LOOP STRUCTURES ON HOMOTOPY FIBRES OF SELF MAPS
OF A SPHERE
CARLOS BROTO AND RAN LEVI
1.Introduction
The problem of deciding whether or not a given topological space X is homotopy
equivalent to a loop space is of classical interest in homotopy theory. Moreove*
*r, given
that X is a loop space, one could ask whether the loop space structure is uniqu*
*e.
A celebrated example of this is given in the classical work of Rector [23] wher*
*e it
is shown that the 3sphere S3 admits infinitely many nonequivalent loop struct*
*ures
(see also [21]) and of Dwyer, Miller and Wilkerson [11], where the authors show*
* that
all such loop structures collapse to a single one if the sphere is localized at*
* a prime p.
The first problem is equivalent to the question whether given a space X, there
exist a space BX and a homotopy equivalence of BX to X. The second problem
is equivalent to asking whether the homotopy type of the space BX is uniquely
determined by the requirement that its loop space is homotopy equivalent to X.
In this paper we study a family of spaces of classical interest in homotopy t*
*heory.
Let Sm {d} denote the homotopy fibre of the degree d self map of Sm . Our aim i*
*s to
study the possible loop structures supported by Sm {d} for various values of m *
*and
d. We restrict attention to the case where, d = 2r. Our main theorem follows.
Theorem 1.1. Let m > 1 be an integer. The space Sm {2r} is a loop space if a*
*nd
only if m = 3 and r > 2. Furthermore, for each value of r > 2, there is a uniqu*
*e loop
space structure on S3{2r}.
The answer to the question, whether Sn{d} is a loop space for d odd has been
known for a while. The uniqueness question for d odd is simpler than the case w*
*here
d is a power of 2 and will be handled below. As a consequence we obtain a compl*
*ete
answer to the problems considered here for this family of spaces.
Corollary 1.2. Let m > 1 be an integer. Then the space Sm {d} is a loop space *
*if
and only if
1. d is odd, m = 2n  1 and np  1 for every prime p dividing d or
2. 8d and m = 3.
Moreover, Sm {d} admits at most one loop structure.
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q5*
*2.
Key words and phrases. Classifying spaces, loop spaces, cohomology, Steenrod *
*squares, Bockstein
spectral sequence.
C. Broto is partially supported by DGICYT grant PB940725.
1
2
The techniques by which Theorem 1.1 is proved and the relations with various *
*other
topics may be of independent interest. Our problem turns out to be closely rela*
*ted
to the study of the homotopy type of pcompleted classifying spaces of certain *
*finite
groups. The tools we use are chosen respectively. Indeed we employ Lannes T fu*
*nctor
technology, homology decomposition methods and various other homotopy theoretic
tools. Many of the arguments are reminiscent of techniques, successfully used i*
*n the
study of pcompact groups and pcompleted classifying spaces of finite groups. *
*The
paper is thus a combination of various theoretical elements which produce what *
*we
regard as a torsion analogue of the results of [11].
For r 3 there is a well known model for BS3{2r} given by the 2completed cla*
*s
sifying space for the special linear group SL2(Fq), where q is chosen so that 2*
*r is the
order of the 2Sylow subgroup. Furthermore, it turns out that the homotopy type
of the loop space BSL2(Fq)^2depends only on the mod2 cohomology of BSL2(Fq)
together with its Bockstein spectral sequence. Thus our problem amounts to show
ing that the specified cohomological information determines the homotopy type of
BSL2(Fq)^2. Along the same lines we show that the homotopy type of BP SL2(Fq)^2,
where P SL means the projective special linear group, is determined by its mod2
cohomology.
An interesting aspect of our study involves showing nonrealizability of cert*
*ain
spaces with a given cohomological structure, as an algebra over the Steenrod al*
*gebra
and a specified Bockstein spectral sequence. From a purely algebraic point of v*
*iew the
structures assumed on these spaces are by all means compatible with each other.*
* Thus
we are lead to employ homotopy theoretic constructions to show that the Steenrod
algebra structure and Bockstein spectral sequence cannot exist in the same spac*
*e.
In [4] the authors study the question to what extent does cohomology inform of
the homotopy type of BG, where G is a finite pgroup. One of the examples studi*
*ed
there is of the quaternion groups Q2n. It is shown that for n 5 the homotopy t*
*ype
of BQ2n is determined by its cohomology. The question of whether or not the same
holds for higher values of n is reduced to asking if there is a free action of *
*the group
Q16on a classifying space for S3{2r} for an appropriate r. Thus one obtains a r*
*elation
between the current project and what may appear to be an unrelated problem. The
authors actually expect that the results here will imply that the homotopy type*
* of
BQ2n is determined by its mod2 cohomology for every n. An account of this, whe*
*re
the problem is discussed in a more general setting is to appear elsewhere.
1.1. Reduction of the problem and the known cases. Some observations are
immediate. First notice that the problem we are looking at only makes sense for*
* m 2
as S1{d} is homotopy equivalent to the cyclic group on d elements. Furthermore,*
* for
m 2, it is easy to compute the integral homology of Sm {d} using the Wang long
exact sequence or equivalently the Serre spectral sequence for the fibration de*
*fining
Sm {d}. Indeed the nth homology of Sm {d} is isomorphic to the cyclic group Z=*
*d if
n is nonzero and divisible by m  1 and is trivial otherwise. In particular al*
*l reduced
homology groups of Sm {d} are torsion groups and thus by [3] there is a homotopy
equivalence of Sm {d} to the product of all its pprimary localizations for pri*
*mes p
3
dividing d. It is easy to see that for a prime p the localization (or completi*
*on) of
Sm {d} at p yields a space homotopy equivalent to Sm {pr}, where r is the highe*
*st
power of p dividing d. Hence there is a homotopy equivalence
Yr
' m ji
Sm {d} ____ S {pi }:
i=1
Moreover, by naturality of the localization functors it is clear that Sm {d} is*
* a loop
space if and only if each factor on the right hand side of the above equation i*
*s a
loop space. Our problem is thus reduced to the study of possible loop structure*
*s on
Sm {pj}, where p is a prime number.
The answer to our first question for odd primes is known. Also at p = 2 there
exist partial answers in the literature. We summarize the state of knowledge in*
* the
subject.
The case where p is an odd prime was treated by Cejtin and Kleinerman [6] and*
* a
bit more generally by Kono and Oshita [19]. In that case they show that Sm {pr}*
* is a
loop space if and only if m = 2n  1 and n divides p  1. It is not hard to see*
* (Lemma
2.1) that independently of the loop space structure on Sm {pr} the cohomology o*
*f its
classifying space is given as follows
(1) H*(BSm {pr}; Fp) ~=P [um+1 ] E[xm ];
with an rth order Bockstein connecting the generators. With this observation o*
*ne
is able to conclude that the homotopy type of a space with cohomology as above *
*is
unique up to pcompletion. A proof of this is included as the last section of t*
*he paper.
The case p = 2 is as usual more complicated. However if Sm {2r} is a loop spa*
*ce
then the mod2 cohomology of its classifying space is still given by Equation (*
*1). A
work of Aguade [1] puts strong restrictions on the possible values of m and r. *
* It
follows that if Sm {2r} is a loop space then m = 3 and r > 1. As mentioned abov*
*e,
for r 3 there is a space which gives a classifying space for S3{2r}, the 2com*
*pleted
classifying space BSL2(Fq)^2. The 2Sylow subgroup of BSL2(Fq) is a generalized
quaternion group of order depending on q. Moreover for every r 3 there exists
a q, such that Q2r is the 2Sylow subgroup of SL2(Fq). The mod2 cohomology of
BSL2(Fq) is given by Equation (1) with m = 3 (cf. [15]) and an observation of F.
Cohen [8] is that BSL2(Fq)^2is homotopy equivalent to S3{2r} for the appropriate
r.
The remaining problems are whether S3{4} is a loop space and moreover, whether
the loop space structure on S3{2r} for r 3 and possibly on S3{4} is unique. Th*
*is
is answered in full by Theorem 1.1.
1.2. Notation and Terminology. By the word space we mean a simplicial set or
a topological space of the homotopy type of a CW complex. By pcompletion we
mean the BousfieldKan pcompletion. Throughout the paper we use the phrase
Lannes theory to mean the theory of Lannes T functors and the homotopy theory
4
of mapping spaces out of BV , where V is an elementary abelian pgroup. We shall
assume the reader is familiar with [20] and omit elementary calculation respect*
*ively.
Let K denote the category of unstable algebras over the Steenrod algebra. By
Kfiwe denote the category of unstable algebras together with a Bockstein spectr*
*al
sequence. A precise definition of this category appears in [4]. By H*fi() we*
* mean
modp cohomology as an object of Kfi.
