1x2a+22! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

xx4a+21

xx4a1 x2

xx4a!21 x2

2...

xx21x

2a2

xx2a+12

""

"

0

" 0

"

"

"

###xx4a+31

xx4a+11 x2

xx4a!11 x2

2

xx4a!31 x3

2...

xx31x

2a2

xx1x2a+12

###

###

xx4a+41

xx4a+21 x2

xx4a1 x2

2

xx4a!21 x3

2...

xx21x

2a+12

xx2a+22

The top and bottom portions of the chart (delimited by the horizontaldotted line) correspond to R and x ·R, respectively. Horizontal arrowsindicate Sq1-images, which are easily computable from (33) and (I):

Sq1(xixi11 xi22 ) = 0

when i+ i1 + i2 is even, while

Sq1(xixi11 xi22 ) = xixi1+1

1 xi22

when i + i1 + i2 is odd—here i ! {0, 1} in view of (I) above. Thereare only two basis elements, in dimensions 4a + 3 and 4a + 4, whoseSq1-images are not indicated in the chart: xx4a+2

1 ! (x · R)4a+3 andx4a+21 x2 ! R4a+4. The second conclusion in the proposition is evident

from the bottom part of the chart—no matter what the Sq1-image ofxx4a+2

1 is. On the other hand, the top portion of the chart implies that,in dimension 4a + 4, Ker(Sq1) and Im(Sq1) are elementary 2-groupswhose ranks satisfy

rk(Ker(Sq1)) = rk(Im(Sq1)) + !

with ! = 1 or ! = 0 (depending on whether or not Sq1(x4a+21 x2)

can be written down as a linear combination of the elements x4a!11 x32,

x4a!51 x52, . . . , and x31x

2a+12 —this of course depends on the actual bino-

mial coe!cients in (37)). But the possibility ! = 0 is ruled out by (35)and (36), forcing ! = 1 and, therefore, the first assertion of this propo-sition.


7 The CLSS for B(P3, 2)

Here is the chart for the E2-term of the spectral sequence for m = 3through filtration degree 13:

0 1 2 3 4 5 6 7 8 9 10 11 12 13Z !2" !1" {2} !2" !4" !3" {4} !4" !6" !5" {6} !6" · · ·0

1

2

3

4

5

!1" {0} !2" !2" !3" {2} !4" !4" !5" {4} !6" !6" !7" · · ·

Z !2" !1" {2} !2" !4" !3" {4} !4" !6" !5" {6} !6" · · ·

!1" {0} !2" !2" !3" {2} !4" !4" !5" {4} !6" !6" !7" · · ·

Since H5(B(P3, 2)) = Z2 (Corollary 3.2), there must be a nontrivialdi!erential landing at node (5, 0). The only such possibility is

(39) d2,23 : E2,23 = Z4

!Im(d0,32 ) # E5,0

3 = Z2 $ Z2

which, up to a change of basis, is the composition of the canonical

projection Z4

!Im(d0,32 ) # Z2 and the canonical inclusion !1 : Z2 "#

Z2$Z2. In particular, as in the conclusion of the second paragraph of theproof of Lemma 5.5, the di!erential d0,32 : E0,3

2 = Z # E2,22 = Z4 cannot

be surjective (otherwise (39) would be the zero map) and, therefore, itsonly options are:

d0,32 is trivial, or(40)

as in (23), d0,32 is twice the canonical projection.(41)

The goal in this example is to discuss how neither of these two op-tions leads to an apparent contradiction in the behavior of the spectralsequence. As a first task we consider the situation where (40) holds,noticing that if d0,32 vanishes, then the H!(BD8)-module structure inthe spectral sequence implies that the whole (q = 3)-line consists of d2-cycles, so the above chart actually gives the E3-term. Furthermore,using again the H!(BD8)-module structure, we note that every d3-di!erential from the (q = 2)-line to the (q = 0)-line would have to repeatvertically as a d3-di!erential from the (q = 5)-line to the (q = 3)-line.


Under these conditions, let us now analyze d3-di!erentials. Theproof of Proposition 6.2 already discusses the d3-di!erential (and itscokernel) from node (1, 2) to node (4, 0). On the other hand, the d3-di!erential from node (2, 2) to node (5, 0) is (39) and has been fullydescribed. Note that the behavior of these two initial d3-di!erentialscan be summarized by remarking that they yield monomorphisms aftertensoring with Z2. We now show, by means of a repeated cycle of threesteps, that this is also the case for all the remaining d3-di!erentials.

