ON THE HOMOTOPY TYPE OF BG FOR CERTAIN FINITE
2-GROUPS G
CARLOS BROTO AND RAN LEVI
Abstract. We consider the homotopy type of classifying spaces BG, where *
*G is a
finite p-group and study the question, whether or not the mod p cohomolo*
*gy of BG, as
an algebra over the Steenrod algebra together with the associated Bockst*
*ein spectral
sequence determine the homotopy type of BG. This article is devoted to p*
*roducing
some families of finite 2-groups, where cohomological information determ*
*ines the
homotopy type of BG.
1. Introduction
Let G and H be finite p-groups, which agree on some of their cohomological in*
*vari-
ants. The question whether G and H are isomorphic groups has a negative answer
in general. For instance mod-2 cohomology algebra cannot differ between dihedr*
*al
2-groups of various orders, even if the action of the Steenrod algebra is taken*
* into
account. A similar fact is known for various metacyclic 2-groups, among which *
*one
counts the quaternion and semidihedral groups [15]. In a recent paper, I. Leary*
* exhib-
ited examples of distinct 3-groups having isomorphic integral cohomology rings *
*[12].
Further examples of distinct finite 2-groups, which mod-2 cohomology algebra ca*
*nnot
tell apart are given in [16]. However if one requires, in a sense that will be *
*made precise
later, that the given cohomological data on G includes its Bockstein spectral s*
*equence
then, although we are not able to prove the contrary, we are not aware of an ex*
*ample
of two distinct p-groups which cohomology cannot tell apart.
The question becomes significantly more complicated when one considers a fini*
*te p-
group G and a p-complete space X such that BG and X have the same cohomological
invariants and asks whether X and BG have the same homotopy type. This last
question is reduced to the previous one if one can show for a particular exampl*
*e that
X has the homotopy type of BH for "some" finite p-group H. One might thus wonder
to what extent does cohomological data determine isomorphism type for finite p-*
*groups
G, or more generally, the homotopy type of BG.
In this note we shall mainly be interested in the second question. We restric*
*t atten-
tion to the prime 2 and study some examples of finite 2-groups G, where the hom*
*otopy
___________
Date: March 28 1995.
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q5*
*2.
Key words and phrases. Classifying spaces, finite 2-groups, cohomology, Steen*
*rod squares, Bock-
stein spectral sequence.
C. Broto is partially supported by DGICYT grant PB94-0725.
R. Levi is supported by a DFG grant.
1
2 Carlos Broto and Ran Levi
type of BG is determined by its cohomology. Further examples where the results *
*are
not as conclusive are also given.
Unless otherwise specified all spaces are assumed to have the homotopy type of
a 2-complete CW complex. The cohomology of a space is considered as an object
in the category Kfi. It is, roughly speaking, the category of unstable algebra*
*s over
the Steenrod algebra, in which the higher Bockstein operators are being taken i*
*nto
account. It is immediate, for instance, that every finite abelian 2-group G giv*
*es rise to
a unique object of Kfi, characterizing BG up to homotopy and hence the group G,*
* up to
isomorphism. We say that two spaces X and Y are comparable if X and Y give rise*
* to
isomorphic objects in Kfi. We say that the homotopy type of a space X is determ*
*ined
by its cohomology if any space Y , comparable to X, is homotopy equivalent to i*
*t.
The first family of groups, we consider, is the dihedral 2-groups.
Theorem 1.1. Let D2n denote the dihedral group of order 2n. Then the homoto*
*py
type of BD2n is determined by its cohomology.
A considerably larger family of 2-groups, containing the dihedral and quatern*
*ion
groups of order 8, are the extra special 2-groups which we consider next.
Theorem 1.2. Let G denote an extra special 2-group. Then the homotopy type o*
*f BG
is determined by its cohomology. In fact in almost all cases, the homotopy type*
* of BG
for an extra special 2-group G is determined by the cohomology algebra of BG.
The examples we present next appear less favorable. Indeed a basic question i*
*n the
subject is whether in general cohomology forces at least the fundamental group *
*of a
space X, given that X is comparable to BG for some finite p-group G. In the fol*
*lowing
examples we were not able to determine a complete answer to this question. Thus
one might suspect that an example of a space X comparable to BG yet not homotopy
equivalent to it might arise from studying the groups G considered below.
Proposition 1.3. Let G = Q2n denote the generalized quaternion group of order *
*2n.
Then for n 5 the homotopy type of BG is determined by its cohomology. For n > 5
let X be a space comparable to BG. Then either ss1X ~=Q2k for some 4 k n - 2
or X ' BG.
The question of whether or not the homotopy type of BQ2n is determined by its
cohomology for n > 5 is reduced in section 5 below to understanding free action*
*s of the
group Q16 on certain simply-connected spaces Y such that Y is homotopy equivale*
*nt
to S3{2r}, the homotopy fibre of a degree 2r map on the sphere S3. In particul*
*ar
showing that such actions do not exist would imply that the homotopy type of BQ*
*2n
is determined by its cohomology for all n.
Proposition 1.4. Let G = SD2n denote the semidihedral group of order 2n. Let X*
* be
a nilpotent space, comparable to BG. Then ss1X ~=SD2k for some k 4.
In section 2 below we give a precise definition of what it means for a space *
*X to
be determined by its cohomology. If one considers cohomology as a functor from *
*the
homotopy category to the category Kfi, then the class of spaces which are deter*
*mined
On the homotopy type of BG for certain finite 2-groups G *
* 3
by cohomology is understood to be the subclass of the homotopy category, on whi*
*ch
the restriction of this functor is 1-1 when considered only as an operation on *
*objects.
Thus one might wonder whether this class is closed under simple operations such*
* as
products, wedges etc. It is hard to believe that one could produce successful a*
*nswers
to those questions in general. However in our more rigid context we are able to*
* prove
the following.
