On The homotopy theory of p-completed classifying spaces
Frederick R. Cohen and Ran Levi
0. Introduction
Let G be a discrete group and let BG denote its classifying space. Recall th*
*at
a group is said to be perfect if it is equal to it's own commutator subgroup. I*
*f G
is an arbitrary group, then write BG+ for the Quillen "plus" construction appli*
*ed
to BG with respect to the unique maximal normal perfect subgroup G of G [32].
The space BG+ can be obtained by attaching 2 and 3 cells to BG and has the
defining properties:
1. There is a natural map BG ____- BG+ , which induces a homology isomor-
phism (with any simple coefficients), and
2. ss1(BG+ ) ~=G=G.
Since Quillen's first defined the higher algebraic K-groups of a ring R, usi*
*ng
the "plus" construction, and computed the K-theory of finite fields, the homoto*
*py
type of BG+ , for G finite and perfect has been a subject of interest as the tr*
*ansition
between the homotopy theory of BG and BG+ is dramatic. Two important classical
examples illustrate this transition.
1. G = 1 is the colimit of the n-th symmetric group and BG+ is the space
Q0S0. [2, 11, 31]
2. G = SL(q) is The colimit of the special linear groups SL(n; q) over the f*
*ield
of q elements Fq and BG+ gives, after localization at a suitable prime, t*
*he
space "image of J" [32].
In addition, Kan and Thurston showed that any path-connected CW -complex
has the homotopy type of BG+ for a suitable group G (which is an extremely
"large" infinite group in their construction) [15]. However, if G is assumed to*
* be
finite, then there are strong restrictions placed on the homotopy type of BG+ .
One feature is that the homotopy groups are entirely torsion and are non-trivia*
*l in
arbitrarily large degrees [22]. In addition, in the cases for which G is finite*
*, it will
be seen below that BG+ behaves through the eyes of homotopy groups as if it were
a finite complex.
Furthermore, the homotopy groups of BG+ for G finite frequently have direct
summands given by the homotopy groups of various classical finite complexes such
as spheres and mod-pr Moore spaces [8, 22, 23, 18]. In contrast, for finite gro*
*ups
G the space BG+ does not admit an essential map to any simply-connected finite
____________
1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55*
*Q52.
F. R. Cohen is partially supported by the NSF.
1
2 F. R. COHEN AND R. LEVI
complex [27]. Thus one must consider different constructions in order to get a
better hold of these relationships.
One general principle in homotopy theory is to use the classical fact that t*
*he
homotopy groups of any (reasonable) space X are isomorphic to the homotopy
groups of the loop space of X with the degrees of all groups shifted down by one
dimension. Thus if one knew additional features of the loop space, then one cou*
*ld
extract information about X. In case X = BG+ , the loop space or possibly itera*
*ted
loop spaces are frequently homotopy equivalent to a product of more familiar sp*
*aces.
In addition, the homology of a loop space supports additional and informative
structure; namely the homology with field coefficients is a Hopf algebra.
This view is illustrated by the next two examples. Start with G = SL(2; 5),
the special linear group over the field of 5 elements. Here the loop space of B*
*G+
is homotopy equivalent to the homotopy theoretic fibre of a degree 120 map from
the 3-sphere to itself, as recalled in section 1 here. Thus the homotopy theory
associated to SL(2; 5) is given in terms of standard spaces and maps.
The space BG+ here does not split as a product in the above example .
However, after looping twice more, it splits into two factors given by the loca*
*lization
at 2, 3, and 5 of 30S3 and 40S3. Here by k0we mean the component of the constant
map of the respective k-fold loop space. Thus the homotopy groups of BG+ are
given by the homotopy groups of the 3-sphere at the primes 2,3, and 5.
A second example, originally discussed in [23], is given by the groups D(p),
defined for a certain family of primes p. The loop space of BG+ is homotopy
equivalent to a product, where one factor is given by the loop space on a mod-p
Moore space. Thus the the mod-p homology of BG+ contains a tensor algebra
on two generators.
This is an example of a different nature and it is not clear what the other
factors are. It will be illustrated in section 1 below. A similar example, wher*
*e a
complete splitting of BG+ in terms of atomic (indecomposable) spaces will also
be discussed in a separate section. The atomic factors are all related to Moore
spaces and homotopy theoretic fibres of self maps of spheres.
We warn the reader at this point that there is an abuse of terminology here.
Namely, the last two families of examples mentioned above, involve groups that *
*are
not perfect but rather p-perfect. Thus the "plus" construction does not mean the
usual Quillen "plus" construction but rather the p-local analogue discussed bel*
*ow.
These examples illustrate the fact that the homotopy theory of the loop spac*
*es
may be given by more familiar and sometimes "handleable" spaces. Thus one useful
principle in our analysis is to consider the structure of BG+ and its iterated *
*loop
spaces, rather than to deal directly with BG+ itself. .
Connections with classical homotopy theory go further as will be illustrated
next. Using the standard fibration obtained from a faithful representation of G*
* in
SU(n), it will be seen below that the loop space of BG+ is homotopy equivalent
to the homotopy theoretic fibre of a map from SU(n) to a Poincare complex Z,
where the map is a rational equivalence. Thus the homotopy theory of BG+ is very
closely related to that of both SU(n) and Z, ( It is not clear whether Z has the
homotopy type of a closed manifold in general).
The features of Z impinge on both the homotopy theory of classical groups
and, an as yet unsolved problem known as the Moore finite exponent conjecture
(this latter conjecture is that if the rational homotopy groups of a simply-con*
*nected
finite complex are a totally finite dimensional vector space, then the p-torsio*
*n in the
F. R. COHEN AND R. LEVI 3
homotopy of this space has a universally bounded order). For example, the Moore
conjecture for Z is satisfied if and only if BG+ has an exponent for its homoto*
*py
groups. One wonders whether the group theory available will then impinge direct*
*ly
on the homotopy theory of either the finite complex Z above or SU(n).
Let us illustrate this last question more precisely. It is not known whether*
* the
homotopy groups of the Lie group SU(3) have any elements of order 8 (although it
is easy to see that there are infinitely many elements of order 4 and no elemen*
*ts of
order 32).
Consider the case of G = M11 or SL(3; 3). A theorem due to J. Martino and
S. Priddy implies that the 2-completed classifying spaces of these groups Coinc*
*ide.
Using work in [22], it is easy to see that if BG+ has an element of order 32 in
its homotopy, then there must exist elements of order 8 in the homotopy groups
of SU(3). On the other hand it is believed that in fact 4 is an exponent for the
homotopy of SU(3). This conjecture would be supported if 16 turns out to be a
bound for the order of the 2-primary component of ss*(BG+ ). Thus the homotopy
theory associated to the "plus" construction has a close relation to the homoto*
*py
theory of classical groups together with basic questions about Lie groups which
seem, at the moment, to be somewhat tricky.
In dealing with this subject a p-local, preferably functorial, analogue of t*
*he
"plus" construction is useful. Such a construction does exist, namely the part*
*ial
R-completion functor of Bousfield and Kan [4], which is functorial and defined *
*with
respect to any ring R (commutative with unit). Thus in 1 and 2 above one has to
replace integral homology theory by homology with coefficients in R and take fo*
*r the
analogue of G, the maximal normal R-perfect subgroup of G. In particular, using
the terminology of [4], if a space X is R-good then the partial R-completion and
the usual R-completion on X yield homotopy equivalent spaces. A special case of
non-simply-connected spaces which are nevertheless, R-good are those spaces with
R-perfect fundamental group, i.e. such that the first mod-R homology vanishes.