We shall abbreviate the phrases Serre spectral sequence, EilenbergMoore spec*
*tral
sequence and Bockstein spectral sequence by Sss, EMss and Bss respectively. We *
*write
P [xi1; xi2; . .].to denote a polynomial algebra on generators xijand E[xi1; xi*
*2; . .].for
an exterior algebra. The subscript on a generator will normally indicate its de*
*gree,
and may be omitted when no confusion can arise.
1.3. Organization of the paper and Acknowledgements. The paper is orga
nized as follows. In section 2 we give a proof of our main theorem based on sec*
*tions
to follow. Sections 3 and 4 are technical and include cohomological calculation*
*s and
homology decompositions that will be required for proving homotopy uniqueness of
the spaces under consideration. In Sections 5 and 6 we prove that the homotopy
type of BP SL2(Fq)^2, q odd, is determined by its cohomology as an object of Kf*
*i. In
Section 7 we study certain spherical fibrations with a special property. The re*
*sults
of section 7 allow us to prove in Section 8 the incompatibility of some algebra*
*s over
the Steenrod algebra with the given Bss. This is used in the proof of Theorem *
*1.1
to conclude homotopy uniquenes of BS3{2j}, j 3. Section 9 deals with homotopy
uniquenes of BS2n1{pr}, when it exists for p odd. Some background material on
homological algebra is written in an appendix.
We wish to thank Centre de Recerca Matematica and Northwestern University for
providing several opportunities for the authors to meet.
2. Proof of the main theorem
We prove theorem 1.1. The essential tools we use in the course of doing so wi*
*ll be
developed in the following sections.
Our first lemma is well known and has already been mentioned in the introduct*
*ion.
It determines the unique possible structure of the modp cohomology of BSm {pr}*
* if
it exists.
Lemma 2.1. Suppose that Sm {pr}, m > 1, is a loop space. Then
H*(BSm {pr}; Fp) ~=P [um+1 ] E[xm ];
as an algebra with fir(x) = u. Furthermore, if X is any pcomplete space with *
*this
cohomological structure then X is homotopy equivalent to Sm {pr}.
Proof.The first statement follows at once by inspection of the Serre spectral s*
*equence
for the pathloop fibration over BSm {pr}. For the second, notice that assumin*
*g p
completeness, X is simply connected. Thus there is a map fl : (Sm )^p____X, wh*
*ich
induces an isomorphism in modp homology up to dimension m. The homotopy fibre
of fl is easily seen to have the modp cohomology of an msphere and since it i*
*s p
complete, it is equivalent to the pcompletion of an msphere. Thus X is equiva*
*lent
5
to the fibre of a self map of (Sm )^p. One easily computes the degree of this s*
*elf map_
to be precisely pr, up to a padic unit, and the proof is complete. *
* __
We now restrict attention to the case where m = 3 and p = 2. From this point *
*and
on H*() will denote mod2 cohomology. Thus if BS3{2r} exists then
(2) H*(BS3{2r}) ~=P [u4] E[x3]; with fir(x) = u.
Let X be a 2complete space, whose cohomology agrees with the above. Then the*
*re
is a morphism in K
ff : H*(X) ____H*(BZ=2)
with ff(x) = 0 and ff(u) = z4, where z 2 H*(BZ=2) is the generator. By Lannes
theory there exists a map g : BZ=2 ____X, inducing ff on cohomology.
The objects H*(X) and H*(BSL2(Fq)), for some q, are isomorphic in K. Moreover,
ff coincides with the the map induced on cohomology by the inclusion i of Z=2 a*
*s the
center of SL2(Fq). Thus
TZ=2(H*(X); g*) ~=TZ=2(H*(BSL2(Fq)); Bi*) ~=H*(BSL2(Fq)):
Since the right hand side vanishes in dimension 1, it follows that the T funct*
*or
computes the cohomology of the mapping space and so the evaluation map
Map(BZ=2; X)g ____X
is a homotopy equivalence. Thus we have obtained a model of X with an action of
BZ=2. Consider the associated principal fibration
g ss
(3) BZ=2 ____Map(BZ=2; X)g x EBZ=2 ____Map(BZ=2; X)g xBZ=2EBZ=2:
Proposition 2.2. Let X be a 2complete space such that H*(X) ~= P [u4] E[x3]
with fir(x) = u. Let Er be a space such that there is a principal fibration
g ss
BZ=2 ____X ____Er
and the map g is essential. Then r 3 and the cohomology of E3 is given as an
object of Kfiby structure 1 below. Furthermore, for r 4 the cohomology of Er *
*is
isomorphic in Kfito one of the objects given in 2 and 3.
1. P [w2; w3; x3]=(x23+ w23+ w3x3 + w32), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3,
Sq1x3 = w22and Sq2x3 = w2w3 + w2x3.
2. P [w2; w3; x3]=(x23+ w3x3), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = 0,
Sq2x3 = w2x3 and fir2x3 = w22.
3. P [w2; w3; x3]=(x23), Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = Sq2x3 =*
* 0
and fir2x3 = w22.
Moreover, let Y be any simply connected space with either one of the cohomolog*
*ical
structures given above and let Y <2> denote its 2connected cover. Then Y <2> '
S3{2r}.
6
Structure 1 in the proposition is isomorphic to H*(BP SL2(Fq)), q 3 (mod 8),
or equivalently to H*(BA4), where A4 is the alternating group on four letters. *
*Struc
ture 2 is isomorphic to H*(BP SL2(Fq)), q 1 (mod 8), r > 3 [22]. Structure 3 *
*is
isomorphic as an object of K to H*(BSO(3) x S3). The proof of the proposition w*
*ill
be carried out in a separate section.
Corollary 2.3. The space S3{4} is not a loop space.
Proof.If S3{4} is a loop space, its classifying space X = BS3{4} fits in the pr*
*incipal
fibration (3), with base space E2 = Map(BZ=2; X)g xBZ=2 EBZ=2, but then, the *
* __
cohomology of X described in (2) contradicts Proposition 2.2. *
* __
Proof of Theorem 1.1.Assume that BSm {2r} is a classifying space for Sm {2r}, m*
* >
1. According to Lemma 2.1, H*(BSm {pr}; Fp) ~= P [um+1 ] E[xm ], as an algebra
with fir(x) = u. In [1] it is proved that a necessary condition for this algebr*
*a to be
the cohomology of a space is that m = 3 and r > 1. By Corollary 2.3 we rule out
the case r = 2 as well. For the remaining cases, the spaces BSL2(Fq)^2, for var*
*ious
odd prime powers q, provide enough models for classifying spaces of Sm {2r}. T*
*his
completes the proof of the first statement of the theorem.
By Lemma 2.1, the second part of the theorem amounts to proving that for r 3
the homotopy type of BS3{2r} is determined by its cohomology as an object of Kf*
*i.
In the sequel we will show that if a 2complete space Y has the cohomology of
BP SL2(Fq) as objects in the category Kfithen Y ' BP SL2(Fq)^2(see Theorem 6.2).
Now, suppose that X is a space such that H*fi(X) ~=H*fi(BS3{2r}). Construct t*
*he
space Er with respect to X as described above. Then H*fi(Er) is isomorphic to o*
*ne of
the three possible structures specified in Proposition 2.2. By the remarks foll*
*owing
the proposition either
1. H*fi(Er) ~=H*fi(BP SL(Fq)) for an appropriate odd prime power q or
2. H*fi(Er) coincides with structure 3 of Proposition 2.2.
Assume the first case. Then Er ' BP SL2(Fq)^2for a suitable value of q and X
is homotopy equivalent to its 2connected cover, given by BSL2(Fq)^2. Homotopy
uniqueness of BSm {2r} follows by showing the nonrealizability of structure 3 *
*in
Proposition 2.2. This is done in section 8, Proposition 8.1, where it is shown*
* that
the proposed Steenrod algebra action is incompatible with the specified higher_*
*order_
Bockstein operations. _*
*_
3. Proof of Proposition 2.2
Fix a positive integer r and let X be a classifying space for S3{2r}. Assume *
*we are
given a principal fibration which up to homotopy has the form
g ss
(4) BZ=2 ____X ____Er
with non trivial g. Our aim is computing the cohomology of Er as an object of K*
*fi.
Consider the Sss for the delooping of (4)
ss w2
(5) X ____Er ____K(Z=2; 2):
7
The E2 page takes the form
Ep;q2~=Hp(K(Z=2; 2)) Hq(X) ~=(P [2; Sq12; Sq2;12; . .].)p (P [u4] E[x3])q:
We claim that the spectral sequence is determined by the requirement that x3 is*
* a
permanent cycle and d5(u4) = Sq2;12 + 2Sq12.