Proposition 1.5. Let G be a finite p-group such that the homotopy type of BG is
determined by cohomology. Then for any elementary abelian p-group E, the same
holds for B(G x E).
Next we restrict our attention to the isomorphism type of the groups under co*
*nsid-
eration.
Definition 1.6. Let Dp denote the class of finite p-groups which are determine*
*d among
finite p-groups by their mod-p cohomology as objects of Kfi. Namely if G 2 Dp a*
*nd ss is
another finite p-group, then comparability of BG and Bss implies that G is isom*
*orphic
to ss.
With this definition we are now able to state a corollary of our results.
Corollary 1.7. Finite abelian 2-groups, extra special 2-groups, dihedral quate*
*rnion
and semidihedral groups belong to the class D2. Furthermore, for any prime p, *
*the
class Dp is closed under direct products with finite elementary abelian p-group*
*s.
It is not hard to observe that in fact, the isomorphism type of G, for G an e*
*xtra
special p-group, p odd, is determined by the cohomology algebra of BG. This res*
*ult can
be extended to any p-group of nilpotency class 2 under certain favorable hypoth*
*eses.
The main task of this paper is the observation that cohomology as an object o*
*f the
category Kfican sometimes determine the homotopy type of BG for finite non-abel*
*ian
2-groups G and indeed, where considering it as an object in the categories K or*
* U of
unstable algebras or modules over the Steenrod algebra fails to do so.
The authors are grateful to the Centre de Recerca Matematica, Barcelona, and
the Mathematisches Institut der Universit"at Heidelberg for their kind hospital*
*ity and
support while part of this research was done. We also thank Jaume Aguade, Car-
les Casacuberta, Richard Kane, Ian Leary, Bj"orn Schuster and Sa"id Zarati for *
*useful
conversations.
2. Conventions and Terminology
All spaces considered are assumed to have the homotopy type of a CW complex a*
*nd
to be 2-complete in the sense of Bousfield and Kan [6]. We denote H*(-; F2) by *
*H*(-).
Let A denote the mod-2 Steenrod algebra. Let K denote the category of unstable
algebras over A. By "EMSS" we abbreviate the Eilenberg-Moore spectral sequence
and by "SSS" the Serre spectral sequence, both for mod-2 cohomology, unless oth*
*er
coefficients are specified. The Bockstein spectral sequence for mod-2 cohomolo*
*gy is
abbreviated by "BSS". The following definitions are included here in order to p*
*rovide
a systematic background for our discussion. The full generality of the concept*
*s we
4 Carlos Broto and Ran Levi
define will not be used in this note but they may become useful in a more gener*
*al
study of the problems suggested here
Definition 2.1. Let K be an unstable algebra over A. A BSS for K is a spectral*
* se-
quence {Ei(K); fii}1i=1of differential graded algebras where the differentials *
*have degree
one and such that (see [7, 11])
1. E1(K) = K and fi1 = Sq1 is the primary Bockstein operator.
2. If x 2 Kevenand x2 6= 0 in E2(K), then fi2(x2) = x Sq1x + Sq|x|Sq1x.
3. If x 2 Ei(K)evenand x2 6= 0 in Ei+1(K), i 2 then fii+1(x2) = x fii(x).
Let Kfidenote the category whose objects are pairs (K; {Ei(K); fii}1i=1), whe*
*re K is
an unstable A-algebra and {Ei(K); fii}1i=1is a BSS for K. We will frequently ab*
*breviate
an object (K; {Ei(K); fii}1i=1) of Kfiby Kfi. A morphism f : Kfi-! K0fiin Kfii*
*s a
family of morphisms {fi}1i=1, where f1 : K - ! K0 is a morphism of A-algebras a*
*nd
for each i 2, fi : Ei(K) -! Ei(K0) is a morphism of differential graded algebr*
*as,
which as a morphism of graded algebras, is induced by fi-1. The mod-2 cohomology
of a space X together with its BSS is a typical object of the category Kfithat *
*will be
denoted by H*fi(X). Thus mod-2 cohomology can be thought of as a functor from t*
*he
category of spaces to the category Kfi.
This category has been considered in [2] and [3]. Examples of spaces X with H*
**(X) ~=
P [x2] E[y3], realizing every BSS, which could possibly be associated to this *
*unstable
algebra by our set of axioms are discussed in [2]. Thus we regard (1), (2) and*
* (3)
above as a reasonable set of axioms for a BSS associated to an unstable algebra*
* over
the Steenrod algebra.
Definition 2.2. Two spaces X and Y are said to be comparable if H*fi(X) and H**
*fi(Y )
are isomorphic objects of the category Kfi. We say that X is determined by coho*
*mology
if given a space Y comparable to X, there is a homotopy equivalence X ' Y .
Definition 2.3. Let Kfibe an object in Kfi. Let K be the underlying unstable a*
*lgebra.
We say that Kfiis weakly generated by x1; . .;.xn 2 K if any endomorphism f of *
*Kfi,
such that the restriction of f1 to the vector subspace of K spanned by x1; . .;*
*.xn is an
isomorphism, is an isomorphism in Kfi. In that case we shall say that x1; . .;.*
*xn is a
set of generators for Kfiin Kfi.
Throughout this note we shall be calculating the EMSS of certain fibrations s*
*uch
that the base space has a finite 2-group as a fundamental group. Thus we shall*
* use
the fact that the action of a finite 2-group on a finite dimensional F2-vector *
*space is
always nilpotent and so the EMSS is granted to converge to the right target by *
*[8].