Step 1. To begin with, observe that the argument in the final paragraphof the proof of Lemma 5.5 does not lead to a contradiction: it onlyimplies that both di!erentials d3 : E

3,23 ! E6,0

3 and d4 : E2,34 ! E6,0

4

must be injective—this time wiping out E2,3! , E3,2

! , and E6,0! .

Step 2. In view of our discussion of the first nontrivial d3-di!erential,the last assertion in the paragraph following (41) implies that the group"1# at node (1, 5) does not survive to E4; indeed, the di!erential

d3 : E1,53 = "1# ! E4,3

3 = {2}

is injective with cokernel E4,34 = {1}. Such a situation has two conse-

quences. First, that the discussion in the previous step applies wordfor word when the three nodes (2, 3), (3, 2), and (6, 0) are respectivelyreplaced by (3, 3), (4, 2), and (7, 0). Second, that there is no room for anonzero di!erential landing in E5,2

i or E4,3j for i $ 3 and j $ 4 (of course

we have detected the nontrivial di!erential d3 landing at node (4, 3)),so that both d5,23 and d4,34 must be injective (recall H7(B(P3, 2)) = 0).Actually, the only way for this to (algebraically) hold is with an injectived5,23 % Z2.

Step 3. Note that the di!erential d6,23 : E6,23 = {2} ! E9,0

3 = "4# has atleast a Z2-group in its kernel. But the kernel cannot be any larger: theonly nontrivial di!erential landing at node (6, 2) starts at node (2, 5)and, as we already showed, E2,5

4 = Z2. Consequently, d6,23 % Z2 isinjective.

The arguments in these three steps repeat, essentially word for word,in a periodic way, each time accounting for the (& % Z2)-injectivity ofthe next block of four consecutive d3-di!erentials. This leads to thefollowing chart of the resulting E4-term (again through filtration degree13):


0 1 2 3 4 5 6 7 8 9 10 11 12 13Z !2" !1" {1} !1" !2" !1" {1} !1" !2" !1" {1} !1" · · ·0

1

2

3

4

5

!1" !1" !1" · · ·

Z !2" !1" {1} !1" !2" !1" {1} !1" !2" !1" {1} !1" · · ·

!1" !1" !1" · · ·

At this point further di!erentials are forced just from the fact thatH i(B(P3, 2)) = 0 for i # 6. Indeed, all possibly nontrivial di!erentialsdp,q4 must be isomorphisms for p # 2, whereas the H!(BD8)-module

structure implies that the image of the di!erential d0,34 : E0,34 = Z $

E4,04 = {1} is generated by an element of order 4. Thus, the whole

E5-term reduces to the chart:

0 1 2 3 4 5

Z !2" !1" !1" !1"0

1

2

3

!1"

Z

This is also the E"-term for dimensional reasons, and the resultingoutput is compatible with the known structure of H!(B(P3, 2))—notethat the only possibly nontrivial extension (in total degree 4) is actuallynontrivial, in view of [12, Theorem 1.5]. This concludes our discussionof the first task in this section, namely, that (40) leads to no apparentcontradiction in the behavior of the spectral sequence (alternatively:the breakdown in the proof of Lemma 5.5 for m = 3, already observedin Step 1 above, does not seem to be fixable with the present methods).

The second and final task in this section is to explain how, justas (40) does, option (41) leads to no apparent contradiction in the be-havior of the spectral sequence. Thus, for the remainder of the sectionwe assume (41). In particular, the H!(BD8)-module structure in thespectral sequence implies that the conclusion of Lemma 5.5 holds. Then,as explained in the paragraph following Remark 5.6, the resulting E3-


term now takes the form

0 1 2 3 4 5 6 7 8 9 10 11 12 13Z !2" !1" {2} !2" !4" !3" {4} !4" !6" !5" {6} !6" · · ·0

1

2

3

4

5

!1" !1" !2" !2" !3" !3" !4" !4" !5" !5" !6" !6" !7" · · ·

Z !2" !1" !3" !2" !4" !3" !5" !4" !6" !5" !7" !6" · · ·

!1" {0} !2" !2" !3" {2} !4" !4" !5" {4} !6" !6" !7" · · ·

where again only dimensions at most 13 are shown.

At this point it is convenient to observe that the last statement in theparagraph following (41) fails under the current hypothesis. Indeed, thegenerator of E0,3

3 is twice the generator of E0,32 , breaking up the vertical

symmetry of d3-di!erentials holding under (40)—of course, the groupsin the current E3-term already lack the vertical symmetry we had inthe case of (40). In order to deal with such an asymmetric situation weneed to make a di!erential-wise measurement of all the groups involvedin the current E3-term (we will simultaneously analyze the possibilitiesfor the two horizontal families of d3-di!erentials).