The first differential, d2, is determined by its value on the class u4. Assum*
*e it is
non trivial; that is, d2(u4) = 2x3. It would follow that x3 is a permanent cycl*
*e, for
d3(x3) = 22would imply d3(2x3) = 326= 0 in the E3term of the spectral sequence.
But 2x3 = 0 in E3, which gives a contradiction. This calculation suffices to co*
*nclude
the structure of H*(Er) in low dimensions if this were the case. Namely, we wou*
*ld
have the classes represented by 2 in degree 2, Sq12 and x3 in degree 3, 2 in de*
*gree
4 and 2Sq12 and Sq2;12 in degree 5. Using naturality of the Bss, one finds that*
* x3
would in that case be a permanent cycle in the Bss for Er, thus a non torsion c*
*lass in
the 2adic cohomology of Er. This in turn is impossible, since Er is the total *
*space
in a fibration with base and fibre, which are torsion spaces. Hence d2 is trivi*
*al.
d3 is zero by degree reasons. The next possibly nontrivial differential is d*
*4, which
may be given by d4(x3) = 22. But if x3 transgresses to 22then Sq222= (Sq12)2 sh*
*ould
also be in the image of the transgression, which is of course impossible. Hence*
* d4 is
also trivial.
Next we compute d5. Notice that u4 is now transgressive. Since it restricts n*
*on
trivially to the cohomology of BZ=2 by hypothesis, it follows by naturality that
d5(u4) = Sq2;12 + A2Sq12, for some A 2 F2. But Sq1;2;12 = (Sq12)2, which would
imply a contradiction if A = 0. Hence A = 1 and d5(u4) = Sq2;12 + 2Sq12. It now
k
follows at once that each element of the form u24 is transgressive to a nonzer*
*o class
and so the E1 term of the spectral sequence is isomorphic to P [2; Sq12] E[x3*
*] as
an algebra. We obtain an isomorphism of P [w2; w3]modules
H*(Er) ~=P [w2; w3] E[x3];
with Sq1w2 = w3.
Since Er is a torsion space, there must exist some positive integer j such th*
*at
fij(x3) = w22. This enables an easy calculation of the Bss for Er in low dimens*
*ions. In
particular it follows that H4(Er; Z^2) is cyclic and that H5(Er; Z^2) vanishes.*
* In turn,
this implies that in the 2adic Sss for (5), the transgression
d5: E0;45~=H4(X; Z^2) ~=Z=2r ____E5;05~=H5(K(Z=2; 2); Z^2) ~=Z=4
is an epimorphism with a nontrivial kernel. It then follows that r 3 and
H4(Er; Z^2) ~=Z=2r2:
Assume r = 3 so that Sq1(x3) = w22. Write Sq2(w3) = Aw2w3+ Bw2x3, A; B 2 F2.
Then
w23= Sq1;2(w3) = Aw23+ B(w3x3 + w32):
Hence A = 1; B = 0 and Sq2(w3) = w2w3. Similarly write Sq2(x3) = Aw2w3+Bw2x3.
Then as before
x23= Aw23+ B(w3x3 + w32):
8
One has Sq2(x23) = w42on one hand and on the other hand
Sq2(x23) = Sq2(Aw23+ B(w3x3 + w32)) = B(w2w3x3 + w3Sq2(x3) + w23w2 + w42):
Manipulating with this equation one obtains A = B = 1 so Sq2(x3) = w2x3 + w2w3
and x23= w23+ w3x3+ w32. This completes the description of H*(Er) in the case w*
*here
r = 3, namely case 1 of Proposition 2.2.
Assume next r > 3. Then Sq1(x3) = 0 and fir2(x3) = w22. One checks as before
that Sq2w3 = w2w3 and writes Sq2(x3) = Aw2w3+ Bw2x3. Then x23= Aw23+ Bw3x3.
Applying Sq2 one obtains that either A or B must be 0. If A = 1 then change the
basis and take x3 = x3 + w3. Then Sq2x3 = 0 and we get the structure of case 3.
Hence we may assume without loss of generality that to begin with either A = 0,
B = 1 or A = B = 0. These two cases correspond to cases 2 and 3 of Proposition *
*2.2
respectively.
To complete the proof we need to calculate the cohomology of the 2connected
covers for spaces with the cohomological structures given in the proposition. C*
*onsider
the principal fibration
(6) BZ=2 ____Y <2> ____Y;
where Y such that H*(Y ) ~=P [w2; w3]E[x3] as a P [w2; w3]module with Sq1w2 = *
*w3,
Sq2w3 = w2w3 and fir2(x3) = w22. Then in the Sss for (6) one has d2(z) = w2,
d3(z2) = w3 and d5(z4) = 0. The spectral sequence thus collapses at the E6 page
without any extension problems and so H*(Y <2>) ~=P [u4] E[x3]. Since all spac*
*es
involved are torsion there must exist a positive integer t, such that fit(x3) =*
* u4. The
integral Sss now gives at once that t = r and so by Lemma 2.1, Y <2> is a class*
*ifying
space for S3{2r}.
4. Homology decomposition of BP SL2(Fq)
Let P SL2(Fq) denote the projective special linear group of rank 2 over the f*
*ield of
q elements Fq. Preparing to prove homotopy uniqueness of BP SL2(Fq)^2, we study
homology decompositions of this space as in [17, 18, 10]. For the terminology w*
*e use,
the reader is referred to [10].
21)
Let q be an odd prime power. Then the order of P SL2(Fq) is q(q___2. Notice t*
*hat
21)
the highest power of 2 dividing q(q___2is precisely 4 if q 3 (mod 8) and that*
* if
21) s
q 1 (mod 8) then q(q___2is divisible by 2 for s 3 and, depending on q, every
such s may appear. We will write 2skn if 2s is the highest power of 2 dividing *
*n. The
2*
*1)
2Sylow subgroup of P SL2(Fq) is known to be the dihedral group D2s if 2skq(q__*
*_2.
The respective cohomology algebras are also known [22] and are described below.
All groups P SL2(Fq) have the same mod2 cohomology algebra and only elementa*
*ry
torsion in their integral cohomology if q 3 (mod 8):
H*(BP SL2(Fq)) ~=P [w2; w3; x3]=(x23+ w23+ w3x3 + w32);
9
with Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = w22and Sq2x3 = w2w3 +
w2x3. As we observe later every 2complete space with this cohomology is homoto*
*py
equivalent to (BA4)^2, where A4 is the alternating group on 4 letters.
The case q 1 (mod 8) is a bit more complicated. Here too, the mod2 cohomol
ogy as an algebra over the Steenrod algebra is the same in all case. The Bss th*
*ough
varies according to the highest power of 2 dividing the order of P SL2(Fq). Ind*
*eed if
21)
q 1 (mod 8) and 2skq(q___2then
H*(P SL2(Fq)) ~=P [w2; w3; x3]=(x23+ w3x3);
with Sq1w2 = w3, Sq1w3 = 0, Sq2w3 = w2w3, Sq1x3 = 0, Sq2x3 = w2x3 and
fis1x3 = w22.
Below we prove that among 2complete spaces there is a unique homotopy type
corresponding to the above cohomology structure for every s 3. Hence, for our
purposes, we will only need to consider prime powers q with q +1 (mod 8), for*
*, as
a consequence of a theorem of Dirichlet's, these are enough to cover all cases.
Lemma 4.1. For every s 3 there is an odd prime q with q +1 (mod 8) such
21)
that 2sk(q  1) and hence 2skq(q___2.
Proof.For s 3 consider the arithmetic progression ns;j= 2s2j  2s3: Let
qs;j= 8ns;j+ 1 = 2s+1j  (2s 1):
Since the greatest common divisor of 2s+1 and 2s  1 is 1, Dirichlet's theorem *
*[16]
implies the existence of infinitely many primes of the form qs;j. Notice that e*
*ach such
prime is congruent to 1 modulo 8 and that
qs;j 1 = 8ns;j= 2s(2j  1):
Thus for integers j such that qs;jis a prime, the highest power of 2 dividing t*
*he_order
of P SL2(Fqs;j) is 2s. *
*__
Throughout this section we restrict attention to the case of odd prime powers*
* q,
where q 1 (mod 8) and consider certain collection of subgroups of P SL2(Fq). *
*Let
s be such that 2skq  1 and let " be a primitive root of unity in Fq of order 2*
*s. Then
the matrices X = "00"1and Y = 0110satisfy the relation
s1 2 2
X2 = Y = (XY ) = 1;
modulo the center of SL2(Fq), generated by the matrix I. Hence the classes [X]
and [Y ] of these matrices in P SL2(Fq) generate a dihedral group of order 2s, *
*which
gives the 2Sylow subgroup.