Next we point out that in order to grant that a mod-2 cohomology isomorphism
through dimension 2 induces an isomorphism on fundamental groups, one might be
tempted to assume that all spaces considered are nilpotent. This assumption is*
* not
necessary in most cases. Indeed [6, I.6.2] gives a slightly more restricted re*
*sult, still
suitable for our purposes, without the nilpotency assumption. We will have to a*
*ssume
nilpotency however in section 5 below where this lemma of Bousfield and Kan can*
*not
be applied. A final remark is that in our examples the 2-completeness assumptio*
*n of
On the homotopy type of BG for certain finite 2-groups G *
* 5
the spaces X under consideration is not essential to the construction of maps f*
*rom X to
suitable classifying spaces BG. One can define the concept of comparability dro*
*pping
the 2-completeness assumption and then, assuming that X is comparable to BG for
some finite 2-group G, one might be able to construct a map from X to BG and th*
*en
examine its cohomological properties. If the map happens to induce an isomorphi*
*sm
on cohomology then our assertions would hold for the 2-completion of X.
There are odd primary analogues of all concepts we have defined in this secti*
*on.
Only definition 2.1 need the obvious reformulation at odd primes.
Definition 2.4. Let p be an odd prime and Ap the mod-p Steenrod algebra. Let K
be an unstable algebra over Ap. A BSS for K is a spectral sequence {Ei(K); fii}*
*1i=1of
differential graded algebras where the differentials have degree one and such t*
*hat (see
[7, 11])
1. E1(K) = K and fi1 = fi is the primary Bockstein operator.
2. If x 2 Ei(K)evenand xp 6= 0 in Ei+1(K), i 1 then fii+1(xp) = xp-1fii(x).
3. The Dihedral Groups
This section is devoted to the proof of Theorem 1.1. The dihedral group of or*
*der 8 is
a particular case of an extra special 2-group. However, the calculation here ap*
*plies to
dihedral 2-groups of arbitrary order and is different in nature from the one pe*
*rformed
in the proof of Theorem 1.2 below.
Let D2n denote a dihedral group of order 2n.
n-1 2 -1
D2n = :
Recall from [10] for n 3
H*(BD2n) ~=P [x1; y1; w2]=(x2 + xy);
where the degrees are given by the subscripts. The action of the Steenrod alge*
*bra
is given by Sq1(w) = wy and all the other operations are determined by the regu*
*lar
axioms. The Bockstein spectral sequence is determined by the requirement that
E2 ~=E3 ~=. .~.=En-1 ~=P [w2] E[xw]
and fin-1(xw) = w2. Indeed D2n has a normal cyclic subgroup of order 2n-1, whi*
*ch
detects this higher Bockstein. This calculation is completely routine using the*
* SSS for
the respective group extension.
Notice that the algebra generators for H*(BD2n) are the same as the generator*
*s in
Kfi. Thus for the proof of Theorem 1.1, it suffices to show that any space X co*
*mparable
to BD2n admits a map X -! BD2n, inducing an isomorphism on cohomology up to
dimension 2.
Let X be a space comparable to BD2n. Let D4 denote an elementary abelian 2-
group of rank 2. There is a map OE2 : X -! BD4, classifying the classes x; y 2 *
*H1(X).
Consider the tower of principal fibrations
ssn-1 ssn-2 ss3 ss2
. .-.! BD2n -! BD2n-1- ! . . .-! BD8 -! BD4: (3.*
*1)
6 Carlos Broto and Ran Levi
Each projection ssi corresponds to the central extension
0 -! Z=2 -! D2i+1-! D2i- ! 1
classified by the w 2 H2(BD2i) if i 3 and by x2 + xy 2 H2(BD4) if i = 2.
Lemma 3.1. The map OE2 : X -! BD4 factors through BD2n.
Proof. Clearly the composite
OE2 x2+xy
X -! BD4 - ! K(Z=2; 2)
is null-homotopic. Hence OE2 factors through BD8. Thus we must show that the
composite
OEn-k w
X -! BD2n-k- ! K(Z=2; 2)
is null-homotopic for k 1.
Indeed it is easy to see by applying naturality of Sq1 that OE*n-k(w) = aw fo*
*r some
a 2 F2. If a = 1 then OE*n-kis an isomorphism, contradicting the existence of d*
*ifferent_
Bockstein operators in H*(X) and H*(BD2n-k). Thus OE*n-k(w) = 0 as required. *
* |__|
To complete the proof of Theorem 1.1 we must show that OE*nw = w. Suppose
OE*nw = 0. An easy EMSS calculation for the fibration
j OEn
F -! X -! BD2n
shows that
H*(F ) ~=P [v2] E[ff1];
as a P [v]-module, where v = j*(w) 2 H2(X).
Since fin-1(xw) = w2 in H*fi(X) and j*(x) = 0, it follows by naturality of th*
*e BSS
that fir(ffv) = v2 in H*fi(F ) for some r < n - 1, which in turn easily implies*
* that
fir(ff) = v.
This determines the homotopy type of F to be that of BZ=2r. Hence X ' BH,
where H fits in a group extension of the form
0 -! Z=2r -! H -! BD2n -! 1:
We argue that H ~=D2n+r and having done so, we obtain a contradiction to our
hypothesis, for if X ~=BD2n+r, r 1 then H*(X) gives rise to a different object*
* of Kfi
than the one assumed.
Indeed, consider the fibration
OEn
BZ=2r -! X -! BD2n:
Since OE*nw = 0, it follows that OEn lifts to a map OEn+1 : X -! BD2n+1. One ob*
*serves
immediately that the homotopy fibre of OEn+1 is BZ=2r-1. The same argument as
above implies that OE*n+1w = aw for some a 2 F2 . But if a = 1 then OEn+1 is an
equivalence, which would imply r = 1 and we get the desired contradiction. Thu*
*s if
r > 1 then a = 0 and one proceeds inductively. Notice that the process must sto*
*p for
OEn+r : X -! BD2n+rhas to be an equivalence. This completes the proof of Theorem
1.1.