Also, if X is a space with finite homotopy (or finite integral homology) groups*
* then
the R-completion of X with respect to R = Fp and Z(p)give homotopy equivalent
spaces. Notice that this is precisely the case when one deals with classifying *
*spaces
of finite groups. Thus from this point on we discuss the homotopy type of BG^p,
the p-completion of BG, assuming that G is finite and p-perfect. Notice that if*
* G
happensQto be perfect then it is in particular p-perfect and if it is also fini*
*te then
BG+ ' p|ord(G)BG^p. Thus no information is lost in the transition to the p-lo*
*cal
analogue.
The interest in these questions arose in the summer 1990, during a bar-room
conversation between the first author and Dave Benson, the last posed the quest*
*ion
of what can be said about the homotopy theory of BG+ when G is a finite perfect
group. We both would like to thank Dave for many conversations about finite
groups and homotopy theory.
This paper represents a survey of some work directed toward answering Ben-
son's question. In particular, questions about BG^pfor finite p-perfect groups *
*G,
why they are interesting, and how they fit with other problems are considered h*
*ere.
Some general theorems in the subject are known and will be discussed. However,
there are many questions that are still to be solved. Numerous examples are giv*
*en
which both illustrate the theory and justify related conjectures. Some new resu*
*lts
are included in section 6.
4 F. R. COHEN AND R. LEVI
1. Motivating examples
This section gives two basic examples as motivation for much of what is done
below. The first is of a very simple nature, whereas the second is a bit more t*
*ricky
and in a sense belongs to a different class of examples.
The first example arises at once from the Poincare homology 3-sphere given by
the orbit space SO(3)=A5. Namely, let SO(3) denote the special orthogonal group
of rank 3 and let A5 be the alternating group on 5 letters. There are fibrations
SO(3)=A5 ____- BA5 ____- BSO(3)
and
(SO(3)=A5)+ ____- BA+5____- BSO(3):
The space (SO(3)=A5)+ is now simply-connected and has the homology of the
3-sphere. Thus it is homotopy equivalent to the 3-sphere. On the other hand, the
3-sphere double covers SO(3). A direct check gives the following
Proposition 1.1. The 2-connected cover of BA+5is a classifying space for the
fibre of the degree 120 map from the 3-sphere to itself. Thus BA+5, after loopi*
*ng
sufficiently often, is homotopy equivalent to a product of loop spaces on spher*
*es.
The degree 120 map on the 3-sphere is null-homotopic after passage to triple
loop spaces by [33, 7]. The splitting follows as one has a multiplicative fibra*
*tion
with a cross-section.
Similar, though sometimes more peculiar behavior propagates for other finite
groups. However the example of A5 has the feature that its cohomology and,
apparently as a result, the associated homotopy type are very simple. Examples *
*of
this form inspired the spherical resolvability conjecture due to the first auth*
*or [8],
which we mention again below. All of those example share the common property
that the cohomology of the group under consideration admits a filtration with
a symmetric associated graded. This last observation was only made after the
spherical resolvability conjecture was disproved by the second author and a clo*
*ser
inspection of the positive examples verified the observation above.
This leads us directly to our second example. A family of p-perfect groups, *
*for
particular primes p, for which the associated homotopy type is much larger than*
* in
the previous example.
Consider primes p 13, such that 4|p-1. Then the cyclic group Z=8Z operates
on an elementary abelian p-group V of rank 2. Let D(p) denote the group given by
the semi-direct product. There is an 8-dimensional unitary faithful representat*
*ion
of D(p) and the p-completion of the resulting orbit space (U(8)=D(p))^phas a
16-dimensional mod-p Moore space P 16(p) as a retract. This together with the
fibration
(U(8)=D(p))^p____- BD(p)^p____- U(8)^p
is used in [23] to show the following.
Proposition 1.2. The loop space P 16(p) is a non-multiplicative retract of
BD(p)^p. Thus BD(p)^psplits as a product where one factor is given by P 16(p).
Although the nature of the other factor is not known, this is enough to conc*
*lude
that each group D(p) forms an example of a totally different nature than our fi*
*rst
F. R. COHEN AND R. LEVI 5
example. Indeed by [29], P 16(p) is homotopy equivalent to S15{p}x(_ffP nff(p)).
Hence the homotopy type resulting from this example is much "larger" than the
previous one. Notice that Proposition 1.2 implies that the homotopy of BD(p)^p
has infinitely many elements of order precisely p2.
These two examples turn out to be generic in the subject. Whereas the first *
*is
nicely behaved and can be shown to satisfy nearly any reasonable conjecture one
can make about it, the second is much harder to analyze in general. The unexpec*
*ted
fact is that within the family of all finite p-perfect groups there is a dichot*
*omy of
this sort. Namely, every group G belongs either to the first family or to the s*
*econd.
We shall elaborate more on this in the following sections.
2.Loop space homology for BG^p
In this section we discuss some results about the loop space homology of spa*
*ces
of the form BG^pfor G finite and p-perfect. A large part of the section however*
* is
devoted to discussion of observations and problems which appear intriguing.
2.1. Torsion in loop space homology. The first type of general behavior for
spaces of the form BG^pwe discuss is that their homology groups have exponents,
a fact that is analogous to the homology of the group having an exponent bounded
above by the order of the group [22].
This type of behavior, where an infinite dimensional space X and its loop sp*
*ace
have exponents for their homology groups appears to be relatively rare in natur*
*e.
The only other class of examples, of which we are aware is those spaces which
are suspensions and which have an exponent for their homology groups. It is also
known that if X is a finite torsion complex then its loop space homology has an
exponent [20]. The family of p-completed classifying spaces of non trivial fin*
*ite
groups has an empty intersection with both classes mentioned above.
An explicit construction, which approximates the singular chain complex for
BG^pwas given in [22]. The model is built as an algebraic "plus"-construction
for the classical bar construction on the group G. More precisely, given a cha*
*in
complex C, one divides out by the 1-skeleton and then adds algebraic cells in
dimension 2 to kill the extra homology created by the first step. This construc*
*tion
in fact does not depend on the original complex having trivial first homology. *
*The
result is a new complex P1(C) together with a map C ____- P1(C) inducing a
homology isomorphism in dimensions larger than 1. In case the first homology was
trivial to begin with, the map induces a homology isomorphism in all dimension*
*s.
This algebraic "plus" construction works in the category of connected different*
*ial
graded coalgebras over a principal ideal domain. In the particular case, where *
*C is
the reduced bar construction on G and the ground ring is Z(p)or Z^p, the result*
*ing
differential graded coalgebra gives a model for the cellular chains on BG^p. T*
*his
construction is of course a direct analogue of the geometric "plus" constructio*
*n due
to Quillen.
The construction P1(-) on arbitrary connected differential graded coalgebras
has a geometric analogue. If one looks closely at how Quillen describes the "pl*
*us"
construction and tries to perform it on a general connected CW complex X, drop-
ping the perfectness assumption, one obtains a space, which we denote P1(X), and
a map X ____- P1(X), which induces an isomorphism on homology groups in di-
mensions larger than one. It is the cellular chains on P1(X) that the algebraic
construction P1(-) models. We summarize all this in the following theorem. For
6 F. R. COHEN AND R. LEVI
any CW complex Y let C(Y ) denote the cellular chains on Y over a fixed princip*
*al
ideal domain R.