Notice that if q 1 (mod 8) the 2Sylow subgroup is of order at least 8. Con*
*sider
the following elementary abelian 2subgroups of P SL2(Fq).
s2
Z = <[X]2 > ~=Z=2
s2
V = <[X]2 ; [Y ]> ~=Z=2 x Z=2
s2
W = <[X]2 ; [X][Y ]> ~=Z=2 x Z=2
10
According to [17, Thm. 7.7], the set E2 = {Z; V; W } is an ample collection o*
*f el
ementary abelian 2subgroups of P SL2(Fq) (which, roughly speaking means that it
can be used to obtain a homology decomposition). The associated conjugacy cate
gory AE2 has E2 as objects, with morphisms among them induced by inclusions and
conjugations in P SL2(Fq).
Our next statement describes the category AE2. First, let G1 and G2 be finite
groups with a specific common subgroup H. Consider the category
__oe_______ _______
A(G1; G2; H) = G2 _2oe___oe__0____1G1:
G2=H G1=H
__ __ __ __
It has three objects _0, _1and _2. The_automorphism group of _iis given*
* by_Gi_for
i =_1;_2 whereas the automorphism group of _0is trivial. Each morphism set fr*
*om _0
to _i, i = 1; 2, admits a natural Gi action and is isomorphic as a Giset to *
*Gi=H.
Proposition 4.2. For q 1 (mod 8), the set E2 = {Z; V; W } is an ample collec
tion of elementary abelian 2subgroups of P SL2(Fq) and the E2conjugacy catego*
*ry
AE2 is the image of the functor from_the category_A(3; 3; Z=2)_to the category *
*of
elementary abelian 2groups taking _0to Z, _1to W and _2to V .
Proof.Let G denote BP SL2(Fq) with q 1 (mod 8). Notice that the automorphisms
of a subgroup U in the conjugacy category are given by the quotient NG (U)=CG (*
*U)
of the normalizer by the centralizer of U in G. One easily verifies the followi*
*ng.
i 0 *
1. CG (Z) = NG (Z) = < 0i1 ; 0110> ~=D2q2, where i is generator of Fq a*
*nd
D2q2is the dihedral group of order 2q  2. Notice that D2s < D2q2and the
mod2 restriction map isfanfisomorphism.i
2. CG (V ) = V and NG (V ) V ~= 3 is generated by the classes A and B in *
*G,
p_ s2 p _
of A = _1_i111iiand B = i0p_ii0respectively. Here i = "2 , so by iwe
s3 2s2 2s2
mean "2 . Conjugation by A sends X to Y and Y to X Y ,
2s2 2s2
whereas conjugation by BfflipsfiY and X Y and fixes X .
3. CG (W ) = W and NG (W ) W ~= 3 generated by the classes C and D in
1 " p_
G of C = _1_i1i"1iand D = ip0_i"i"10respectively. Conjugation by C
2s2 2s2+1
sends X to X Y and X Y to X Y , whereas conjugation by
2s2+1 2s2
D flips X Y and X Y and fixes X .
Thus, the automorphisms in AE2are as described, and the other morphisms follo*
*w __
immediately. This completes the proof of the proposition. *
* __
We can draw the E2conjugacy category as
oe____ _____
AE2: 3 W oe____oe_Z ______V 3 :
3=2 3=2
The centralizer diagram
ffE2: AopE2___Spaces
11
is the functor defined in [17], which associates with each object U a model for*
* the
classifying space of its centralizer in G. Namely, to an object U it associates*
* the orbit
space EG xG (G=CG (V )). Up to homotopy and 2completion the diagram in our case
is given by ____ oe___
3 BW _____BD2s oe___oe_BV 3:
Notice that CPSL2(Fq)(Z) = D(q1), but (BD(q1))^2= BD2s.
One checks immediately that this diagram satisfies the conditions of [17, Thm*
* 7.7]
and so, using the terminology of [9], it is a sharp diagram. Specifically, this*
* means
that the natural map
aE2: hocolimffE2 ____BP SL2(Fq)
induces a mod2 cohomology isomorphism. Moreover,
lim0H* O ffE2~= H*(BP SL2(Fq));
AE2
whereas all the higher limits vanish. We summarize this in the following
Proposition 4.3. For q 1 (mod 8) there is a homotopy equivalence
^ ^
hocolim ffE2 ' BP SL2(Fq)2
AE2 2
and an isomorphism
lim0H* O ffE2~= H*(BP SL2(Fq))
AE2
*
* __
while limiAEH* O ffE2= 0 for all i 1. *
* __
2
5.pSylow subgroups
A question that could be considered of general interest is the following. Let*
* X be a
pcomplete space. Is there an appropriate concept of a pSylow subgroup for X? *
*For
general spaces X, it is not even clear what these objects ought to satisfy or w*
*hether
they could exist. However, if X has the modp cohomology of a finite group G, t*
*hen a
sensible definition for a pSylow subgroup for X appears to be a map f : Bss __*
*__X,
where ss is the pSylow subgroup of G, which realizes the restriction map in p*
*local
cohomology.
One way to attack this problem starts by finding an elementary abelian psubg*
*roup
V contained in ss, the pSylow of G, such that CG (V ) = ss. Of course, this su*
*bgroup V
does not always exist, which is the main reason why, in our specific case, we w*
*ork first
with P SL2(Fq) rather than directly with SL2(Fq). In those favorable cases wher*
*e one
can find a subgroup as above, one may approach further technical difficulties. *
* The
main complication is in the use of Lannes' T functor, which is not guaranteed t*
*o do
its work. This makes it essential to use an interpretation of T due to E. Farjo*
*un and
J. Smith [14]. We define a class of groups which behaves nicely with respect to*
* this
interpretation of the T functor and call those groups Tpgroups. We show that c*
*yclic
pgroups and dihedral 2groups are Tpgroups, a fact which is then used to obta*
*in a
2Sylow subgroup for a space X with the same cohomology as P SL2(Fq).
12
Definition 5.1. Let ss be a finite pgroup. We say that ss is a Tpgroup if fo*
*r any
tower of fibrations {Fs} such that
1. ssi(Fs) is a finite pgroup for every i and s and
2. lim!H*fi(Fs; Fp) ~=H*fi(ss; Fp)
there is a homotopy equivalence limFs ' Bss.
We now give some elementary examples of groups which are Tp. We know of furth*
*er
examples, which are irrelevant to this paper. One might like to ask whether ev*
*ery
finite pgroup is Tp as a positive result might have interesting implications.
Proposition 5.2. Every cyclic pgroup is a Tpgroup. Also, every dihedral 2gr*
*oup
is a T2group.
Proof.The proof for the case of a cyclic pgroup is identical to that of Lemma *
*6.4 in
[2] and we omit it. For dihedral 2groups, our statement is a stronger version *
*of [4,
Thm. 1.1]. The proof combines methods fromf[4,fx3]iand [2, x6].
Recall that H*fi(BD2n) ~=P [x1; y1; w2] (x2 + xy), where degrees are given by*
* sub
scripts, Sq1(w) = wy and fin1(xw) = w2.
Let {Ys} be a tower of fibrations satisfying 1 and 2 in Definition 5.1 with r*
*espect
to D2n. Let sss : Ys ____Ys1 denote the projection. For a sufficiently large *
*s there
are elements y1;s2 H1(Ys) such that
1. ss*s+1(y1;s) = y1;s+1and
2. the sequence {y1;s} represents the element y1 2 lim!sH1(Ys) ~=H*(BD2n).
Let 's : Ys ____BZ=2 be a map classifying y1;s. Then there is the following ho*
*motopy
commutative diagram of fibrations.
: :_:__Fs+2_____ Fs+1______Fs
  
js+2 js+1 js
? sss+2 ? sss+1 ?
: :_:__Ys+2______Ys+1______Ys

's+2  's+1 's
? ? ?
: :=:==BZ=2 ==== BZ=2 ==== BZ=2
Since the homotopy groups of all spaces involved are finite, it follows that *
*the
sequence
limFs ____ limYs ____BD2n
is a quasi fibration, namely it gives rise to a long exact sequence in homotopy.
The EilenbergMoore spectral sequence for each fibration in the diagram colla*
*pses
at its E2 page, being a fibration over BZ=2. Thus
H*(Fs) ~=T orH*(BZ=2)(F2; H*(Ys)):
13
Since T or commutes with direct limits, we have that
(7) P [v2] E[ff1] ~=T orH*(BZ=2)(F2; H*(D2n)) ~=
~=T orH*(BZ=2)(F2; lim H*(Y )) ~=lim H*(F )
! s s ! s s
as P [v2]modules. Notice that the edge homomorphism
lim!H*fiYs ! lim!H*fiFs
is a morphism in Kfiwhich at the level of the first page of the Bss is an epimo*
*rphism
ffi2
P [x1; y1; w2] (x + xy) ___P [v2] E[ff1]
sending x1 to ff1, y1 to zero and w2 to v2.