On the homotopy type of BG for certain finite 2-groups G *
* 7
4. The extra special 2-Groups
An extra special 2-group is a central extension of an elementary abelian 2-gr*
*oup by
a cyclic group of order 2. The cohomology of extra special 2-groups was compute*
*d by
Quillen [14]. We start by recalling Quillen's result. A good general referenc*
*e for the
subject is [4].
Let G be an extra special 2-group. Thus G fits in a central extension
0 -! Z=2 -! G -! E -! 0: (4.*
*1)
where E is an elementary abelian 2-group of rank n. Write
H*(BE) ~=P [x1; x2; . .;.xn]:
Then the extension 4.1 is classified by some quadratic form q 2 H2(BE).
Theorem 4.1 (Quillen). Let G be as above. Then
H*(BG) ~=P [x1; x2; . .;.xn]=(q0; q1; . .;.qh) P [i]
i-1
with q0 = q, q1 = Sq1q0 and qi = Sq2 qi-1. The elements x1; x2; . .;.xn are de*
*gree 1
elements inflated from H1(BE). The number 2h can be interpreted as the index o*
*f a
maximal elementary abelian subgroup of G. The element i has degree 2h and can *
*be
chosen to be any element of this degree, restricting non-trivially to the kerne*
*l in the
extension 4.1 above.
The following theorem deals with the Bockstein spectral sequence for H*(BG) a*
*nd
is proven in [4].
Theorem 4.2. Let G be an extra special 2-group with h 2. Then the E2 page o*
*f the
Bockstein spectral sequence for H*(BG) is given by
E*2= E[fl1; fl3; fl4. . .; flh] P [i]
with |fl1| = 3, |flj| = 2j-1 for j 3 and |i| = 2h.
For 3 r h we have,
E*r= E[fl1fl3. .f.lr; flr+1; . .;.flh] P [i]:
Finally, E*h+1= E*hand E*r= F2 for r h + 2.
The differentials are given by fi2(fl1) = fl3, fir(fl1fl3. .f.lr) = flr+1, fi*
*r(fli) = 0 for
r < i, fir(i) = 0 for all r and fih+1(fl1fl3. .f.lh) = i.
Notice that the case h = 0 corresponds to elementary abelian 2-groups and h =*
* 1 to
Z=4 x E or D8 x E where E is elementary abelian. The first case is abelian and *
*thus
obvious. We will remark about D8 x E later.
Lemma 4.3. Let G be an extra special 2-group with h 2. Then H*fi(BG) is wea*
*kly
generated by H1(BG).
Proof. Let Kfidenote H*fi(BG). Let f be an endomorphism of Kfi, such that the
corresponding endomorphism f1 of K, restricted to dimension 1, is an isomorphis*
*m.
We must show that f is an isomorphism.
8 Carlos Broto and Ran Levi
Assume that G is an extra special 2-group, given by an extension on the form *
*(4.1),
whose cohomology is given by Theorems 4.1 and 4.2. Without loss of generality *
*we
may assume that
f1(xi) = xi for all i. (4.*
*2)
Indeed, f1 restricted to dimension 1 being an isomorphism, it is obviously an i*
*somor-
phism when restricted to P [x1; x2; . .;.xn]=(q0; q1; . .;.qh). In particular,*
* the unique
extension of f1 to P [x1; x2; . .;.xn] ~=H*(BE) preserves the quadratic form q *
*that clas-
sifies the extension (4.1). Regarding f1, restricted to dimesnion 1, as the dua*
*l of an au-
tomorphism "f1of E, one sees that "f1extends to an automorphism "f1:G ! G. The *
*in-
duced map on cohomology "f*1is an isomorphism with the property that "f*1(xi) =*
* f1(xi).
Hence the map f0 := (f"-11)* O f is an endomorphism of H*fi(BG), satisfying con*
*dition
(4.2) above.
So, assuming (4.2), it follows easily that fr maps the exterior generators in*
* E*r(K)
to themselves for all r. Thus we get at the h + 1 page of the spectral sequence
fh+1(i) = fh+1fih+1(fl1fl3. .f.lh) = fih+1fh+1(fl1fl3. .f.lh) = i:
On the other hand fj is induced by fj-1. Thus we must have f1(i) = i + p; for s*
*ome
polynomial p = p(x1; . .;.xn). By Theorem 4.1 the self map of K which takes xi*
* to __
itself for every i and i to i + p is an isomorphism, hence the proof is complet*
*e. |__|
We are now ready to prove Theorem 1.2. Assume that X is a space with
H*(X) ~=H*(BG)
as algebras, where G is an extra special 2-group of order 2n+1 classified by th*
*e quadratic
form q and with maximal elementary abelian 2-subgroup of index 2h. We will show:
1. If h 3, X ' BG.
2. if h 1 and H*fi(X) ~=H*fi(BG), then X ' BG.
We start by proving (1). Suppose h 3 and let X be a space such that H*(X) ~=
H*(BG) as algebras. Let G be given by the central extension 4.1. The argument a*
*bove
p 1
yields that any map X - ! BE inducing an isomorphism on H (-), lifts to a map
* h
X -! BG such that is an isomorphism up to dimension 2 - 1. Furthermore, it*
* is
possible to choose such that under a suitable change of basis, *(i) = ffli f*
*or some
ffl 2 F2. If ffl = 1 then * is an isomorphism. Thus assume that ffl = 0.
The EMSS for the fibration
F -! X -! BG
is easily shown to collapse at E2 and
E2 ~=E1 ~= P [v] E[ff]
with |v| = 2h and |ff| = 2h - 1, as a P [v]-module. By [1] it follows that eith*
*er |v| = 2
or |v| = 4. Hence we are in the case h 2, contradicting our hypothesis.