To begin with, note that the arguments dealing, in the case of (40),with the two di!erentials E1,2

3 # E4,03 and E2,2

3 # E5,03 apply with-

out change under the current hypothesis to yield that these two dif-ferentials are injective, the former with cokernel E4,0

4 = Z2 $ Z4 (i.e.,both yield injective maps after tensoring with Z2). Note that any othergroup not appearing as the domain or codomain of these two di!er-entials must be eventually wiped out in the spectral sequence, eitherbecause H i(B(P3, 2)) = 0 for i % 6, or else because the already ob-served E5,0

4 = Z2 accounts for all there is in H5(B(P3, 2)) in view ofCorollary 3.2. This observation is the key in the analysis of further dif-ferentials, which uses repeatedly the following three-step argument (thereader is advised to keep handy the previous chart in order to followthe details):

Step 1. The groups Ep,q3 not yet considered and having smallest p+q are

E3,23 and E2,3

3 . Both are isomorphic to !2"; none can be hit a di!erential.

Since E6,03 = !4", we must have injective di!erentials d3 : E

3,23 # E6,0

3

and d4 : E2,34 # E6,0

4 , clearing the E!-term at nodes (2, 3), (3, 2), and


(6, 0). Look now at the groups not yet considered and in the nextsmallest total dimension p+ q. These are E1,5

3 , E3,33 = !1", and E4,2

3 =

!2". Again the last two cannot be hit by a di!erential and, since E7,03 =

!3", the two di!erentials d3 : E4,23 # E7,0

3 and d4 : E3,34 # E7,0

4 must beinjective, now clearing the E!-term at nodes (3, 3), (4, 2), and (7, 0).

Step 2. The only case remaining to consider with p+q = 5 is E1,53 = !1".

We have seen that there is nothing left in the spectral sequence for thisgroup to hit with a d6-di!erential, so it must hit either E4,3

3 = !3" or

E5,23 = !3". Therefore, in these two positions there are 25 elements

that will have to inject into (quotients of) E8,03 = {4}, a group with

cardinality 26. The outcome of this situation is two-fold:

(i) the E!-term is now cleared at positions (1, 5), (4, 3), and (5, 2);

(ii) there is a Z2 group at node (8, 0) that still needs a di!erentialmatchup.

But (i) implies that the only way to kill the element in (ii) is with ad6-di!erential originating at node (2, 5), where we have E2,5

3 = Z4.

Step 3. The above analysis leaves only one element at node (2, 5) stillwithout a di!erential matchup. Since everything at node (8, 0) has beenaccounted for, the element in question at node (2, 5) must be clearedup at either of the stages E3 or E4 with a corresponding nontrivialdi!erential landing at nodes (5, 3) or (6, 2), respectively. But E5,3

3 = !2"while E6,2

3 = !3". Thus, the last di!erential will leave 24 elements thatneed to be mapped injectively by previous di!erentials landing at node(9, 0). Since E9,0

3 = !4", our bookkeeping analysis has now cleared upevery group Ep,q

! with either

• q = 0 and p $ 9;

• q = 2 and p $ 6;

• q = 3 and p $ 5;

• q = 5 and p $ 2.

These three steps now repeat to cover the next four cases of p. Forinstance, one starts by looking at E3,5

3 = !2", whose two basis elementsare forced to inject with di!erentials landing either at node (6, 3) or(7, 2). Since E6,3

3%= E7,2

3%= !4", this leaves 26 elements that must be

mapping into node (10, 0) through injective di!erentials. But E10,03 =


!6", clearing the appropriate nodes—the situation in Step 1. At theend of this three-step inductive analysis we find that there is just theright number of elements, at the right nodes, to match up throughdi!erentials—the opposite of the situation that we successfully exploitedin the previous section to deal with cases where m #$ 3 mod 4.

From the chart we note that d4 : E0,34 = Z % E4,0

4 = Z2 & Z4 is theonly undecided di!erential, and that its cokernel equals H4(B(P3, 2))—since E2,2

! = 0 = E1,3! . The two possibilities (indicated at the end of

the proof of Proposition 6.2) for this cokernel are Z2 and Z4, but [12,Theorem 1.5] implies that the latter option must be the right one underthe present hypothesis (41).