Theorem 2.1. Let X be a connected CW complex. Assume that C(X) has a
strictly associative coproduct, making it a differential graded R-coalgebra. Th*
*en
1. the differential graded R-coalgebras P1(C(X)) and C(P1(X)) are quasi iso-
morphic.
2. if X = BG for G finite and p-perfect and R = Z(p)or Z^pthen C(X) is
the mod-R bar construction B[G] on G and P1(B[G]) is quasi isomorphic to
C(BG^p).
The idea is naive and the construction P1(-) suffers from lack of functorial*
*ity
both on the algebraic and the geometric level. However it provides a tool in do*
*ing
certain calculations. The construction is given explicitly in [22] and [20].
Let G be a finite p-perfect group. In order to study the loop space homology
of BG^p, one applies the loop space algebra functor to the differential graded
coalgebra P1B[G] to get a model for the chains on BG^p. The first proof of
the following theorem was obtained by using this differential graded algebra to
construct an explicit null-homotopy for the homomorphism given by multiplication
by the order of G.
Theorem 2.2. Let G be a finite p-perfect group. Then the highest power of p
dividing the order of G is an exponent for "H*(BG^p; Z^p).
Going through the details of these constructions and calculations fall beyond
the scope of this article. However a different and easier proof of this theorem*
* was
suggested to the authors by Bill Dwyer shortly after the result was first annou*
*nced.
His proof has never appeared in print and is thus given here for the first time.
Consider a finite p-perfect group G. The natural map BG ____- BG^pinduces
an isomorphism on homology with coefficients in Z^p. Thus its homotopy fibre,
which we denote by ApG is a mod-p acyclic space. Now consider the principal
fibration
(1) BG^p____- ApG ____h-BG;
obtained by pulling back the path-loop fibration over BG^palong the completion
map. Let EG denote the universal bundle over BG. Taking the pull-back once
more, this time of the universal covering EG ____- BG along the map h, we obtain
a covering
LpG ____- ApG;
where LpG is homotopy equivalent to BG^p. Since the fibration (1) is principal
and its fibre connected, the action of ss1(BG) = G on the homology groups of LpG
is trivial. Consequently the restriction
res : H*(LpG; Z^p) ____- H*(ApG; Z^p)
followed by the homology transfer is given by the following formula
X X
tr O res(x) = g*x = x = |G|x
g2G g2G
We have thus factored multiplication by the order of the group through the homo*
*l-
ogy of Z^p-acyclic space. The theorem follows immediately.
F. R. COHEN AND R. LEVI 7
Dwyer's observation is the core of another general result on BG^p, namely
that it has a stable homotopy exponent. This will be discussed below.
It is a classical fact that the exponent for the homology of a finite group *
*G is
bounded above by the order of G. By the theorem above, the same result is corre*
*ct
for the p-local loop space homology of BG^p. However, the actual bounds which
occur for certain choices of examples vary. Consider for instance the special c*
*ase
when the Sylow 2-subgroup of G is the semi-dihedral group of order 2n and G is
2-perfect. The homology of G has exponent 2n-1. However the situation changes
in loop space homology. Namely the order of torsion in the loop space homology
of BG^2grows by a factor of two and is exactly 2n. Thus looping increases the
exponent although the universal bounds are the same [22]. More examples could
be given here, but we omit them as a detailed calculation might appear a bit too
lengthy..
Our model for the chains on BG^premains useful in at least two ways. The
first way is that it provides an explicit chain complex for computing the loop *
*space
homology of BG^pin much the same way as the classical bar construction provides
an explicit chain complex for computing the homology of G. Furthermore, like the
bar complex, this chain complex depends entirely on the structure of G.
A second use of the model is that it can be used to generalize Theorem 2.2.
Indeed, consider a finite, not necessarily perfect group G. Then the algebraic *
*model
for P1(BG) given by P1(B[G]) is exploited in [20] to prove the following
Theorem 2.3. Let G be an arbitrary finite group. If pr is the highest power
of p, dividing the order of G, then p3r annihilates the reduced p-local homolog*
*y of
P1(BG).
The method of proof is identical to the original proof of Theorem 2.2, except
for one extra observation. Namely, consider the endomorphism OE of the graded
module underlying the cobar construction P1(B[G]), given by multiplication by
the order of G. In [20] an obstruction is constructed, which measures the failu*
*re of
OE to be null-homotopic. However, the third iteration of this obstruction is sh*
*own
to vanish, thus proving the theorem.
2.2. Spherically resolvable spaces. A different aspect of the homological
behavior for BG^pis discussed next. As mentioned above, a conjecture in the
subject, due to the first author, was that BG^pis spherically resolvable of fin*
*ite
length for G finite and p-perfect [8]. The conjecture was disproved by the seco*
*nd
author in [23], using a homological method which we next describe.
For a space X, the property of being spherically resolvable of finite length*
* is
roughly defined by the requirement that X is the total space of a fibration, wh*
*ich
can be obtained by iteratively fibering the base and fibre in terms of spheres *
*and
their loop spaces. Finite length here means, of course, that this decomposition
is required to be finite. Although this may seem somewhat artificial, we confi*
*ne
ourselves in the sequel to spaces, which are spherically resolvable of finite l*
*ength
with the resolving spaces being of the form nSn+k, rather than allowing arbitra*
*ry
iterated loop spaces.
If X is spherically resolvable of finite length, then it is easy to see that*
* its mod-p
homology cannot grow "too fast". More precisely, if one considers the (well kno*
*wn)
mod-p homology of any potential resolving space nSn+k, one sees that the rate
of growth of this graded vector space is constant if n = 1 and hyper-polynomial*
* if
8 F. R. COHEN AND R. LEVI
n > 1. By hyper-polynomial growth we mean that the the rank grows faster than
polynomially but not as fast as exponentially. Thus the same restriction on the
growth applies to X, by iteratively using the Serre spectral sequence to bound *
*the
growth.
Let G be a finite p-perfect group and consider a faithful representation of *
*G in
SU(n) for some n. Then we obtain a fibration
(SU(n)=G)^p____- BG^p____- BSU(n)^p:
Since the space (SU(n)=G)^pis the p-completion of a finite complex, a result of
Felix, Halperin and Thomas [13] gives that its mod-p loop space homology grows
either polynomially or sub-exponentially (i.e. as fast as d1=n, where > 1 and
n 1). For those spaces X whose loop space homology grows polynomially, it is
also proven in [13] that H*(X; Fp) is a finitely generated nilpotent Hopf algeb*
*ra.
Now back to our context, notice that since SU(n) is both finite and spherically
resolvable, the fibration above yields the following theorem, originally proven*
* in
[18]
Theorem 2.4. For a finite p-perfect group G
1. BG^pis spherically resolvable if and only if (SU(n)=G)^pis.
2. H*(BG^p; Fp) grows either polynomially or sub-exponentially.
3. If H*(BG^p; Fp) grows polynomially then it is a finitely generated nilpot*
*ent
Hopf algebra.
Unfortunately, BG^pis not spherically resolvable of finite length in general.