On the homotopy type of BG for certain finite 2-groups G *
* 9
Next we turn to the proof of (2). Consider the case h 2. Indeed, there is a *
*map
p ss
X -! BE corresponding to the projection BG -! BE. Since the composite
p q
X -! BE -! K(Z=2; 2)
is null-homotopic, there is a map X - ! BG, lifting p. Moreover, since p indu*
*ces
an isomorphism in 1-dimensional cohomology and h 2, it follows that any such l*
*ift
induces an isomorphism in cohomology up to dimension 2h - 1 3. The result foll*
*ows
by Lemma 4.3.
For h = 1 it remains to consider the case, where G ~= D8 x E with E elementary
abelian. This case is a consequence of Theorem 1.1 proved in the previous secti*
*on and
Proposition 1.5 which is proved in section 7.
This completes the proof of Theorem 1.2.
Remark 4.4. Notice that the conclusion X ' BG is not true for h = 1 unless
H*fi(X) ~= H*fi(BG), for dihedral groups of different orders produce counter ex*
*amples.
We are not aware of a suitable example for h = 2.
5. The Quaternion Groups Q2n
The quaternion groups are given as follows
n-2 2 2n-1 -1 -1
Q2n = :
Recall the cohomological structure of BQ2n.
H*(BQ2n) ~=P [x1; y1; v4]=(ff; fi)
where ff = x2 + xy + y2, fi = x2y + y2x if n = 3 and ff = x2 + xy, fi = y3 if n*
* 4.
The Bockstein spectral sequence is determined by the regular axioms for Sq1 and*
* the
requirement that fin(c) = v, where c 2 H3(BQ2n) is the unique generator. Notice*
* that
the n - th Bockstein is defined here with no indeterminacy and that the class c*
* is a
product of 1-dimensional generators. Thus H*(BQ2n) is weakly generated by x and*
* y
as an object of Kfi.
The central extension
0 -! Z=2 -! Q2n -! D2n-1- ! 0
is characterized by the class q2 2 H*(BD2n-1), where q2 = x2 + xy + y2 if n = 3*
* and
q2 = w + y2 if n 4 (see for example [9]).
The group Q8 is an extra special 2-group. Thus we will assume n 4.
Proposition 5.1. Suppose X is comparable to BQ2n for some n 4. Then ss1X is
isomorphic to Q2k for some 4 k n. In particular if n = k then X ' BQ2n.
Proof. There is a map OE2 : X - ! BD4, where D4 = Z=2 Z=2, classifying x; y 2
H*(X). Consider the tower of principal fibrations
ssn-1 ssn-2 ss3 ss2
. .-.! BD2n -! BD2n-1- ! . . .-! BD8 -! BD4: (5.*
*1)
Recall that the map ss2 is classified by the class x2+ xy 2 H2(BD4) and that ea*
*ch map
ssi for i 3 is classified by w 2 H2(BD2i).
10 Carlos Broto and Ran Levi
Since OE*2(x2 + xy) = 0, OE2 lifts to OE3 : X - ! BD8 inducing isomorphism in*
* 1-
dimensional cohomology. Suppose that for some i 3 there is a map OEi: X -! BD2i
factoring OE2. Then OE*iis an isomorphism in dimension 1 and by applying natura*
*lity of
Sq1 one easily observes that OE*i(w) = by2 for some b 2 F2. If b = 0 then OEi l*
*ifts further
to a map OEi+1: X -! BD2i+1and the argument may be iterated. On the other hand
if for some k 4, OE*k-1(w) = y2 then OE*k-1(w + y2) = 0 and the composite
OEk-1 w+y2
X -! BD2k-1- ! K(Z=2; 2)
is null-homotopic. The class w+y2 is known to be the extension class for the qu*
*aternion
group Q2n (see for instance [9]). Thus OEk-1 lifts to a map k: X ! BQ2k, induci*
*ng an
isomorphism on cohomology in dimension 1 through 3. Hence in this case ss1(X) ~*
*=Q2k
for some k 4 ([6, I.6.2].) It remains to show that the lifting process actuall*
*y finishes
at some k n. Thus suppose that it doesn't. Then OEi is defined for every i 2 *
*and
OE*i(w) = 0. Define Fi to be the homotopy fibre of OEi, thus obtaining a seque*
*nce of
fibrations
. . ._____-Fk-1 ______-. ._._____-Fi _______-. . .______-F3 _______-F2
| | | |
| | | |
| |j | |
| | | |
| | | |
= |? = = |? = = |? = |?
. . .______-X _______-. ._._____-X _______-. . .______-X ________X-
| | | |
| | | |
|OE |OE |OE |OE
| k-1 | i | 3 | 2
| | | |
|? |? |? |?
. . .____BD2k-1- _____-. ._.____-BD2i ______.-. ._____BD8- _____-BD4
An EMSS calculation yields
H*Fi~= E[a1; a2] P [v4]
as P [v4]-modules, where v4 is the restriction of v4 2 H4(X). Moreover by natu*
*ral-
ity of the EMSS, the map Fi+1 -! Fi induces an isomorphism in even dimensional
cohomology and it is trivial in odd dimensions.
Now consider each fibration separately. Since fin(x3) = v in H*fi(X), we obta*
*in that
firi(a1a2) = v in H*(Fi) for some ri < n and therefore fisi(a1) = a2 for some s*
*i ri.
If si = ri for some i then Fi is comparable to BZ=2ri and thus F ' BZ=2ri. In
that case a similar argument to the one presented in the proof of Theorem 1.1 s*
*hows
that X ' BQ2i+ri, which in turn implies that i + ri = n by considering the acti*
*on of
the higher Bockstein operators, distinguishing between classifying spaces of di*
*fferent
quaternion groups. Assume now si < ri for all i. Comparing cohomology of Fi+1 a*
*nd
Fi, we observe that ri+1 < ri. Hence, at the (k - 1)-st step we have a sequenc*
*e of
inequalities:
1 sk-1 < rk-1 < . .<.r3 < r2 < n
*
* __
from which it follows that the lifting process has to stop at some k, 4 k n. *
* |__|
Corollary 5.2. The homotopy type of BQ16 is determined by cohomology.