Next, we compute lim!H*fiFs as an object of Kfi. From the description of the*
* edge
homomorphism above it follows that Sq1 acts trivially on P [v2] E[ff1] and the
morphism induced on the second page of the Bss
P [w22] x1w2P [w22]____P [v2] E[ff1]
is monic.
The next and last nontrivial differential in the Bss of the source is determ*
*ined
by fin1(x1w2) = w22. Thus in the Bss of the target we must have fin1(ff1v2) =*
* v22.
By the P [v2]module structure of lim!sH*(Fs), it follows that fin1(ff1) = v2*
* and so
fin1(ff1vk2) = vk+12. Moreover, since homotopy groups of Ys are finite for eve*
*ry s, it
follows that the homotopy groups of each Fs are finite and since cyclic 2group*
*s are
T2 by the first part of the proposition, we have holimFs ' BZ=2n1.
We have thus obtained a quasi fibration
(8) BZ=2r ____ limYs ____BD2n:
In particular Y = limYs is the classifying space of a finite 2group and so (8*
*) is in
fact a fibration. It follows that the map of towers
{Y } ____{Ys};
where {Y } is the constant tower, is a weak prohomotopy equivalence. Consequen*
*tly
the map of towers {Hn(Y )} ____{HnYs} is a proisomorphism by [3, III, 3.4] an*
*d so
H*fiY ~= lim!H*fiYs ~=H*fiBD2n:
Now [4, Thm. 1.1] applies,
Y ' BD2n
__
and the proof is complete. _*
*_
Lemma 5.3. Let X be a pcomplete space with H*fi(X) ~=H*fi(BG) where G is a f*
*inite
pperfect group. Then:
i) ssi(X) is a finite pgroup for all i > 0.
ii)For any elementary abelian pgroup V , and an nth Postnikov section PnX o*
*f X,
the homotopy groups of any component of Map (BV; PnX) are finite pgroups.
14
Proof.By Hypothesis H1(X) = 0 and X is pcomplete. Hence X is 1connected
and [2, 5.7 (1)] applies so that Hj(X; ^Zp) is a finitely generated ^Zpmodule *
*for all j.
Furthermore, by our assumption on the Bss all padic cohomology groups are fini*
*te
pgroups, hence the first statement follows by [2, 5.7 (2)].
The second statement follows as well by an argument of [2, x6]. For an Eilenb*
*erg
MacLane space we have
Y
Map (BV; K(ss; n))' K Hnj(BV; ss); j:
1jn
so if ss is a finite pgroup, then the homotopy groups of Map (BV; K(ss; n)) ha*
*ve_the
same property. Proceed by induction. _*
*_
We are now ready to produce a 2Sylow subgroup for a space X with the same
cohomology as P SL2(Fq). More precisely, we have the following
Proposition 5.4. Let X be a 2complete space with
H*fi(X) ~=H*fi(BP SL2(Fq)):
i
Let D2s____P SL2(Fq) be the inclusion of a 2Sylow subgroup. Then there is a m*
*ap
f :BD2s____X
such that f*, coincides with the composition
(Bi)* *
H*fi(X; F2) ~=H*fi(BP SL2(Fq)) ____Hfi(BD2s):
Proof.If q 3 (mod 8) then the 2Sylow subgroup of P SL2(Fq) is D4, an elemen
tary abelian 2group of rank 2. Then our claim follows directly from Lannes the*
*ory
[20]. Namely, in this case one has
[BD4; X] ~=[BD4; BP SL2(Fq)^2] ~=Hom K (H*(BP SL2(Fq)); H*(BD2s)):
Suppose q 1 (mod 8). Because of Lemma 4.1 it is enough to consider the case
q 1 (mod 8). Let s satisfy 2sk(q 1) so in particular s > 2. Fix a 2Sylow su*
*bgroup
ss ~=D2sand let Z ~=Z=2 be its center. Let j denote the inclusion of Z in P SL2*
*(Fq).
By Lannes theory there is a map fZ :BZ ____X, inducing the composition
(Bj)* *
H*(X) ~=H*(BP SL2(Fq)) ____H (BZ)
on cohomology.
ev
Consider the evaluation map Map (BZ; X)fZ ____X. We proceed by showing that
Map (BZ; X)fZ ' BD2sand that the map thus obtained has the correct cohomological
effect.
We start by computing the Lannes T functor. By hypothesis on the cohomology
of X and the calculations of section 4,
(9) T (H*(X); f*Z) ~=T (H*(BP SL2(Fq); (Bj)*)~=
~=H* BCPSL2(Fq)(Z) ~=H*(BD(q1)) ~=H*(BD2s)
15
Furthermore, one obtains as well that the augmentation
(10) ": H*(X) ____T (H*(X); f*Z)
coincides with (Bi)*: H*(BP SL2(Fq)) ____H*(D2s) under the identification of t*
*he
sources given in terms of the assumed isomorphism. It is then, explicitly descr*
*ibed as
ffi ffi
" : P [w2; w3; x3] (x23+ w3x3)___ P [x1; y1; w2] (x21+ x1y1)
w2 ___ w2 + y21
(11)
w3 ___ w2y1
x3 ___ w2x1
Let {PnX}n be the Postnikov Tower for X and fZ(n): BZ _____PnX be the
factorization of fZ through the tower. Applying the functor Map (BV; ), we get*
* a
tower, {Map (BV; PnX)fZ(n)}n and a homotopy equivalence
(12) holimMap (BZ; PnX)fZ(n)' Map (BZ; X)fZ:
Since the space X is 2complete nilpotent and of finite type, the FarjounSmi*
*th
theorem states
(13) T (H*(X); f*z) ~=lim!H* Map (BZ; PnX)fZ(n) :
n
The map in the theorem is given as the obvious map
(14) !limT (H*(PnX); f*Z(n)) ____!limH* Map (BZ; PnX)fZ(n)*
* ;
n n
using the fact that for a space X as above the map!limnH*(Pn(X)) ____H**
*(X) is
an isomorphism and T commutes with direct limits.
Notice that in our particular case the left hand side in (13) is isomorphic to
H*(BD2s) as an A*2algebra.
There are homomorphisms H*(PnX) _____H*(Map (BZ; PnX)fZ(n)) induced by
evaluation maps. Taking the limit on both sides and using the isomorphism (14),*
* the
resulting map
ev* *
(15) H*(X) ____!limH (Map (BZ; PnX)fZ(n))
n
coincides with the augmentation map given in (11). But being induced by maps
of spaces, this monomorphism is compatible with higher Bockstein operations. One
observes at once that it should induce an isomorphism in the second page of the*
* Bss
and therefore there is only one structure of H*(BD2s) as an object of Kfithat is
compatible with such a monomorphism, namely
*
!limH*fiMap (BZ; PnX)fZ(n) ~=Hfi(BD2s) :
n
Now, since X is 2complete and H*fi(X) ~=H*fi(P SL2(Fq)), Lemma 5.3 imply that
ssj(X) is a finite 2group for all j and also that the homotopy groups of the m*
*apping
spaces ssj(Map (BZ; PnX)fZ(n)) are finite 2groups for all j and n.
16
By Proposition 5.2 the dihedral group D2s is a T2group and so we conclude th*
*at
holimnMap (BZ; PnX)fZ(n)' BD2s. This combined with (12) gives
Map (BZ; X)fZ ' BD2s:
Taking the evaluation, we obtain a map
f :BD2s____X;
such that the induced homomorphism on cohomology coincides with the composition
(Bi)* * __
H*fi(X) ~=H*fi(BP SL2(Fq)) _____Hfi(BD2s): __
6. Homotopy type of the two completion of BP SL2(Fq)
In this section we prove that if X is a 2complete space with the mod2 cohom*
*ology
of BP SL2(Fq) as an object of Kfithen X ' BP SL2(Fq)^2.
Proposition 6.1. Let X be a 2complete space such that
H*fi(X) ~=H*fi(BP SL2(Fq))
Let s be the highest power of 2 dividing the order of P SL2(Fq). Assume s > 2 a*
*nd let
f :BD2s____X
be the map constructed in Proposition 5.4. Then, the induced map
f]
BZ=2 ' Map (BD2s; BD2s)id____ Map (BD2s; X)f
is a homotopy equivalence.