In [23, 18] the second author produced examples of groups G, such that the loop
space homology of BG^pgrows exponentially. Thus the conjecture is not satisfied
for these examples.
One family of counter examples was already given in the preceding section.
Indeed the groups D(p), defined above do not satisfy the spherical resolvability
conjecture, since, as we pointed out, the loop space on a Moore space splits of*
*f as a
retract of BD(p)^pand the first has exponentially growing mod-p homology. An
even simpler family of counter examples is given in [18]. For every prime p, th*
*ere
is an action of the cyclic group of order 3 on the abelian group Z=pZ x Z=pZ. L*
*et
E(p) denote the respective semi direct product. Then for p 6= 3, the group E(p)
is p-perfect and has a rather simple mod-p cohomology ring. However for p 5,
the mod-p homology of the loop space contains a tensor algebra on more than one
generator and thus grows exponentially. A complete splitting of BE(p)^pwas
obtained by the second author and J. Wu [24] and is given in a separate section
below. From this splitting it becomes apparent that BE(p)^pis a very "large"
space in spite of the naive appearance of H*(BE(p); Fp).
These examples suggest that the spherical resolvability conjecture fails bec*
*ause
there are not enough symmetries in the group to make the cohomology "nice". This
last sentence is very vague. However, we do not yet have enough data to replace*
* it
with a more precise phrase.
One could consider the following questions.
1. What is the subset of the set of all finite p-perfect groups, consisting *
*of
groups G such that the homology of BG^psatisfies polynomial growth?
2. Is the "spherical resolvability" conjecture satisfied for this class of g*
*roups?
F. R. COHEN AND R. LEVI 9
It would be very interesting if one could find a group theoretic characterizati*
*on
which would provide an answer to either one of these questions.
2.3. Periodic homology. A particularly nice family of groups in this context
is given by those groups G, where the loop space homology of BG^pis periodic,
namely it is a (possibly non-commutative) ring of Krull dimension 1. These of
course satisfy polynomial growth and thus are finitely generated and nilpotent
as Hopf algebras. A classical theorem of R. Swan gives the groups which have
periodic cohomology. At odd primes these are precisely the groups whose Sylow-
p subgroup is cyclic and at the prime 2 the Sylow subgroup is either cyclic or
generalized quaternion. Of course the cohomology rings of those groups all have
Krull dimension 1 and those which in addition are p-perfect, at the respective *
*prime
p, form the first useful examples in analyzing the structure of BG^p.
Indeed if G is a finite p-perfect group with periodic mod-p cohomology then
BG^palso has periodic mod-p homology [8]. The converse is not true as there are
examples of groups G where the loop space homology is periodic, but the mod-p
cohomology of G has Krull dimension 2. One such family of examples is given in
[14]. It is not clear what sort of implications for G are obtained by the assum*
*ption
that BG^phas periodic homology.
2.4. Finite generation and the existence of elements of infinite height.
The next question is whether the mod-p loop space homology of BG^pis always a
finitely generated algebra. Of course, by the remarks above, it is finitely gen*
*erated
if it satisfies polynomial growth. Thus the question remains what happens at the
other extreme, namely when the loop space homology grows semi-exponentially.
In the few examples that have been worked out the answer appears to be yes but
there is no general theorem to that effect. One related example is the loop spa*
*ce
of a mod-p Moore space where the mod-p homology is a tensor algebra on two
generators and is thus finitely generated, but grows quickly. Other examples are
the groups D(p) and E(p) described above. As we shall observe later the mod-p
homology of BE(p)^pis finitely generated as an algebra although its growth is
exponential. The analogous result for D(p) is not known.
Notice that if finite generation holds in H*(BG^p; Fp), then the fact that t*
*his
homology is infinite dimensional implies that there must exist at least one ele*
*ment
of infinite height. In all of the examples, which have been studied successfull*
*y, this
has always been the case, even if the finite generation question has not been s*
*ettled
yet (the groups D(p) for instance). The existence of an element of infinite hei*
*ght
seems likely to be true in general but there is no theorem to that effect when *
*the
loop space homology fails to grow polynomially.
Another related fact is the following. In section 3 we prove that the homoto*
*py
of BG^pis non-trivial in arbitrarily high degrees by using a Moore-Postnikov ty*
*pe
argument. It is thus appropriate to point out that this result is actually impl*
*ied
by the existence of an element of infinite height in H*(BG^p; Fp). Indeed suppo*
*se
that the ssi(BG^p) = 0 for all sufficiently large i . Then BG^pis equivalent to*
* a finite
Moore-Postnikov section. After looping, the homology ring mod-p has no elements
of infinite height as in Moore and Smith [30].
2.5. Loop space homology as a Lie algebra. As the loop space homology
is an associative algebra, it can be thought of as a Lie algebra in the usual w*
*ay,
10 F. R. COHEN AND R. LEVI
defining [x; y] = xy - (-1)|x||y|yx. Being in fact a Hopf algebra, the sub modu*
*le of
primitives becomes a sub Lie algebra with respect to this structure
The structure of the underlying Lie algebra of primitives is part of what fo*
*rces
various spaces to split as products and forces the loop space to not be "resolv*
*able"
by finitely many spheres in some examples. For instance in studying the groups
D(p) and E(p), the respective loop spaces have the property that the underly-
ing mod-p homology Lie algebra of primitives contains a free Lie algebra on two
generators.
On the other hand, the example arising from A5 has an abelian Lie algebra
associated to it, as is also the case in mod-2 homology for all finite simple g*
*roups
of 2-rank two with the possible exception of U(3; 4), the unitary group of rank*
* 3
over F4 [14].
As the dichotomy theorem 2.4 states, the Lie algebra associated to a group G
either grows polynomially, in which case it is nilpotent, or it grows sub expon*
*en-
tially, in which case one may conjecture that the Lie algebra of primitives con*
*tains
a free Lie algebra on at least two generators.
Consider the polynomially growing case first. Then H*(BG^p; Fp) is a nilpo-
tent Lie algebra. However, we have no examples, where the homology is known
to be non-abelian and is nilpotent. In other words all the examples of this ca*
*se
familiar to us are abelian. An interesting question would be what is true in ge*
*neral
here.
One could try to construct a family of finite p-perfect groups G(n) such that
the homology Lie algebra of BG(n)^pis nilpotent of rank exactly n. Indeed, as we
see no reason to believe that the actual loop space has any reasonable homotopy
commutativity properties, we suggest that any nilpotency rank is possible in our
context. It remains to be seen whether or not this is the case.
Another interesting aspect of this question is how does the group theory as-
sociated to G reflects itself in the rank of nilpotency of the associated loop *
*space
homology.
Next look at the sub exponential case. It is not known whether or not this
case can be nilpotent. Our evidence suggest that the answer to that is no, but
there is no general theorem. Notice that these questions are all closely relat*
*ed
to each other. Namely, if an algebra is finitely generated and nilpotent then *
*it
obviously satisfies polynomial growth. Our observation above is that the conver*
*se
also holds for H*(BG^p; Fp). On the other hand an algebra which is growing
sub exponentially and is nilpotent cannot be finitely generated. Thus one might
conjecture that in the sub exponential case the algebras under consideration are
non-nilpotent.