On the homotopy type of BG for certain finite 2-groups G *
* 11
Proposition 5.3. Let X be a space comparable to BQ2n with ss1X ~= Q2k for some
4 < k < n, and let X" be its universal cover, then
(a) H*fi(X") = E[z3] P [v4] with fin-k(z3) = v4
(b) The action of Q2k on H*(X"; Z) is trivial.
(c) The quotient space X"=Q2k-1is comparable to BQ2n-1.
Proof. The EMSS for X" -! X -! BQ2k shows that
H*X" = E[z3] P [v4]
where v4 is the restriction of v4 2 H4(X). Furthermore, by naturality of the B*
*SS,
fit(z3) = v4 for some t < n . It follows that "Xis of finite type [2, prop 5.7]*
* and that the
rings H*(X"; Z) and Z=2t[v4] have isomorphic augmentation ideals. The generato*
*r v4
reduces to v4 mod 2. We proceed in proving (c). The proofs of (a) and (b) will *
*follow.
There is a homomorphism Q2n -! Z=2 given by sending x to the generator of Z=2
and y to 1. The kernel of this homomorphism is a quaternion subgroup of Q2n of *
*order
2n-1. Let X - ff!BQ2k denote the map classifying the universal cover for X. T*
*hen
there is a commutative diagram
"X _________-"X=
| |
| |
| |
| |
| |
| |
|? |?
(A) X1 _________-X ________BZ=2-
| | |
| | |
| | |
|ff1 |ff |=
| | |
| | |
|? |? x |?
(B) BQ2k-1 _____-BQ2k ______-BZ=2
Calculating the EMSS for (A) and (B) and using naturality, it follows that H*(X*
*1) ~=
H*(BQ2n-1) as an algebra. Obviously ss1X1 ~=Q2k-1. Moreover by naturality of *
*the
BSS, fir(x3) = v4 in H*(X1), for some r n - 1. Thus X1 is comparable to BQ2r.
Next consider the integral SSS for (A).
Ep;q2~=Hp(Z=2; Hq(X1; Z)):
In particular Ep;22~=Hp(Z=2; Z=2 Z=2) and by inspection of the spectral sequen*
*ce it
follows that the action of Z=2 is the twisting action. Thus E0;22~=Z=2 and Ep;*
*22= 0
if p 1. Consequently E0;42~=(H4(X1; Z))Z=2 ~=Z=2n-1 and since H4(X1; Z) ~=Z=2r,
r n - 1, we get both that r = n - 1 and that the action is trivial. This compl*
*etes
the proof of part (c).
12 Carlos Broto and Ran Levi
Consider the commutative diagram
X" ___________X"-=
| |
| |
| |
|? |?
(C) X2 __________-X1 _________BZ=2-
| | |
|ff2 |ff1 |=
| | |
|? |? y |?
(D) BZ=2k-2 _____-BQ2k-1 _______BZ=2-
Again the EMSS is employed to compute H*(X2). By naturality we get
H*(X2) ~=P [v4] E[x1] P [u2]=(u2)
where v4 is restriction of the corresponding class in H*(X1) and x1 and u2 are *
*inflated
from H*(BZ=2k-1) via ff*2. The structure given above is easily seen to be both*
* the
P [v4]-module structure and the H*(BZ=2k-1) ~=E[x1]P [u2]-module structure. Usi*
*ng
naturality of the BSS, it follows that fik-1(x) = u and fis(xu) = v, for some s*
* < n - 1.
We proceed in computing the Bockstein spectral sequence for H*(X2). Consider SSS
with integral coefficients for the fibration (C) above.
Ep;q2~=Hp(BZ=2; Hq(X2; Z) ) Hp+q(X1; Z)
In total degree two we have
E2;02= H2(BZ=2; H0(X2; Z)) ~=Z=2
E0;22= H0(BZ=2; H2(X2; Z)) ~=H0(BZ=2; Z=2k-1) ~=(Z=2k-1)Z=2:
Since H2(X1; Z) ~= Z=2 Z=2, it follows that Z=2 acts non trivially on Z=2k-1 a*
*nd
E0;22~=Z=2.
In total degrees three and four we have:
E1;22= H1(BZ=2; H2(X2; Z)) = H1(BZ=2; Z=2k-1)
E4;02= H4(BZ=2; H0(X2; Z)) = H4(BZ=2; Z)
E2;22= H2(BZ=2; H2(X2; Z)) = H2(BZ=2; Z=2k-1)
E0;42= H0(BZ=2; H4(X2; Z)) = H0(Z=2; Z=2r) = (Z=2s)Z=2
Since H3(X1; Z) = 0 it follows that E1;23~=Z=2 and d3 : E1;23-! E4;03is an isom*
*or-
phism.
Next notice that E0;43= (Z=2s)Z=2, where s < n - 1 and that 2 annihilates the
cohomology of BZ=2 with any coefficients. Since H4(X1; Z) ~=Z=2n-1, it follows *
*that
E0;43= Z=2n-2 and E2;23= Z=2. Thus s = n - 2 and the action of Z=2 on H4(X2; Z)*
* is
trivial. This completes the calculation of the BSS for H*(X2).
Now, we are ready to compute the BSS for H*(X"). Again we use the SSS with
integral coefficients for the maps ff, ff1 and ff2 in the diagrams above. In to*
*tal degree
four, these spectral sequences give the following extensions:
ff: Z=2k ! Z=2n ! (Z=2t)Q2k.
On the homotopy type of BG for certain finite 2-groups G *
* 13
ff1: Z=2k-1 ! Z=2n-1 ! (Z=2t)Q2k-1.
k-2
ff2: Z=2k-2 ! Z=2n-2 ! (Z=2t)Z=2 .