Remark 7.1. The previous paragraph suggests that, if our methodsare to be used to understand the CLSS in the remaining case with m $3 mod 4, then it will be convenient to keep in mind the type of 2e-torsionTheorem 1.2 describes for the integral cohomology of B(P4a+3, 2).

8 Case of F (Pm, 2)

The CLSS analysis in the previous two sections can be applied—withG = Z2 ' Z2 instead of G = D8—in order to study the cohomologygroups of the ordered configuration space F (Pm, 2). The explicit de-tails are similar but much easier than those for unordered configurationspaces, and this time the additive structure of di!erentials can be fullyunderstood for any m. Here we only review the main di!erences, sim-plifications, and results.

For one, there is no 4-torsion to deal with (e.g. the arithmetic Propo-sition 5.2 is not needed); indeed, the role of BD8 in the situation of anunordered configuration space B(Pm, 2) is played by P! ' P! for or-dered configuration spaces F (Pm, 2). Thus, the use of Corollaries 2.4and 2.5 is replaced by the simpler Lemma 2.8. But the most importantsimplification in the calculations relevant to the present section comesfrom the absence of problematic d2-di!erentials, the obstacle that pre-vented us from computing the CLSS of the D8-action on Vm+1,2 form $ 3 mod 4. [This is why in Lemma 2.8 we do not insist on describingH"(P! ' P!;Z!) as a module over H"(P! ' P!)—compare to Re-mark 5.6.] As a result, the integral cohomology CLSS of the (Z2 'Z2)-action on Vm+1,2 can be fully understood, without restriction on m,


by means of the counting arguments used in Section 5, now forcing theinjectivity of all relevant di!erentials from the following two ingredients:

(a) The size and distribution of the groups in the CLSS.

(b) The Z2!Z2 analogue of Proposition 3.2 in Remark 3.3—the inputtriggering the determination of di!erentials.

In particular, when m is odd, the Z2 ! Z2 analogue of Lemma 5.5 doesnot arise and, instead, only the counting argument in the proof followingRemark 5.6 is needed.

We leave it for the reader to supply details of the above CLSS andverify that this leads to Propositions 1.7 and 1.8 in the case G = Z2!Z2,as well as to the computation of all the cohomology groups in Theo-rem 1.1.

Jesus GonzalezDepartamento de Matematicas,Centro de Investigacion y de Estu-dios Avanzados del IPN,Apartado Postal 14-740,07000 Mexico City, [email protected]

Peter LandweberDepartment of Mathematics,Rutgers University,Piscataway, NJ 08854, USA,[email protected]

References

[1] A. Adem and R. J. Milgram, Cohomology of Finite Groups, sec-ond edition. Grundlehren der mathematischenWissenschaften, 309.Springer-Verlag, Berlin, 2004.

[2] D. Barden, “Simply connected five-manifolds”, Ann. of Math. (2)82 (1965) 365–385.

[3] P. V. M. Blagojevic and G. M. Ziegler, “The ideal-valued index fora dihedral group action, and mass partition by two hyperplanes”,Topology Appl. 158 (2011) 1326–1351. A longer preliminary versionis available as arXiv:0704.1943v4 [math.AT].

[4] H. Cartan, “Espaces avec groupes d’operateurs. I: Notionspreliminaires; II: La suite spectrale; applications”, Seminaire HenriCartan, tome 3, exposes 11 (1–11) and 12 (1–10) (1950-1951), bothavailable at http://www.numdam.org.


[5] C. Domınguez, “Cohomology of pairs of points in real projectivespaces and applications”, Ph.D. thesis, Department of Mathemat-ics, Cinvestav, 2011.

[6] C. Domınguez, J. Gonzalez, and Peter S. Landweber, “The inte-gral cohomology of configuration spaces of pairs of points in realprojective spaces”, to appear in Forum Mathematicum.

[7] E. Fadell and S. Husseini, “An ideal-valued cohomological indextheory with applications to Borsuk-Ulam and Bourgin-Yang theo-rems”, Ergod. Th. and Dynam. Sys. 8! (1988) 73-85.

[8] S. Feder, “The reduced symmetric product of projective spacesand the generalized Whitney theorem”, Illinois J. Math. 16 (1972)323–329.

[9] E. M. Feichtner and G. M. Ziegler, “The integral cohomology al-gebras of ordered configuration spaces of spheres”, Doc. Math. 5(2000) 115–139.

[10] E. M. Feichtner and G. M. Ziegler, “On orbit configuration spacesof spheres”, Topology Appl. 118 (2002) 85–102.