One might like to find a direct relation between the cohomology of G and
the loop space homology of BG^p. An intuitive guess is the following. Suppose
that the mod-p cohomology of G is Cohen-Macaulay, namely that there exists a
polynomial subalgebra P of H*(BG; Fp) over which it is a finitely generated free
module. Then one can consider the algebra quotient = H*(BG; Fp)==P . It is a
finite dimensional algebra and one might guess that if this algebra contains di*
*stinct
indecomposable elements whose cup product is zero, then the loop space homology
Lie algebra contains a free Lie algebra on at least two generators.
F. R. COHEN AND R. LEVI 11
2.6. The loop space homology functor. The functor on the category of
finite groups given by loop space homology seems to be interesting enough to de-
serve its own name. Rather then defining this functor on the category of finite
p-perfect groups, we define it on the category of all finite groups and specify*
* a ring
of coefficients in the definition as follows.
Let G denote the category of all groups and let FG denote the full sub categ*
*ory
consisting of all finite groups. Let R be a commutative ring with a unit R. L*
*et
MR denote the category of R-modules. Let AlgR denote the category of connected
graded R-algebras. If R is a field, let HR denote the category of connected Hopf
algebras over R. For a fixed prime p let HAp denote the category of connected H*
*opf
algebras over the mod-p Steenrod algebra. Finally let Lp denote the category of
graded restricted Lie algebras over Fp.
For a group G 2 G define
LS H*(G; R) := H*(CR BG<1>; R);
where CR (-) denotes the partial R-completion functor of Bousfield and Kan, men-
tioned in the introduction.
If G is finite and R = Fpthen CR BG ' BG^pand BG^p<1> ' BpG^p. Thus in
that case LS H*(G; Fp) = H*(BpG^p; Fp). We summarize this in the following
theorem.
Theorem 2.5. Let R be a commutative ring with a unit. There is a bifunctor
LSH*(-; -) defined on the product category G x MR which takes values in MR .
Furthermore,
1. The functor LS H*(-; R) takes values in AlgR .
2. If R = Fp then the values of LS H*(-; R) can be thought of as objects of
either HAp or Lp.
Furthermore, with respect to both categories in (2) the functor LS H*(-; Fp) ta*
*kes
products into coproducts and preserves inductive colimits.
Proof. The only points which requires proof are that the functor LSH*(-; Fp)
takes products into coproducts and preserves inductive colimits. First observe *
*that
B(G x H)^p' BG^px BH^p. Thus
LS H*(G x H; Fp) ~=LSH*(G; Fp) LSH*(H; Fp):
Since the tensor product is the coproduct in both HAp and Lp, the first stateme*
*nt
follows.
Next, notice that LS H*(-; Fp) is a composition of functors which all preser*
*ve_
inductive colimits. The second statement follows. |__|
An immediate corollary of the Kan-Thurston theorem is that for R = Z the
homology of every connected loop space is of the form LS H*(G; Z) for a suitable
group G. One may wonder about abstract properties of this functor. In particular
it would be interesting if one could come up with a purely algebraic description
of it. The major obstacle at this point to doing so is the fact that the algebr*
*aic
"plus" construction is not functorial. However we believe that this problem can*
* be
solved. Notice that a purely algebraic construction for a differential graded a*
*lgebra
whose homology gives LSH*(G; R) would be desirable as it would imply that all t*
*he
algebraic invariants discussed above, the Hopf algebra, Lie algebra, growth fea*
*tures,
nilpotence questions etc., are somehow encoded completely in the structure of t*
*he
groups under consideration.
12 F. R. COHEN AND R. LEVI
3. The stable and unstable homotopy of BG^p
Spaces of the form BG^pappear to have some intriguing homotopy proper-
ties. In this section some known general facts and examples of the behavior of *
*the
homotopy of BG^pare given. Relations to the existence of exponents in homotopy
are discussed.
3.1. The stable homotopy theory of BG^p. Here one can prove the fol-
lowing.
Theorem 3.1. Let G be a finite p-perfect group. Then the highest power of p
dividing the order of G is an exponent for the stable homotopy of BG^p.
As mentioned above, Dwyer's alternative proof of Theorem 2.2, inspired the
proof of 3.1. The reader will recognize the similarity at once.
Let G be a finite p-perfect group. Consider the principal bundle discussed in
the preceding section
LpG ____ss-ApG;
where ApG is mod-p acyclic and LpG is homotopy equivalent to BG^p. Consider
the stable adjoint of the Kahn-Priddy stable transfer map
tr: ApG ____- Q(LpG):
This map is null-homotopic, since ApG is mod-p acyclic and Q(LpG) is a loop spa*
*ce
on a p-local space.
Thus consider the composite trO ss : LpG ____- Q(LpG). On one hand it is
obviously null-homotopic since tris. On the other hand it is not hard to show t*
*hat,
due to the fact that G operates trivially up to homotopy on LpG, this compositi*
*on
gives precisely the adjoint of the degree |G| map on 1 BG^p. Thus |G| annihilat*
*es
the stable homotopy of BG^p.
In contrast to the above theorem, the following is proven in [22]
Theorem 3.2. Let G be a finite group. Then the p-completed classifying space
BG^pdoes not have a stable homotopy exponent.
3.2. Unstable homotopy theory. We now turn to the unstable homotopy
of BG^p. A first question one could ask is how much of it is there? Throughout
this article we consistently stress the point that in many respects BG^pbehaves
like it is trying to be a finite complex (but for technical reasons cannot be a*
* finite
complex). Indeed one can show that the homotopy of BG^phas a common feature
with that of a finite complex.
Theorem 3.3. Let G be a finite group containing a non-trivial p-perfect sub-
group of order divisible by p. Then BG^phas infinitely many non-vanishing k-
invariants; in particular infinitely many non-trivial homotopy groups.
The proof for this is rather simple. Namely, let G be a finite p-perfect gro*
*up
and consider a faithful representation of G in SU(n) for a suitable n. Then from
the fibration
SU(n)=G ____- BG ____- BSU(n);
it follows that BG^pis the fibre of the p-completion of the projection from SU(*
*n)
to SU(n)=G. Thus by [16] BG^phas a quasi-bounded cohomology module;
F. R. COHEN AND R. LEVI 13
that is the cyclic module over the Steenrod algebra generated by every element
x 2 H*(BG^p; Fp) is a finite dimensional vector space. By Miller's theorem, the
Sullivan conjecture, spaces with this property have the feature that they do not
accept any essential map from BG where G is a finite group.
This property of BG^pis exploited as follows. Recall that BG^pis up to
homotopy a finite cover of an acyclic space ApG. One observes that ApG cannot be
a product of a finite Postnikov section and a generalized Eilenberg MacLane spa*
*ce.
Then one assumes that ApG is a finite Postnikov section and observes that in th*
*is
case BG^phas to be the target of an essential map from K(ss; n), where ss is the
top homotopy group of ApG. Notice that ss is a finite group. Thus by looping n *
*- 1
times a contradiction to Miller's theorem is obtained and the proof is complete.
A second view of this last theorem is to ask whether the mod-p homology ring
of BG^phas an element of infinite height. If that were the case, then this theo*
*rem
would follow at once from [30].
The question now becomes, what can be said about the homotopy of BG^p.
An easy observation gives an upper bound on the connectivity of BG^p.