It follows that
k-2 n-k
(Z=2t)Q2k ~=(Z=2t)Q2k-1~= (Z=2t)Z=2 ~=Z=2 ;
where the groups Z=2k-2 Q2k-1operate on Z=2tas subgroups of Q2k. But Aut(Z=2t)
is abelian and then the action of Q2k factors through its abelianization: (Q2k*
*)ab ~=
Z=2 Z=2. Since Z=2k-2, generated by x2 2 Q2k, is contained in the kernel of t*
*he
abelianization, its action is trivial. But the action of Q2k on Z=2t has the s*
*ame in-
variants as the action of Z=2k-2. Hence t = n - k and Q2k operates trivially. *
* This_
completes the proof of (a) and (b) *
* |__|
Corollary 5.4. Let X be comparable to BQ2n with ss1X ~= Q2k for some 4 < k < n.
Then for every 1 j k - 4, there exists a space Xj comparable to BQ2n-j with
ss1Xj ~=Q2k-j.
Corollary 5.5. Let X be comparable to BQ2n for some n 5. Then ss1X 6= Q2n-1.
Consequently the homotopy type of BQ32 is determined by its cohomology.
Proof. By Proposition 5.3 assuming the contrary implies that the universal cove*
*r X" of
X satisfies
H*(X") ~=P [t4] E[x3]
with Sq1x = t. By an argument due to Aguade [1], a space with such cohomology d*
*oes_
not exist. *
* |__|
Finally we remark that proving that the homotopy type of BQ2n is determined by
cohomology for all n amounts to showing the non-existence of a faithful action *
*of Q16
on spaces of the form Yr, r 2, where Yr is a simply-connected 2-local space su*
*ch that
H*(Yr) ~=P [t4] E[x3]
with firx = t. For r = 2 it is not clear whether or not a space such as Y2 exis*
*ts. Of course
non-existence of Y2 would imply that if X is comparable to BQ2n then ss1X 6= Q2*
*n-2
and hence the homotopy type of BQ64 could be determined by its cohomology. For
r 3 spaces of the form Yr arise from the 2-completion of BSL2(Fq) for q odd. H*
*owever
there is no suitable free action of Q16 on BSL2(Fq) because in this case the or*
*bit space
would be the classifying space of a finite group and comparable to BQ2n for som*
*e n.
This implies a contradiction for groups with periodic mod-2 cohomology are comp*
*letely
classified. It might still be the case that the 2-completion BSL2(Fq)^2admits *
*a free
Q16-action. Nonetheless we find it reasonable to conjecture that the homotopy t*
*ype of
BQ2n is determined by cohomology for all n.
6.The Semidihedral Groups
The semidihedral group SD2n is given as follows
n-1 2 2n-2-1
SD2n = :
14 Carlos Broto and Ran Levi
Recall from [9]
H*(BSD2n) ~=P [x; y; u; t]=(x2 + xy; xu; x3; u2 + (x2 + y2)t)
where |x| = |y| = 1, |u| = 3 and |t| = 4. The action of the Steenrod algebra is*
* given
i 2i
by Sq1u = Sq1t = 0, Sq2u = (x + y)(t + yu), Sq2t = u2 and Sq2 u = Sq t = 0 for
i 2. The Bockstein spectral sequence is determined by the regular axioms for *
*Sq1
together with the requirement that fin-1u = t in En-1. The calculation of fin-1*
*u easily
follows by observing that the unique subgroup of SD2n, isomorphic to Q2n-1, det*
*ects
the classes u and t in mod-2 cohomology [9]. Thus H*(BSD2n) is weakly generated*
* by
x, y and u.
Consider the central extension
0 -! Z=2 -! SD2n -! D2n-1- ! 1 (6.*
*1)
characterized by the class w + x2 2 H2(BD2n-1). Consider the tower of principal
fibrations
ssn-1 ssn-2 ss3 ss2
. . .-! BD2n -! BD2n-1- ! . . .-! BD8 -! BD4 (6.*
*2)
Define BD1 as the inverse limit of the tower 6.2. It turns out to be an extens*
*ion
BZ^2! BD1 ! BD4 (6.*
*3)
and by comparison with any other BZ=2n-2 ! BD2n ! BD4 it follows that the only
possible differential in the SSS for 6.3 is d2(u) = x2 + xy, where u is the gen*
*erator of
the mod 2 cohomology of BZ^2, and therefore
H*(BD1 ) ~=P [x; y]=(x2 + xy)
Let X be a space comparable to BSD2n. There is an obvious map OE2: X ! BD4
classifying the classes x; y 2 H1(X). Lifting OE2 through the tower 6.2 and 6.*
*1 we
will obtain a map f :X ! BG, where G is either semidihedral or D1 , such that t*
*he
induced map in mod 2 cohomology is an isomorphism in dimensions one and two.
Recall from [5] that a map f is called an HZ=2-equivalence whenever the induc*
*ed
map on mod 2 cohomology is an isomorphism in dimension 1 and a monomorphism in
dimension 2 . There exists a localization theory EZ=2 on the category of groups*
* that
invert those maps. Finite 2-groups and inverse limits of finite 2-groups are ex*
*amples
of EZ=2-local groups.
Proposition 6.1. Let X be a space comparable to BSD2n. Then X is HZ=2-equivale*
*nt
to either BSD2k for some k or BD1 .
Proof. The map ss2 is classified by x2 + xy 2 H2(BD4). But x2 + xy = 0 in H2(X).
Hence there is a map OE3 : X - ! BD8 with ss2OE3 = OE2. Assume that there is a *
*map
OEk : X -! BD2k lifting OE3. Clearly OE*kx = x, OE*ky = y. Using naturality of *
*Sq1 we find
that OE*kw = ax2 for some a 2 F2.