Proof.Without loss of generality, we can assume that q 1 (mod 8). Consider the
sequence of maps
BD4 BD8 : : :BD2r : : :BD2s;
induced by the natural inclusions. Denote fr = f O (Bir): BD2r! BD2s! X. We
prove by induction that f induces homotopy equivalences
f]: Map (BD2r; BD2s)Bir____ Map (BD2r; X)fr:
The case r = 2, follows by Lannes theory [20]. Since D4 ~= (Z=2)2 and is sel*
*f
centralizing in both D2sand P SL2(Fq), a T functor computation shows that BD4 '
Map (BD4; BD2s)Bi2' Map (BD4; X)f2.
By induction assume the conclusion is true for r  1. The space Map (BD2r; X)
may be written as the homotopy fix point space Map (ED2r=D2r1; X)hZ=2. The cor
responding construction holds for BD2sreplacing X. Now, the map
f] : Map (ED2r=D2r1; BD2s)Bir1____ Map (ED2r=D2r1; X)fr1
is a Z=2equivariant homotopy equivalence, so we obtain,
(16) Map (BD2r; BD2s)g ' Map (ED2r=D2r1; BD2s)hZ=2Bir1'
' Map (ED2r=D2r1; X)hZ=2fr1' Map (BD2r; X)f
17
where g = {g  gBD2r1' Bir1} and f = {f  fBD2r1' fr1}. This completes the
induction step.
Finally, notice that for r = s, the left hand side of the equation above is c*
*onnected.
Hence the same hold for the right end side and the component of the mapping spa*
*ce __
there is essentially the one of f. This completes the proof. *
* __
Theorem 6.2. Let X be a 2complete space with
H*fi(X) ~=H*fi(BP SL2(Fq)):
Then there is a homotopy equivalence
X ' BP SL2(Fq)^2
Proof.If q 3 (mod 8) then the 2Sylow subgroup of P SL2(Fq) is D4, i.e. eleme*
*n
tary abelian of rank 2. In this case one has by Lannes theory a map
f : BD4 ____X
inducing the restriction map under the given identification of the algebras H*(*
*X)
and H*(BP SL2(Fq)). The T functor is now employed to compute the cohomology
of Map (BD4; X)f. Indeed, since D4 is the 2Sylow subgroup, it is immediate th*
*at
H*(Map (BD4; X)f) ~=H*(BD4) as A*2algebras and so Map (BD4; X)f is homotopy
equivalent to BD4. The evaluation map thus gives a map BD4 _____X, which is
equivariant with respect to any automorphism of D4, in particular with respect *
*to
the natural Z=3 action. This gives a map
f": BD4 xZ=3EZ=3 ____X;
which one easily checks to induce an isomorphism on mod2 cohomology and so a
homotopy equivalence after 2completion,
X ' (BD4 xZ=3EZ=3)^2' (BA4)^2:
Next restrict attention to the case where the 2Sylow subgroup is nonabelian*
*. In
particular q 1 (mod 8). And we can restrict our attention to the case q 1
(mod 8) without loss of generality. The discussion above gives a homotopy comm*
*u
tative diagram
______3=2______ oe________3=2___
3 BW _________________________BD2soe_______oe_______BV 3
HH 
HH 
(17) HH f0
f2 HHj ? ss f1
X
One can regard this diagram as a natural transformation, defined only up to hom*
*o
topy, from the diagram ffE2 for BP SL2(Fq) (see section 4) to the constant diag*
*ram
on our space X
f :ffE2 ____X:
The map f0: BD2s ! X is given by Proposition 5.4. The maps f1: BV ! X,
and f2: BW ! X, as well as homotopy commutativity in the diagram are provided
by Lannes theory [20]. Indeed one only needs to check that these maps can be
18
defined on the level of cohomology (which of course they can since they exist f*
*or
X = BP SL2(Fq)).
By Proposition 4.2 we can identify AE2 with A(3; 3; Z=2). To simplify the no
tation let A denote A(3; 3; Z=2) and let ff denote the diagram ffE2.
Diagram (17) defines a map from the 0skeleton of hocolimAopff to X. The ob
structions to extending this map to the actual homotopy colimit lie in the grou*
*ps
limj+1ssj(Map (ff; X)f)
A
for j 1 [26]. By Lemmas 6.3 and 6.4 below these obstruction groups vanish. Thus
we obtain a map
f
(BP SL2(Fq))^2' hocolimff ____X
Aop
where the homotopy equivalence on the left was stated in Proposition 4.3. Moreo*
*ver,
the diagram
BD2s P
 PP P
 PP Pf0
Bi  P PP
? P Pq
(BP SL2(Fq))^2__________________X
f
commutes up to homotopy. It follows at once that f induces an isomorphism in mo*
*d2_
cohomology and is therefore a homotopy equivalence. *
*__
To complete the proof of the theorem we need to prove vanishing of the obstru*
*ction
groups. For each j 1 define a functor j: A ____Ab by
__ __
j(_i) = ssjMap (ff(_i);fX)i:
Lemma 6.3. The values of the functors defined above are given as follows.
__
1. j(_0)_= ssj(BZ=2)
2. j(_1)_= ssj(BV )
3. j(_2) = ssj(BW )
Proof.First notice that
__ __
j(_0) = ssjMap (ff(_0);fX)0= ssj(Map (BD2s; X)f0) = ssj(BZ=2)
by Proposition 6.1.
Next since V is self centralizing in P SL2(Fq) it follows from Lannes theory *
*that
__ __
j(_1) = ssj(Map (ff(_1); X)f1)= ssj(Map (BV; X)f1) = ssj(BV ):
*
*__
The third case follows by the same argument. *
*__
Lemma 6.4. limiAj = 0 for all i 0; j 1 except for i = j = 1 in which case
lim1A1 = Z=2.
Proof.We only need to take care of the case j = 1. The functor 1 is
oe____ _____
3 V oe____oe_Z=2 ______W 3
3=2 3=2
19
and according to Proposition 10.3 we have an exact sequence
0 ____lim01 ____Z=2 ____V 2 W 2 ____lim11 ____0
A A
and limiA1 = 0 if i > 1. The result is forced by the fact that V 2 W 2 ~=_
Z=2 Z=2. __
7.Spherical fibrations
Our argument in the proof of nonexistence of a space with the cohomological
structure, given in 3 of Proposition 2.2, involves investigations of certain sp*
*herical
fibrations. We say that the fibration
F ____E ____B
is cohomologically trivial if the associated mod2 Sss collapses at its E2 page*
* and
H*(E) ~=H*(B) H*(F )
as algebras over the Steenrod algebra. By a spherical fibration we mean a Serre
fibration, where the fibre is a sphere. We shall assume as always that all spac*
*es are
2complete.
The case of interest to us is when the base space of the given fibration is B*
*O(2) and
the fibre is a 3sphere. this section is devoted to the proof of the following *
*statement.
Theorem 7.1. Any 2complete, cohomologically trivial spherical fibration
f g
= S3 ____E() ____BO(2)
is trivial.
Proof.Let i: Z=2_____O(2) be the inclusion of the central element of order two
in O(2). The quotient is homeomorphic to O(2) and we have an induced principal
fibration
Bi Bss
(18) BZ=2 ____BO(2) ____BO(2)
w2 2
which is classified by the second StiefelWhitney class BO(2) ____B Z=2. Thus
(19) Bss*(w1) = w1 and Bss*(w2) = 0
in mod2 cohomology, while Bi*(w1) = 0 and Bi*(w2) = z2, the class z being the
1dimensional generator of H*(BZ=2).
Furthermore, since the fibration is cohomologically trivial, one has by Lann*
*es
theory [20] that there is a lifting of Bi to a map
h: BZ=2 ____E();
which is unique up to homotopy. Thus the map
g* : Map (BZ=2; E())h ____ Map (BZ=2; BO(2))Bi
is equivariant with respect to the action of the topological group BZ=2, acting*
* in
both mapping spaces by translation in the source.
20
i ev
Since Z=2 ____O(2) is central, the evaluation Map (BZ=2; BO(2))Bi ____BO(2)
is a mod2 equivalence. Also, an easy computation of the respective Lannes T fu*
*nctor
ev
shows that Map (BZ=2; E())Bi ____E() is a mod2 equivalence. Denote
E(1) = Map (BZ=2; E())h xBZ=2EBZ=2:
Taking the respective Borel constructions one gets a diagram of principal fib*
*rations
BZ=2 === BZ=2
h Bi
? ?
S3w____E() ____BO(2)
ww
(20) ww  Bss
? ?
S3 ____E(1) ___BO(2)
 
 w2
? ?
B2Z=2 == B2Z=2
where all spaces are assumed 2complete. Let 1 denote the quotient spherical fi*
*bra
tion S3 ____E(1) ____BO(2).