Proposition 3.4. Let G be a finite p-perfect group and let n be the minimal
dimension of a faithful unitary representation of G. Then BG^pis at most (2n-1)-
connected.
Let a faithful representation of G in U(n) be given. Then, as usual, we obta*
*in
a principal fibration
U(n)^p____- (U(n)=G)^p____- BG^p:
Since the fibration is principal and since the top dimensional indecomposable i*
*n the
fibre line of the associated Serre spectral sequence appears in dimension 2n-1,*
* it fol-
lows at once that if BG^pis 2n-connected then the spectral sequence collapses. *
*But
this implies a contradiction, since BG^pis infinite dimensional, whereas (U(n)=*
*G)^p
is finite dimensional.
Notice that one obtains better estimates on the first non-vanishing group by
using the Steenrod algebra. We omit the precise statement.
In all the examples where one has a reasonable hold on the homotopy type
there is an exponent for the homotopy of BG^p. There are basically two families*
* of
examples. The first is the family of groups which satisfy the spherical resolva*
*bility
conjecture. Since spheres are known to have homotopy exponents, so does any spa*
*ce
which is spherically resolvable of finite length. A list of such examples appea*
*rs in
[22] and includes finite groups of classical Lie type and many more examples.
The second family of groups consists of those which fail to satisfy the sphe*
*rical
resolvability conjecture (and, as observed above, there are such groups). In ex*
*am-
ples there is not much known. But for at least one family of finite groups G it*
* is
known that BG^psplits as a product of familiar pieces, all of which have been
shown by Cohen, Moore and Neisendorfer to have homotopy exponents. Thus it
seems to be reasonable to conjecture that the existence of exponents is a gener*
*al
feature of BG^pfor G finite. We shall collect all known examples in an appendix*
* at
the end of the paper.
4. The Postnikov tower of BG^p
A major viewpoint in our subject is that the functor (B(-)^p) applied to a
p-perfect group G gives a loop space whose structure is encoded in a large part*
* in
14 F. R. COHEN AND R. LEVI
the group structure of G. Thus, since our groups G are always assumed to be fin*
*ite,
we could ask what finiteness properties does BG^phave? Several such properties
were already mentioned or conjectured to be true in the discussion above. Here
we provide another finiteness condition on BG^p, namely we discuss the fact that
for many groups G, any space X with the same mod-p cohomology as BG^pand
the same Moore-Postnikov tower through a range has the property that X is
homotopy equivalent to BG^p.
Recall that a commutative algebra with a unit A is said to be Cohen-Macaulay
if its Krull dimension is equal to its depth. Another way of saying the same th*
*ing
is that there exists a finitely generated polynomial subalgebra P in A such tha*
*t A
is a finitely generated free module over P . Many finite groups have the proper*
*ty
that their mod-p cohomology is Cohen-Macaulay. By abuse of terminology we say
in this case that the group itself is Cohen Macaulay at p. For instance all abe*
*lian
groups have this property. Every group G such that the cohomology of its Sylow
p-subgroup is Cohen-Macaulay has the same property itself and some groups whose
Sylow p-subgroups are not Cohen-Macaulay are still Cohen-Macaulay. We should
point out though that being Cohen-Macaulay is a rather strong condition on G and
if a finite group is chosen at random, chances are its cohomology won't have th*
*is
property (for instance most symmetric groups are not Cohen-Macaulay at most
primes dividing their orders).
From now on use the phrase G is a CMp group to say that H*(BG; Fp) is
Cohen-Macaulay. For finite CMp groups G the following well-known lemma holds.
A reference for a slightly more general version is [9]. For a finite group G, *
*the
p-rank, rkp(G) is the maximal rank of an elementary abelian subgroup of G.
Lemma 4.1. Let G be a finite group of p-rank r. Then there is a faithful un*
*itary
representation ae of G with the following property:
1. there are precisely r non-zero Chern classes ci1; . .c.irfor ae and
2. H*(BG; Fp) is a finitely generated module over the polynomial subring gen-
erated by ci1; . .c.ir.
In particular if G is CMp and a representation as in the lemma is given then
H*(BG; Fp) is a finitely generated free module over the polynomial subring gen-
erated by the non-zero Chern classes. Representation with the properties granted
by the lemma will be called an admissible representations for G. The existence *
*of
admissible representations for any CMp group is an essential ingredient in the *
*proof
of the theorem below. Details are in [21], where the theorem is proven assuming
the existence of admissible representations rather than having it granted for e*
*very
CMp group.
For any space X and a positive integer d, let X[d]denote the d-th stage of t*
*he
Postnikov tower for X. We say that X and Y have the same d-th Postnikov stage
if there is a map lY : Y -! X[d], inducing an isomorphism on ssi(-) for 0 i d.
Definition 4.2. Two spaces X and Y are said to be comparable of degree d if
1. X and Y have the same d-th Postnikov stage X[d].
2. H*(X) ~=H*(Y ) as algebras.
3. There is an isomorphism OE : H*(X) -! H*(Y ) realizing 2, such that OE O
l*X= l*Yin mod-p cohomology, up to dimension d.
F. R. COHEN AND R. LEVI 15
Theorem 4.3. Let G be a finite p-perfect CMp group. Assume that G has an
admissible representation in SU(n). Suppose that X and BG^pare comparable of
degree n2 - 1. Then BG^p' X.
A sketch of the proof follows. First, let X[n] be the n-th stage of the Post*
*nikov
tower for a space X. Then for any space C such that dimC n + 1, the canonical
map l : X -! X[n] induces an epimorphism of sets
l* : [C; X] -! [C; X[n]]:
Furthermore, if dimC n, then l* is an isomorphism.
Next, as a consequence of the big collapse theorem of L. Smith [34], we get
that if G is a finite p-perfect group which admits an admissible representation*
* ae in
SU(n) for some n, then
H*(SU(n)=G) ~=H*(BG)=(P ) E
as an H*(BG)-module, where P is the polynomial subalgebra of H*(BG), generated
by the non-zero Chern classes of ae, E is a trivial H*(BG)-module and is isomor-
phic to an exterior algebra on generators corresponding to the regular sequence
generating Ker(ae*). Moreover, if p 6= 2 then this isomorphism is an isomorphism
of algebras.
Using the first observation, one obtains the following commutative diagram of
fibrations.
FX ________-X _______-B[n2 - 1]
| | |
| | |
?| ?| ?|
FG _________-L___________-S _______-BG^p
| | |
| | | |
?| ?| j ?| ?|
BG^p _____-SU(n)^p ___-(SU(n)=G)^p ______-BG^p
| | ss|| |
?| ?| ?| ?|
B[n2 - 1] __- X_______-X ________-B[n2 - 1]
Since FX is the (n2-1)-connected cover of X and SU(n) is (n2-1)-dimensional,
it follows that ss O j is null homotopic. Hence j lifts to a map l : SU(n)^p___*
*_- S.
This in turn induces a map f : BG^p____- X, which is obviously a homotopy
equivalence if and only if l is a homotopy equivalence.