If a = 1 then the composite
OEk w+x2
X -! BDk -! K(Z=2; 2)
On the homotopy type of BG for certain finite 2-groups G *
* 15
is null homotopic. Thus OEk lifts to a map
k+1 : X -! BSD2k+1;
which induces an isomorphism on cohomology up to dimension 2. Thus it is an HZ=*
*2-
equivalence. On the other hand if a = 0 then OEk can be lifted further. If OE*k*
*w = 0 for
all k we get a map X - ! BD1 inducing as well isomorphism on cohomology up to *
* __
dimension 2, which is therefore an HZ=2-equivalence. *
* |__|
Lemma 6.2. Let f :X ! BG be an HZ=2-equivalence where G and its subgroups are
HZ=2-local. Then the induced map f# :ss1X ! G is an epimorphism. Moreover, if G
is a finite 2-group then ss1X ~=G.
Proof. The induced map f# :ss1X ! G is an HZ=2-equivalence as well, and therefo*
*re
EZ=2(ss1X) ~=G. Let H denote the image of f# . Since H is EZ=2-local by hypothe*
*sis,
the above isomorphism factors through H and therefore H = G; that is, f# is an
epimorphism.
Assume now, that G is a finite 2-group. In particular the previous hypotheses*
* are
satisfied. Let F be the fibre of f and
E*;*2~=H*(BG; H*(F ; F2)) =) H*(X; F2)
the SSS for the fibration F ! X ! BG. Since f induces isomorphism on mod 2
cohomology in dimension one and a monomorphism in dimension two, every element
of E1;02and E2;02must be a permanent cycle and E0;12~=H1(F ; F2)G must be trivi*
*al.
Being this last the group of invariants of an F2 vector space by the action of *
*a finite 2-
group, it is trivial if and only if H1(F ; F2) is itself trivial. But this mean*
*s that_ss1F = 0
because F is 2-complete. It follows that ss1X ~=G. *
* |__|
Corollary 6.3. Let X be a nilpotent space comparable to BSD2n. Then ss1X is
isomorphic to SD2k for some k. Moreover, the natural map k : X -! BSD2k
classifying the universal cover for X is either a homotopy equivalence or sati*
*sfies
*k(u) = *k(t) = 0.
Proof. In Proposition 6.1 we constructed an HZ=2-equivalence f :X ! BG, where
G is either semidihedral of D1 . It follows from lemma 6.2 that the induced map
ss1X ! G is an epimorphism. Then, G cannot be D1 for in that case X would not *
*be
nilpotent. Hence G is semidihedral, which is a finite 2-group, so again by lemm*
*a 6.2 f
induces ss1X ~=G.
Finally one uses the action of the higher Bockstein operators as well as the *
*action
of the Steenrod algebra in order to show that k could only have the cohomologi*
*cal __
effect described in the proposition, in case it fails to be a homotopy equivale*
*nce. |__|
7. products with elementary abelian groups
This section is motivated by the will to understand our subject matter from a*
* differ-
ent point of view. Namely, we have exhibited examples of finite non-abelian 2-g*
*roups
G such that the homotopy type of BG is determined by its cohomology. A natural
question at that point is whether one can construct new examples of groups G wi*
*th this
16 Carlos Broto and Ran Levi
property out of old ones. More precisely, given two finite 2-groups (or more ge*
*nerally
p-groups) G and H such that the homotopy type of BG and BH is determined by
cohomology, can one perform certain group theoretic constructions on G and H, t*
*hus
obtaining a new group K, possibly infinite, whose classifying space BK has the *
*same
property. One might have in mind constructions such as Cartesian products, amal*
*ga-
mated products, fibre products etc. The simplest possible construction is of co*
*urse the
Cartesian product. Thus one might wonder whether the homotopy type of B(H x G)
is determined by its cohomology if the same holds for each one of the factors.
Proposition 1.5 handles the simplest possible version of this question. We no*
*w turn
into its proof. Assume that p is any prime. H*fiis thus assumed to be mod-p coh*
*omology.
Let E be an elementary abelian p-group and suppose that X is a space comparable*
* to
B(G x E), that is
H*fi(X) ~=H*fi(B(G x E)) ~=H*fi(BG) H*fi(BE):
We can obtain a classifying map for one dimensional classes
OE: X ! BE
inducing the obvious inclusion H*(BE) ,! H*(BG) H*(BE). Let F be the fibre
of OE. Since H*(X) is free as a module over H*(BE), the EMSS for this fibratio*
*n is
concentrated in the zero vertical line and hence collapses with no extension pr*
*oblems.
Thus H*(F ) ~= H*(BG). By naturality of the BSS and the structure of H*(X) it
follows at once that F is comparable to BG. By hypothesis, G is determined by
cohomology, hence F ' BG and therefore X is the classifying space of a p-group *
*K,
given as an extension
1 ! G ! K ! E ! 1: (7.*
*1)
Central elements of order p in K are cohomologically detectable by means of t*
*he
Lannes' T functor [13]. Since H*(BK) ~=H*(B(G x E)) as algebras over the Steenr*
*od
algebra, the central elements of order p in K and GxE are in one to one corresp*
*ondence.
Hence K contains a central subgroup isomorphic to E and an inclusion s can be c*
*hosen
so that the induced map on cohomology is the obvious projection. Thus s is a se*
*ction
for OE and the extension 7.1 is therefore split. Since the section s is central*
*, the action
of E on G is trivial and so K ~=G x E and X ' BK ' B(G x E).
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Departament de Matematiques, Univeritat Autonoma de Barcelona, E-08193 Bel-
laterra, Spain
E-mail address: broto@mat.uab.es
Mathematisches Institut, Universit"at Heidelberg, INF 288, Heidelberg 69120, *
*Ger-
many
E-mail address: rlvi@vogon.mathi.uni-heidelberg.de