[11] Y. Felix and D. Tanre, “The cohomology algebra of unordered con-figuration spaces”, J. London Math. Soc. 72 (2005) 525–544.

[12] J. Gonzalez, “Symmetric topological complexity as the first ob-struction in Goodwillie’s Euclidean embedding tower for real pro-jective spaces”, to appear in Trans. Amer. Math. Soc. (currentlyavailable at arXiv:0911.1116v4 [math.AT]).

[13] J. Gonzalez and P. Landweber, “The integral cohomology groupsof configuration spaces of pairs of points in real projective spaces”,initial version of the present paper available at arXiv:1004.0746v1[math.AT].

[14] B. Grunbaum, “Partitions of mass-distributions and of convex bod-ies by hyperplanes”, Pacific J. Math. 10 (1960) 1257–1261.

[15] D. Handel, “An embedding theorem for real projective spaces”,Topology 7 (1968) 125–130.

[16] D. Handel, “On products in the cohomology of the dihedralgroups”, Tohoku Math. J. (2) 45 (1993) 13–42.


[17] A. Hatcher, Algebraic Topology. Cambridge University Press, Cam-bridge, 2002.

[18] S. Kallel, “Symmetric products, duality and homological dimensionof configuration spaces”, Geom. Topol. Monogr., 13 (2008) 499–527.

[19] M. A. Kervaire and J. W. Milnor, “Groups of homotopy spheres: I”,Ann. of Math. (2) 77 (1963) 504–537.

[20] H. F. Lai, “On the topology of the even-dimensional complexquadrics”, Proc. Amer. Math. Soc. 46 (1974) 419–425.

[21] J. McCleary, A User’s Guide to Spectral Sequences, second edition.Cambridge Studies in Advanced Mathematics, 58. Cambridge Uni-versity Press, Cambridge, 2001.

[22] J. R. Munkres, Elements of Algebraic Topology. Addison-WesleyPublishing Company, Menlo Park, CA, 1984.

[23] V. V. Prasolov, Elements of homology theory. Translated fromthe 2005 Russian original by Olga Sipacheva. Graduate Studiesin Mathematics, 81. AMS, Providence, RI, 2007.

[24] A. Ranicki, Algebraic and Geometric Surgery. Oxford Mathemat-ical Monographs, Oxford Science Publications. Oxford Univer-sity Press, 2002. Electronic version (August 2009) available athttp://www.maths.ed.ac.uk/!aar/books/surgery.pdf.

[25] H. Seifert andW. Threlfall, A Textbook of Topology, translated fromthe German 1934 edition by Michael A. Goldman, with a prefaceby Joan S. Birman. Pure and Applied Mathematics, 89. AcademicPress, Inc. New York-London, 1980.

[26] W. A. Sutherland, “A note on the parallelizability of sphere-bundles over spheres”, J. London Math. Soc. 39 (1964) 55–62.

[27] P. Teichner, Slice Knots: Knot Theory in the 4th Di-mension. Lecture notes by Julia Collins and MarkPowell. Electronic version (October 2009) available athttp://www.maths.ed.ac.uk/!s0681349/#research.

[28] B. Totaro, “Configuration spaces of algebraic varieties”, Topology35 (1996) 1057–1067.


[29] C. T. C. Wall, “Killing the middle homotopy groups of odd dimen-sional manifolds”, Trans. Amer. Math. Soc. 103 (1962) 421–433.

[30] G. W. Whitehead, Elements of Homotopy Theory. Graduate Textsin Mathematics, 61. Springer-Verlag, New York-Berlin, 1978.


An upper bound on the size of irreduciblequadrangulations !

Gloria Aguilar Cruz Francisco Javier Zaragoza Martınez

Abstract

Let S be a closed surface with Euler genus !(S). A quadrangu-lation G of a closed surface S is irreducible if it does not haveany contractible face. Nakamoto and Ota gave a linear upperbound for the number n of vertices of G in terms of !(S). Byextending Nakamoto and Ota’s method we improve their boundto n " 159.5!(S) # 46 for any closed surface S.

2010 Mathematics Subject Classification: 05C10.Keywords and phrases: irreducible quadrangulations, Euler genus.

1 Introduction

The orientable closed surface Mg with genus g is the sphere with ghandles attached. The non-orientable closed surface Ng with genus gis the sphere with g cross-caps attached. The Euler genus of thesesurfaces is !(Mg) = 2g for the orientable surface Mg and !(Ng) = g forthe non-orientable surface Ng.