One now proceeds by showing that any lift l of j has the property that the
induced map on Z(p)homology is an isomorphism in dimension i if i n2-2 and a
split monomorphism if i = n2 - 1. One then argues, by using the Eilenberg-Moore
spectral sequence, that S is at most (n2-1)-dimensional. This is done by observ*
*ing
that the spectral sequence for the fibration
SU(n)^p____- (SU(n)=G)^p____- BG^p
collapses at the E2 term and that, by our algebraic assumptions, the E2 term for
the fibration
S ____- (SU(n)=G)^p____- BG^p
16 F. R. COHEN AND R. LEVI
is identical to the one above. Finally one observes that S is rationally equiva*
*lent
to SU(n) and hence is precisely (n2 - 1)-dimensional. This implies that any lif*
*t l
induces an isomorphism on homology and the proof is complete.
5.Homotopy decompositions of BG^pfor certain examples
We say that a space Y is a costable factor of another space X if for some
n 0 Y is a factor of nX. This notion is dual to that of a stable summand.
It is well known that for a finite group G the classifying space BG^pstably spl*
*its
into a finite number of irreducible components. Thus we present the dual questi*
*on,
namely what are the costable building blocks of BG^p. The answer suggested by
numerous examples appears to be that these are related to iterated loop spaces *
*of
finite complexes. Here, once more, there is no general theorem to that effect, *
*at the
moment.
In this section we present an example of a group G, which has the property
that BG^psplits as a product, where the factors are loop spaces on Moore spaces
and fibres of degree maps on spheres. This decomposition is a theorem due to Jie
Wu and the second author. We sketch the proof as a more detailed account will
appear elsewhere.
The groups under consideration are the groups E(p) defined in section 2 above
for p 5.
There is a faithful representation of E(p) in SU(3). The explicit map is giv*
*en
in [18]. One important fact about this representation is the following
Proposition 5.1. For each prime p 5 there is an isomorphism of algebras
H*(SU(3)=E(p); Fp) ~=P [a2; b3; b03; c5; c05; d6]=R;
where degrees are given by subscripts and the set of relations R consists of the
equalities ad = cb = c0b0 together with setting every other product being equal*
* to
zero. The only non trivial Steenrod operations in H*(SU(3)=E(p)) are fira = b a*
*nd
fi1c = d.
Let X denote (SU(3)=E(p))^p. The observation made in [18] is the following
Proposition 5.2. There is a homotopy equivalence
X(6)' P 3(p) _ P 6(p) _ S3 _ S5;
where X(6)is the 6-skeleton of X.
The proof involves a mildly tricky argument, using an improvement on a theo-
rem of I. James, due to J. Harper. Proposition 5.2 gives us enough information *
*to
proceed.
For primes p 3 there is a mod-p homotopy equivalence SU(3) ' S3 x S5.
Thus there is a map ss : X -! SU(3), which is degree 1 on the cells represented*
* by
b0 and c0. Let F denote the homotopy fibre of ss. Notice that the composition of
ss with the projection to each one of the spheres admits a right homotopy inver*
*se.
This observation is used to prove the following
Proposition 5.3. There are homotopy equivalences
1. X ' F x SU(3) and
2. BE(pr)^p' F x S3{p} x S5{p}.
F. R. COHEN AND R. LEVI 17
Thus to complete the decomposition one needs to understand the homotopy
type of F . For any two pointed spaces X and Y we denote by X o Y the half
smash product X x Y= * xY . Indeed it is an easy observation that there is a
homotopy equivalence
F ' (P 3(p) _ P 6(p)) o SU(3):
Finally the formula
(X o Y ) ' X x (Y ^ X)
gives that
F ' (P o S) ' P x (S ^ P ) ' P x (S ^ P );
where P denotes the wedge P 3(p) _ P 6(p) and S denotes SU(3)^p. But since S is*
* a
product of two spheres, S is an infinite wedge of spheres and by [28], S ^ P
is an infinite wedge of mod-p Moore spaces.
The known results of [28] about the homotopy of Moore spaces and spaces of
the form S2n+1{pr} at odd primes now imply the following.
Corollary 5.4. The groups ss*BE(p)^phave an exponent p2 and the p2 power
map on the double loop space 20BE(p)^pis null-homotopic.
6.Relation with certain finite complexes
Here we present some relationships between spaces of the form BG^pfor G fini*
*te
and p-perfect and finite complexes. Part of this appears here for the first tim*
*e.
An unexpected fact that sheds new light on some of what is said above is that
for any finite p-perfect group G, the loop space BG^pis a retract of the loop
space of a finite complex. The statement of the main theorem we discuss here, in
its full generality, requires the use of p-compact groups and maps which we call
homotopy representations into such gadgets. However to simplify the discussion,
we will restrict ourselves to using compact Lie groups instead. Conceptually th*
*ere
is no difference. The reader interested in full details is referred to [19].
Let G be a finite p-perfect group. Let L be a compact connected Lie group, s*
*uch
that there exists a faithful representation of G in L. Then one obtains a fibra*
*tion
L=G ____- BG ____- BL;
which p-completion preserves, since BL is simply-connected. Notice that the as-
sumption that ae is faithful implies that L=G is a compact manifold, in particu*
*lar a
finite complex. Consider the following diagram, where the rows are fibrations
g ^ ff ^
BG^p ___-L^p___-(L=G)p ___-BGp
| OE|| || = |
?| ffi ?| ?| ae ?|
BG^p ___-Fae______-S_____-BG^p:
In the diagram, S is the cofibre of g, the map ae extends ff and Faeis the homo*
*topy
fibre of ae. An old result of Ganea [12] gives a formula for F ae. Indeed there*
* is a
homotopy equivalence
Fae' (L^p^ BG^p):
18 F. R. COHEN AND R. LEVI
In particular the map OE in the diagram is null-homotopic and hence by com-
mutativity so is ffi. It follows that ae has a right homotopy inverse and so
S ' BG^px (L ^ BG^p):
Notice that S is
1. a p-torsion space, since g is a rational equivalence and
2. has finite mod-p homology.
Thus S has the homotopy type of the p-completion of a the finite complex given *
*by
the cofibre of the projection L ____- L=G. This implies that, given a CW struct*
*ure
on BG with BkG denoting the k-skeleton, there is an integer n such that the ske*
*letal
inclusion
BnG^p,i!BG^p
has the property that i has a right homotopy inverse. The minimal such n is
called the p-essential dimension of G. It is obviously an invariant of the grou*
*p and
will be denoted by edp(G). We have thus proven
Theorem 6.1. Every finite p-perfect group has a finite p-essential dimensio*
*n.
The terminology "essential dimension" here is motivated by the fact that if
f : BG ____- X is any map with X p-complete and such that conn(X) > edp(G)
then f is null-homotopic.
Here is a simple example of an application of this concept. Let G be any
proper subgroup of H, where H itself is the normalizer of a Z=pZ in the symmetr*
*ic
group on p letters. If G contains Z=pZ and is p-perfect, then the natural map
BG^p____- BH^pgives an injection in mod-p cohomology. However, is null-
homotopic. Notice that an immediate consequence of this is that induces the ze*
*ro
map on homotopy group. One should really consider this observation in contrast
to the fact that if one suspends at least once it admits a right homotopy inve*
*rse
[6].
Theorem 6.1 explains some of the results above in a new way. In particular
it gives a new proof of the fact that H*(BG^p; Z^p) has an exponent for G finite
and p-perfect. Indeed it is shown in [20] that every finite torsion complex has*
* this
property. A bound is not easy to determine by this method though.
The following is another easy application of Theorem 6.1. Its proof appeared
originally in [19]. Here we present a slightly stronger statement.