Lemma 7.2. 1 is cohomologically trivial.
Proof.The Thom space for 1, T (1), is defined as the homotopy cofibre of the pr*
*o
jection E(1) _____BO(2). Its mod2 cohomology is determined by the Thom iso
morphism theorem
"H*(T (1)) ~=U1 . H*(BO(2))
as H*(BO(2)) modules, where U1 2 "H*(T (1)) is the Thom class. The action of the
Steenrod squares on the Thom class defines the StiefelWhitney classes of 1 by *
*the
equations
Sqi(U1 ) = U1 . wi(1):
The action of the Steenrod algebra on the cohomology of T (1) is thus determine*
*d by
its action on the Thom class, given above, the action on H*(BO(2)) and the Cart*
*an
formula.
We have then a long exact sequence of modules over both H*(BO(2)) and the
Steenrod algebra
(21)
j* q p* q ffi q+1
: : :___H"q(T (1)) ____H (BO(2)) ____H (E(1)) ____H" (T (1)) ____ : : :
where j* is determined by its value on U1 , j*(U1 ) = e1 , the Euler class, whi*
*ch
coincides with the top StiefelWhitney class.
We proceed by computing the StiefelWhitney classes for 1.
Claim 7.3. The StiefelWhitney classes of 1 are trivial.
21
Proof.Write w(1) = 1 + w1(1) + w2(1) + w3(1) + w4(1) with wi(1) 2 Hi(BO(2)),
for the total StiefelWhitney class of 1. Since Bss*(1) = and is cohomologica*
*lly
trivial Bss*(w(1)) = 1.
The action of the Steenrod squares on the StiefelWhitney classes of any sphe*
*rical
fibration is given by the Wu formula. This fact is exploited in [5], where poss*
*ible total
StiefelWhitney classes in certain invariant algebras are classified. In the pa*
*rticular
case, where the base of a spherical fibration is BO(2), only 1, 1 + w1, 1 + w1 *
*+ w2
and their products can form total StiefelWhitney classes (see [5, section 2]).*
* So,
w(1) = (1 + w1)r(1 + w1 + w2)s with r; s 0. One then checks that the only
possibility that satisfies Bss*(w(1)) = 1 is r = s = 0 (see (19)). Hence the t*
*otal_
StiefelWhitney class for 1 is trivial. *
* __
It follows that the long exact sequence (21) collapses to a short exact seque*
*nce
p* * ffi *+1
0 ____H*(BO(2)) ____H (E(1)) ____U1 . H (BO(2)) ____0 :
There is a 3dimensional class x3 in H*(E(1)) with ffi(x3) = U1 . We now compute
the Steenrod algebra action on x3.
Since Sqi(U1 ) = U1 . wi(1) = 0, we have Sqi(x3) 2 p*(H*(BO(2))). Identifying
H*(BO(2)) with its image under p*, write Sq2(x3) = 1w51+ 2w31w2 + 3w1w22,
i2 Z=2.
Without loss of generality 1 = 3 = 0 or otherwise define x3= x3+1w31+3w1w2.
Then Sq2(x3) = w31w2 with = 2 + 3 and ffi(x3) = U1 .
It follows that (x3)2 = Sq3(x3) = Sq1Sq2x3 = 0 and so (Sq1x3)2 = Sq2(x23) = 0.
Hence Sq1x3, being considered as an element of H*(BO(2)), must be zero. Finally,
Sq2Sq2x3 = Sq1Sq2Sq1x3 = 0, and since w31w226= 0, it follows that = 0.
We have thus obtained a choice of x3 2 H3(E(1)) with ffi(x3) = U1 and Sqi(x3)*
* = 0
for all i 1 and therefore H*(E(1)) ~= H*(BO(2)) H*(S3) as algebras over the *
* __
Steenrod algebra and the proof of the lemma is complete. *
*__
Let jn: D2n ____O(2) denote the canonical inclusion of the dihedral group of*
* order
2n in O(2). Notice that jn factors through jn+1 and the direct limit with respe*
*ct to
the ji's gives a discrete approximation for BO(2) at the prime 2 (in the sense *
*of [12]).
Lemma 7.4. For every n 2, the pullback fibration (Bjn)*() is trivial.
Proof.Recall that the class j*n(w2) = w 2 H*(BD2n) classifies the central exten*
*sion
Z=2 ____D2n ____D2n1and so the square
Bjn
BD2n ______BO(2)

 
 Bss
? Bjn1 ?
BD2n1 _____BO(2)
is a pullback diagram. Notice that this means that the orbit spaces O(2)=D2n *
*are
the same for all n. Indeed, an easy diagram chase argument show that these orb*
*it
spaces are, one and all, homotopy equivalent to S1.
22
Observe that the fibration 1 over BO(2) satisfies the same condition as , nam*
*ely
it is cohomologically trivial. Hence the construction can be repeated inductive*
*ly to
obtain a sequence of spherical fibrations
i S3w_____E(i) ____BO(2)
ww
(22) ww  Bss
? ?
i+1 S3 ____E(i+1) ___BO(2)
where each i is a pullback of i+1.
Pulling back Diagram (22) along the successive maps Bjn, we get the following
diagram, where every square on the right is a pullback square.
S3w_____E((Bjn)*()) _____BD2n
ww 
ww  
? ?
S3 ____E((Bjn1)*(1)) ___BD2n1
.. . .
. .. ..
S3w____E((Bj1)*(n1)) ___BZ=2
ww 
ww  
? ?
S3 _____E((Bj0)*(n)) ______*
Obviously, (Bj0)*(n) is a trivial fibration. Since (Bjn)*() is a successive p*
*ullback_
of (Bj0)*(n) it is trivial as well. *
* __
To end the proof of the theorem notice that BO(2) is, up to completion, the
homotopy colimit of the sequence : : :___BD2m ____BD2m+1 ____ : :.:Thus in
the category of fibrations and commutative diagrams between them, the fibration*
* is
the homotopy colimit of : :_:__(Bjm )*() ____(Bjm+1 )*() ____ : :.:But we ha*
*ve
seen that all these fibrations are trivial and furthermore, it is easy to obser*
*ve that the
product decomposition for the respective total spaces commute with the inclusio*
*ns.__
Thus is a trivial fibration and the proof is complete. *
* __
8. homotopy uniqueness of BS3{2r}
We are now ready to prove that any 2complete space with the mod2 cohomology
of BS3{2r} coincides with it up to homotopy, whence the second part of Theorem *
*1.1.
The proof boils down to showing that structure 3 of Proposition 2.2 is not re*
*alizable
as the cohomology of a space.
Proposition 8.1. There does not exist a space W with mod2 cohomology
H*(W ) ~=P [w2; w3; x3]=(x23);
with Sq1w2 = w3, Sq2w3 = w2w3 and with fir2x3 = w22.
23
Proof.Let W be a space with this cohomological structure. Thus H*(W ) is isomor
phic to H*(BSO(3)) H*(S3) as an algebra over the Steenrod algebra, with the
additional higher Bockstein. Let i : Z=2 ____SO(3) be the map sending the gene*
*ra
tor to the diagonal matrix diag(1; 1; 1). Then the centralizer of the image o*
*f i in
SO(3) is O(2). By [20]
TZ=2(H*(BSO(3)); Bi*) ~=H*(BO(2))
and furthermore, by [13],
Map(BZ=2; BSO(3)) ' BO(2):
By Lannes theory there is a map f : BZ=2 ____W inducing the composite
proj * Bi* *
H*(W ) ____H (BSO(3)) ____H (BZ=2):
Thus we have
(23) TZ=2(H*(W ); f*) ~=
TZ=2(H*(BSO(3)); Bi*) TZ=2(H*(S3); *) ~=H*(BO(2)) H*(S3)
as algebras over the Steenrod algebra. Furthermore, since there are no relatio*
*ns
among 1dimensional elements in the object computed above, it follows that the
calculation gives the cohomology of the mapping space
X = Map(BZ=2; W )f:
Also, the evaluation map e : X ____W induces the same map on cohomology as the
obvious inclusion
BO(2) x S3 ____BSO(3) x S3
and it follows that fir2(x3) = w22in H*(X) up to the usual indeterminacy.
Lemma 8.2. The space X defined above fits in a fibration
S3 ____X ____BO(2):
Proof.We will show that the second MoorePostnikov section of X is equivalent to
BO(2). Indeed, since X is a 2complete torsion space, its cohomological struct*
*ure
implies that ss1(X) ~= Z=2. It is easy to calculate the cohomology of the univ*
*ersal
cover, Y of X from the fibration
Y ____X ____BZ=2:
Thus H*(Y ) ~= P [w2] E[x3], with fir(x3) = w22. In particular the second int*
*egral
cohomology of Y is torsion free. Since X is a torsion space, inspection of th*
*e Sss
for the fibration above gives that the Z=2 action on H2(Y ; Z) is given by invo*
*lution.