Let G be a simple graph, that is, a graph without loops or paralleledges. The orientable genus !(G) of G is defined as the least g such

!This work is part of the first author’s Ph.D. thesis to be presented at the Math-ematics Department of CINVESTAV under the supervision of Dr. Isidoro Gitler(CINVESTAV) and Dr. Francisco Javier Zaragoza Martınez (UAM Azcapotzalco).The first author was partly supported by CONACyT Doctoral Scholarship 144571.

otuAdadisrevinUybdetroppusyltrapsawrohtuadnocesehT noma MetropolitanaAzcapotzalco grant 2270314 and by CONACyT-SNI grant 33694. Both authors werepartly supported by Programa Integral de Fortalecimiento Institucional PIFI 3.3 andPIFI 2007.

61

62 G. Aguilar Cruz and F. J. Zaragoza Martınez

that G is embeddable in Mg and the non-orientable genus !(G) of Gis defined as the least g such that G is embeddable in Ng. The Euler

genus !(G) of G is defined to be !(G) = min{2!(G), !(G)}. Note that!(G) = min{!(S) : G is embeddable in S}.

A quadrangulation G of a closed surface S is a graph 2-cell embeddedon S in such a way that all faces of G are quadrangles.

Let G be a quadrangulation of a closed surface S and let abcd bea face of G. We say that abcd is contractible if we can obtain a newquadrangulation by identifying a with c and deleting edges ab and cd,see Figure 1. A quadrangulation G is said to be irreducible if G hasno contractible face. The size of an irreducible quadrangulation can bemeasured in terms of its number of vertices, edges, or faces. By Euler’sformula these are all equivalent and we have chosen to measure the sizein terms of the number n of vertices.

a bb

c dd

a = c

Figure 1: Contracting face abcd.

Nakamoto and Ota [3] gave a linear upper bound for the number nof vertices of an irreducible quadrangulation G in terms of !(S), namelyn ! 186!(S)" 64.

In this paper we prove an upper bound of n ! 159.5!(S)"46 for thesize of our quadrangulations by extending Nakamoto and Ota’s method.

2 Preliminaries

We use the following bound on the Euler genus of a 1" or 2"sum ofgraphs.

Lemma 2.1 (Miller [1]). Let G1 and G2 be two graphs and let G :=G1#G2. If G1 and G2 have at most two common vertices, then !(G) $!(G1) + !(G2).

Irreducible quadrangulations 63

From now on, let S be either Mg or Ng with g ! 1 and let G be anirreducible quadrangulation of S. Let G! be the graph embedded on S

obtained from G by adding a vertex of degree four into each of its facesand joining it to the vertices of the corresponding face, see Figure 2.

v v

G G!

Figure 2: Construction of G!.

For v " V (G) let Hv be the subgraph of G! induced by v, the verticesof the incident faces to v in G and the vertices added to the incidentfaces to v. Let NG(v) be the set of adjacent vertices to v in G.

Nakamoto and Ota [2] proved the following result for degG(v) # 4.

Lemma 2.2. Let G be an irreducible quadrangulation of a closed surface

S and let v be a vertex of G. Then !(Hv) ! 1.

Proof. Since G is irreducible and S is not the sphere, it follows thatG has no vertex of degree less than three. Let v be a vertex of G

of degree d ! 3 and let Wv := v0e0v1e1 . . . v2d"1e2d"1v2d be a closedwalk in G such that v0, v2, . . . , v2d"2 are the neighbors of v in clockwisedirection and v v2iv2i+1v2i+2 is a face of G for i = 0, 1, . . . , d$ 1. SinceG is irreducible every v2i+1 must be adjacent or equal to some v2j withj %= i, i+1. We denote by w2i+1 the new vertex of degree four added tothe face v v2iv2i+1v2i+2 in G. Let vm, vn " Wd, we define dist(vm, vn) :=min{|m$ n|$ 1, 2d$ |m$ n|$ 1}, for m %= n.

Let v!, v" " Wd be vertices such that v! " NG(v), v v""1v"v"+1 "F (G), v! is adjacent or equal to v" and dist (v!, v") > 0 is minimal.Without loss of generality we can assume that " = 1, # = 2k + 2, anddist(v!, v") = 2k.


Since G is irreducible v2k+1 is adjacent or equal to some neighbor ofv, namely v2j , with k+ 1 < j < d because dist(v!, v") = 2k is minimal,see Figure 3.

v0v1v2

w1

v2k

v2k+1

v2k+2

w2k+1

v2jv

Figure 3: Vertex of degree d ! 3.

Therefore we have a subdivision of K3,3 with partition

{v, w2k+1, w1} " {v2k+2, v2k, v2j}.

See Figure 4. !

v0v1v2

w1

v2k

v2k+1

v2k+2

w2k+1v2jv

Figure 4: Subdivision of K3,3.

We say that a set I of vertices of G is face-independent if no twovertices in I are incident to the same face of G. Nakamoto and Otaproved the following result for k = 4 [2].

Lemma 2.3. Let G be an irreducible quadrangulation of a closed surface

S and let k ! 3 be an integer. For each i ! 3 let Vi be the set of vertices


of degree i of G. Then there exists an independent set X ! V3" · · ·"Vk

such that

|X| #k

!

i=3

|Vi|

2i+ 1.

Proof. Let X3 be a maximal face-independent subset of V3. For i =4, . . . , k let Xi be a maximal face-independent subset from

Vi $i!1"

j=3

Ai,j

where Ai,j is the set of vertices of degree i that are incident to a face

which is incident to a vertex in Xj . We claim that X =#k

i=3Xi satisfiesthe required property. Counting the vertices in the incident faces of eachvertex x in Xi (including vertex x) we obtain

(2i+ 1)|Xi| # |Vi|+k

!

j=i+1

|Aj,i|$i!1!

j=3

|Ai,j | for every 3 % i % k.

Therefore

|X| =k

!

i=3

|Xi| #k

!

i=3

|Vi|

2i+ 1+

k!1!

i=3

k!

j=i+1

|Aj,i|

2i+ 1$

k!

i=4

i!1!

j=3

|Ai,j |

2i+ 1.

Observe that Ai,j = & for i % j. Therefore

k!1!

i=3

k!

j=i+1

|Aj,i|

2i+ 1$

k!

i=4

i!1!

j=3

|Ai,j |

2i+ 1=

k!1!

i=3

k!

j=i+1

$

|Aj,i|

2i+ 1$

|Aj,i|

2j + 1

%

# 0

since j > i. We conclude that

|X| #k

!

i=3

|Vi|

2i+ 1. !

3 Main Result

Theorem 3.1. Let G be an irreducible quadrangulation of a closed sur-

face S with n vertices. Then

n % 159.5!(S)$ 46.


Proof. Let m and f be the number of edges and faces of G, respectively.By Euler’s formula

n!m+ f = 2! !(S).

Since G is a quadrangulation we have that 4f = 2m and therefore

4n! 2m = 8! 4!(S).

Since!

i!3 |Vi| = n and!

i!3 i|Vi| = 2m we have that

3n+"

i!3

(1! i)|Vi| = 8! 4!(S).

Let k " 4 be an integer (to be chosen later). By adding and sub-stracting kn = k

!

i!3 |Vi| we obtain

(3! k)n+"

i!3

(k + 1! i)|Vi| = 8! 4!(S).

Thus

(1)k"

i=3

(k + 1! i)|Vi| " (k ! 3)n! 4!(S) + 8.

Let X be an independent set as in Lemma 2.3 and define

Y := {y # V (G)!X|y # NG(x) for some x # X}.

Consider the bipartite graph B with bipartition X and Y , wherexy # E(B) for x # X, y # Y if and only if xy # E(G).

Let X " := {v1, v2, . . . , vr} be a maximal subset of X satisfying thefollowing condition:

#

#

#

#

#

#

$

%

&

'

1#i<j

NB(vi)

(

)

*

$NB(vj)

#

#

#

#

#

#

% 2, for each j = 1, 2, . . . , r.

In other words, for each 2 % j % r, vj has at most two commonneighbors with v1, v2, . . . , vj$1. See Figure 5.

By Lemma 2.1 and Lemma 2.2 we obtain

!

+

r'

i=1

Hvi

,

"r"

i=1

!(Hvi) " r = |X "|.


. . .

. . .

v1 v2 v3 vrX

Y

X !

Y !

Figure 5: The sets X, X !, Y , and Y !.

Since!r

i=1Hvi is a subgraph of G!, it is embeddable in S, thus

!(S) ! !

"

r#

i=1

Hvi

$

,

and it follows that

(2) |X !| " !(S).

Now define Y ! := {y # Y |y # NB(v) for some v # X !}. Let M bethe subgraph of B induced by X $ Y !. Since M is a subgraph of G it isembeddable in S, therefore

|V (M)|% |E(M)|+ |F (M)| ! 2% !(S).