Thus Z=2 acts on ss2(X) by involution. This determines the homotopy type of the
second MoorePostnikov section of X to be the 2completion of BO(2) and the res*
*ult_
follows. __
24
Lemma 8.2 gives an immediate contradiction. On one hand X is a torsion space,*
* but
on the other hand we have established above that H*(X) ~=H*(BO(2)) H*(S3) as
algebras over the Steenrod algebra. By Theorem 7.1 X must be homotopy equivalent
to a product S3 x BO(2), up to completion. This is of course impossible since b*
*oth
S3 and BO(2) have nontrivial rational cohomology. This completes the proof of *
*the_
proposition and with that the proof of Theorem 1.1. *
*__
9. Homotopy Uniqueness for the odd prime case
Let p be an odd prime and let X be a classifying space for S2n1{pr}. Thus
np1 and H*(X; Fp) is given by Equation (1). In particular this cohomology alg*
*ebra
appears as the cohomology of a finite group. Indeed let G denote the semi dire*
*ct
product of Z=pr by Z=n, where the last acts on Z=pr as a subgroup of its automo*
*rphism
group. Then H*fi(X; Fp) ~=H*fi(BG; Fp). Conversely,
Proposition 9.1. Let X be a space with H*fi(X; Fp) ~= H*fi(BG; Fp), where G ~=
Z=pr o Z=n. Then X ' BG up to pcompletion.
Sketch of proof.The argument in this proof is similar to that of Theorem 6.2, s*
*o we
indicate the proof without going into great detail.
Lannes theory gives that there is a map
f : BZ=p ____X;
*
*OE
such that f* coincides with the restriction map induced by the inclusion Z=p __*
*__G
on modp cohomology. Consider the mapping space Map (BZ=p; X)f.
We have to show how to recognize this as a classifying space for Z=pr. This *
*is
done along the lines of the proof of Proposition 5.4 using the fact that Z=pr i*
*s a
Tpgroup (Proposition 5.2). It follows that Map(BZ=p; X) ' BZ=pr: Moreover the
evaluation map ev* : H*(X) _____H*(Z=pr) coincides with the restriction to the
pSylow subgroup.
Now, one has an action of Z=n on Map (BZ=p; X)f, which preserves the componen*
*t,
since it does so if X is replaced by BG. Furthermore, one has that the evaluati*
*on map
is equivariant with respect to the trivial action of Z=n on X. Thus the evalua*
*tion
map extends to a map on the Borel construction
f : Map (BZ=p; X)f xZ=nEZ=n ____X;
and it is a triviality that this map induced a modp cohomology isomorphism. No*
*tice_
that the source is homotopy equivalent to BG. This completes the proof. *
* __
10. Appendix: Some homological algebra
In this appendix we review the techniques used in section 6 in order to compu*
*te
higher derived functors of the inverse limit functor. These techniques are esse*
*ntially
due to Aguade [1] and are adapted here to serve our purposes.
25
Lemma 10.1 (Aguade). Let G be a finite groups and let H < G be a subgroup. Let
R be a commutative ring with a unit. Let K be the R[G] module defined by the sh*
*ort
exact sequence
K ____R[G=H] ____R :
Then for any R[G]module M there is a long exact sequence
(24) 0 ! MG ! MH ! Hom G(K; M) ! H1(G; M) ! H1(H; M) ! : : :
: :!:Hi(H; M) ! ExtiG(K; M) ! Hi+1(G; M) ! Hi+1(H; M) ! : : :
For the proof the reader is referred to [1].
Consider the following categories:
__ oe______ ________
A = G2 _2oe_____oe_______0__________1G1
G2=H G1=H
and
__ ________
B = _0__________1G1
G1=H
Here G1 and_G2_are_groups_and H a common subgroup. The category_A has three
objects _0, _1and _2. The automorphism_groups of _iis given by Gi_for *
*i_=_1; 2
whereas the automorphism group of _0is trivial. The morphism set from _0to*
* _i,
i = 1; 2, admit a natural Gi action and are isomorphic_as Gisets_to Gi=H. The
category B is a full sub category of A with objects _0and _1.
Let Rmod denote the category of Rmodules. For a small category C let CRmod
denote the functor_category_from C to Rmod . Define the functor L 2 ARmod
which takes _0to R and _ito R[Gi=H]. The action of L on morphisms is taken *
*to
be the obvious one. __
For any other M 2 ARmod , let Mi denote M(_i), i = 0; 1; 2. Then
Hom ARmod(L; M) ~=Hom R (R; M0) ~=M0:
hence L is projective in ARmod . Let R 2 ARmod denote the constant functor w*
*ith
value R. Then there is an exact sequence in ARmod :
0 ____K ____L ____R ____0
where K(0) = 0 and K(i) = Ki = ker(R[Gi=H] ____R) for i = 1; 2. This gives an
exact sequence:
(25) 0 ! Hom ARmod(R; M) ! Hom ARmod(L; M) ! Hom ARmod(K; M) !
! Ext1ARmod(R; M) ! 0
and isomorphisms:
(26) ExtiARmod(K; M) ~=Exti+1ARmod(R; M) ; i 1
Lemma 10.2. M
ExtnARmod(K; M) ~= ExtnR[Gi](Ki; Mi)
i=1;2
26
Proof.Let #i : ARmod ! R[Gi]mod , i = 1; 2, be the forgetful functors defined
by #i(M) = Mi. There are functors going_the other direction si : R[Gi]mod !
ARmod , i = 1; 2, defined by si(N)(_j) = N if i = j and 0 if i 6= j. The fun*
*ctors si
are clearly exact and left adjoint to the forgetful functors. Thus
Hom ARmod(siN; M) ~=Hom R[Gi](N; Mi):
It follows that the functors #i preserve injectives; that is, if I 2 ARmod is*
* injective
then Ii is an injective R[Gi]module for i = 1; 2.
So, given_an object of ARmod , M and an injective resolution M ! I* we have
that I(_i)* is an injective resolution for Mi and this together with the defi*
*nition of
the functor K, given above, implies:
(27) Ext*ARmod(K; M) := H (Hom ARmod(K; I*))~=
i M j M
~=H Hom R[Gi]; (Ki; I(___i)*) ~= Ext *R[G(K ; M )*
* : ___
i]i i
i=1;2 i=1;2
For a small category C and a functor F 2 CRmod the nth cohomology group of*
* C
with coefficients in F is defined to be the group ExtnCRmod(R; F ), where R 2 *
*CRmod
is the constant functor. These cohomology groups are usually interpreted as the*
* higher
derived functors of the inverse limit of the functor F over the category C. We*
* are
now ready to prove the following
Proposition 10.3. There is a long exact sequence:
M
(28) 0 ! H0(A; M) ! M0 ! MHi=MGii! H1(A; M) !
i=1;2
M M
! H1(Gi; Mi) ! H1(H; Mi) ! H2(A; M) !
i=1;2 i=1;2
M M
! H2(Gi; Mi) ! H2(H; Mi) ! : : :
i=1;2 i=1;2
and for the cohomology of B:
(29) 0 ! H0(B; M) ! M0 ____MH1=MG11! H1(B; M) !
! H1(G1; M1) ! H1(H; M1) ! H2(B; M) ! H2(G1; M1) ! : : :
Proof.Start by adding the long exact sequences (24) for the pairs of groups (G1*
*; H)
and (G2; H) and use the isomorphisms (26) and Lemma 10.2 to obtain the long exa*
*ct
27
sequence of the bottom row in the diagram:
0____H0(A; M) ________M0 ________Hom ARmod(K;wM) ___
 ww
 ww
M ?
0 _______ MHi=MGii ___Hom ARmod(K; M) ___
i=1;2
_______H1(A; M) ___________0


M ? M
_____ H1(Gi; Mi) ___ H1(H; Mi) ___H2(A; M) ____: : :
i=1;2 i=1;2
Observe that the top row is a rewriting of 25 and the vertical arrows follows
because M0 ! Hom ARmod(K; M) ~= i=1;2Hom Gi(Ki; Mi) clearly factors through
i=1;2MHi=MGii. The sequence (28) follows by diagram chasing. The proof for (29)*
*_is
similar. __
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Departament de Matematiques, Univeritat Autonoma de Barcelona, E08193 Bel
laterra, Spain
Email address: broto@mat.uab.es
Dept. of Mathematics, Northwestern University, 2033 Sheridan Rd. Evanston, IL
60208
Email address: ran@math.nwu.edu