Since M is bipartite each of its faces has at least 4 edges, therefore4|F (M)| " 2|E(M)|. Hence we have

(3) 2|V (M)|% |E(M)| ! 4% 2!(S).

By maximality of X !, each vertex v # X % X ! has at least threeneighbors in Y !. There are at least |Y !| edges between X ! and Y !.Hence

|E(M)| ! 3(|X|% |X !|) + |Y !|.

By replacing |V (M)| = |X| + |Y !| and |E(M)| in inequality (3), weobtain


4! 2!(S) " 2|X|+ 2|Y !|! 3(|X|! |X !|)! |Y !|

" !|X|+ |Y !|+ 3|X !|

" !|X|+ (2k + 3)|X !| since |Y !| " 2k|X !|

" !k

!

i=3

|Vi|

2i+ 1+ (2k + 3)|X !| by Lemma 2.3.

Let nk be the smallest integer such that

nk

2i+ 1# k + 1! i, for every 3 " i " k.

Since (2i+ 1)(k ! i+ 1) has a unique maximum, we take this valueas nk, namely

nk :=

"

#

$

#

%

14 if k = 4

(k + 1)(k + 2)

2if k # 5.

Thus we obtain

4! 2!(S) " !1

nk

k!

i=3

nk

2i+ 1|Vi|+ (2k + 3)|X !|

" !1

nk[(k ! 3)n+ 8! 4!(S)] + (2k + 3)|X !|

and therefore

(4)(k ! 3)n+ 8 + 4nk ! (2k + 3)nk|X !|

4 + 2nk" !(S).

Thus, (2) provides a good bound for !(S) when |X !| is large and (4)provides a good bound when |X !| is small. These two bounds are thesame when their left-hand sides are equal, that is, when

|X !| =(k ! 3)n+ 4nk + 8

(2k + 5)nk + 4.

In particular, from (2) we obtain

(k ! 3)n+ 4nk + 8

(2k + 5)nk + 4" !(S),


that isn ! f(k)!(S)" g(k),

where f(k) =(2k + 5)nk + 4

k " 3and g(k) =

4nk + 8

k " 3. A straightforward

calculation shows that f(k) attains its minimum at k = 5, therefore

n ! 159.5!(S)" 46. !

Observe that for k = 4 we obtain n ! 186!(S) " 64, this is thebound obtained by Nakamoto and Ota [3].

Acknowledgements

We are very grateful for the suggestions, contributions, and bugreports o!ered by Ernesto Lupercio, Isidoro Gitler, and Elıas Micha.

Gloria Aguilar CruzDepartamento de Matematicas,Centro de Investigacion y de Estu-dios Avanzados del IPN,Apartado Postal 14-740,07000 Mexico City, [email protected]

Francisco Javier Zaragoza MartınezDepartamento de Sistemas,Universidad Autonoma Metropoli-tana Unidad Azcapotzalco,Av. San Pablo 180,02200 Mexico City, [email protected]

References

[1] Gary L. Miller, An additivity theorem for the genus of a graph.,J. Combin. Theory Ser. B, 43(1987), 25-47.

[2] Atsuhiro Nakamoto, Triangulations and quadrangulations of sur-

faces, Keio University, (1996), Doctoral Thesis.

[3] Atsuhiro Nakamoto and Katsuhiro Ota, Note on irreducible tri-

angulations of surfaces, J. Graph Theory, 20(1995), 227-233.

Morfismos, Comunicaciones Estudiantiles del Departamento de Matematicas delCINVESTAV, se imprime en el taller de reproduccion del mismo departamento locali-zado en Avenida Instituto Politecnico Nacional 2508, Colonia San Pedro Zacatenco,C.P. 07360, Mexico, D.F. Este numero se termino de imprimir en el mes de febrero de2011. El tiraje en papel opalina importada de 36 kilogramos de 34 ! 25.5 cm. constade 500 ejemplares con pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contenido

A note on distributional equations in discounted risk processes

zelaznoGocehcaP.GsolraCdnaellaVledzednanreHodrareG . . . . . . . . . . . 1

Cohomology groups of configuration spaces of pairs of points in real projectivespaces

rebewdnaLretePdnazelaznoGsuseJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

An upper bound on the size of irreducible quadrangulations

Gloria Aguilar Cruz and Francisco Javier Zaragoza Martınez . . . . . . . . . . . 61

Morfismos, Vol 14, No 2, 2010

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Transcript of Morfismos, Vol 14, No 2, 2010