Theorem 6.2. Let G be a finite p-perfect group. Let E : BG^p____- BG^p
denote the Freudenthal suspension map. Then E is an element of finite order in
the group [BG^p; 2BG^p]. Furthermore, let S be as i-connected p-torsion space
together with a map f : S ____- BG^psuch that f has a right homotopy inverse.
Then a bound on the order of E is given by the order of the identity map in the
group [S; S].
Proof. First notice that a space as in the statement of the theorem always
exists by the proof of Theorem 6.1. Next observe that the adjoint to the compos*
*ite
r>
S ____f-BG^p____E-BG^p____ O E O f is null homotopic. But upon looping f has a right
F. R. COHEN AND R. LEVI 19
homotopy inverse and it follows that O E is null-homotopic, which_proves_o*
*ur
statement. |__|
Corollary 6.3. Let G be a finite p-perfect group. Let F be any representable
cohomology theory. Then the natural map
F n(BG^p) ____- F n-1(BG^p);
given by the looping functor has an image of finite exponent. Furthermore, a bo*
*und
is universally given by the order of E 2 [BG^p; 2BG^p].
Proof. Let Fn denote the n-th space of an -spectrum representing F . Then
any element of x 2 F n(BG^p) is represented by a map x : BG^p___- Fn. Since Fn *
*is
a loop space, x factors through BG^pas (adx) O E, where E is the Freudenthal
suspension. Let x denote (adx). Then under the group homomorphism
[BG^p; BG^p] ____x*-F n(BG^p)
the Freudenthal suspension E is taken to x. By naturality of the looping functo*
*r,
x* : [BG^p; 2BG^p] ____- F n-1(BG^p)
takes E to x. But E is an element of finite order and the result follows. |_*
*__|
Another application of Theorem 6.1 comes next. Let G be a finite p-perfect
group and choose any faithful representation ae of G in U(n). In the examples w*
*hich
have been understood, the map Bae^p: BG^p____- U(n)^phas been nontrivial,
which might lead one to believe that this should always be the case. However, t*
*he
following theorem will imply the existence of many faithful representations ae *
*of a
finite p-perfect group G, where the induced map Bae^pis null-homotopic.
Theorem 6.4. Let G be a finite p-perfect group and let ae be a representati*
*on
of G in L(n), where L(n) is either O(n), U(n) or Sp(n). For a positive integer k
let kae denote the k-fold Whitney sum of ae. Then there exists an integer r suc*
*h that
B(prae) : BG^p____- L(prn)^p
is null-homotopic.
Proof. Let L denote the colimit of the groups L(n). Let n : L(n) ____- L
denote the inclusion. Recall that the k-fold Whitney sum induces the k-th power
map on the infinite loop space BL in the sense that the diagram
BL(n) __-BL(n)xk __-BL(kn)
n|| kn||
?| ?|
BL _________________-BL
commutes. Let wk denote the composite on the top row of the diagram.
Let S be a 1-connected p-torsion space together with a map f : S ____- BG^p,
such that f has a right homotopy inverse. Without loss of generality we may
assume that the representation ae is faithful and and thus that S is of dimensi*
*on at
most one more than the dimension of L(n). To justify this, notice that the proof
of Theorem 6.1 implies the existence of a complex S of this dimension satisfying
our requirement, provided the representation ae is faithful. If ae isn't faithf*
*ul, then
it factors through a faithful representation of a quotient group H of G, and th*
*us it
suffices to prove the claim for H.
20 F. R. COHEN AND R. LEVI
Consider the p-completion of the diagram above. If ps is the order of identi*
*ty
in [S; S] then we have
psnO B(psae)^pO f = psnO wprO Bae^pO f ' O n O Bae^pO f
and the right hand side is null-homotopic. Thus the map B(psae)^pO f lifts to t*
*he
fibre (L=L(psn))^pof psn, which is (psn - 1), (2psn - 1) or (4psn + 1)-connecte*
*d,
according to L being O; U or Sp. On the other hand dim(S) dim(L(n)) + 1.
Thus if r is chosen so that dim(L(n)) + 1 conn(L=L(prn)), then B(prae)^pO f is
null-homotopic. But f has a right homotopy inverse, implying that for such_r,
B(prae)^pvanishes. |__|
Corollary 6.5. Let G be any finite group. Let L(n) be as in Theorem 6.4.
Then there exists a positive integer r such that for any representation ae of G*
* in
L(n), the induced map B(prae)^pis null-homotopic.
Proof. We may assume that G contains a non-trivial p-perfect subgroup or
otherwise the claim is redundant. Thus BG^pis a finite disjoint union of copies
of BpG^p. The result follows at once from Theorem 6.4 and the fact that any
finite group has a finite number of distinct quotient groups, each of which hav*
*ing_
at most a finite number of non-equivalent representations in L(n). |*
*__|
Corollary 6.6. Let G be a finite p-perfect group and let a faithful represe*
*n-
tation ae of G in L(n) be given, where L(n) is as in Theorem 6.4. Then there ex*
*ists
a positive integer r such that
(L(prn)=G)^p' L(prn)^px BG^p;
where G acts on L(prn) via the faithful representation prae.
Proof. This follows at once from the fact that for a suitable r, the map
B(prae)^p r ^
BG^p_______- L(p n)p
is null-homotopic. |___|
The discussion above might motivate one to wonder about the order of the
identity map in the group [S; S], where S is the space produced in the proof of
Theorem 6.1. We are not able at this point to answer that question, but a first*
* step
would be getting an upper bound on the exponent of the integral homology of S.
Theorem 6.7. Let L be a connected compact Lie group with torsion free p-
adic cohomology and let G be a finite p-perfect group. Let ae : G _____- L be a
faithful representation of G in L. Let S denote the cofibre of the natural map
L^p____- (L=G)^p. Then the reduced p-adic homology of S is annihilated by the
maximal power of p dividing the order of G.
Proof. consider the (non-completed) projection L ____- L=G. Let S0 denote
its cofibre. By the proof of Theorem 6.1, S ' (S0)^p. Thus it suffices to bound*
* the
torsion of the homology of S0. But S0 is the cofibre of a fibre inclusion and h*
*ence
Ganea's theorem, mentioned in the proof of Theorem 6.1, applies to compute its
universal cover. Indeed the map L=G ____- BG factors through S0by the universal
property of cofibrations. By Ganea's theorem the fibre of the map resulting map
S0 ____- BG is given by
_
(L ^ BG) ' (L ^ G) ' L:
|G|-1
F. R. COHEN AND R. LEVI 21
Thus there is a G covering spaces over S0, with a homologically torsion free to*
*tal_
space. The usual transfer argument now implies the result. |__|
7.Problems and conjectures
Since many of the problems concerning the homotopy theoretic features of BG^p
remain unsolved, a list of them together with examples is given below.
7.1. Homotopy exponents. The first question we present about the ho-
motopy type of BG^pfor G finite and p-perfect is whether or not there exist an
exponent for the unstable homotopy ss*(BG^p). Evidence suggest that the answer
should be yes and furthermore, that the order of the Sylow p-subgroup of G shou*
*ld
be an upper bound for this exponent. We mention the examples discussed in [22]
of groups G, such that BG^pis spherically resolvable of finite length. For most
of those examples it is possible to show, using the resolution that the order o*
*f the
Sylow p-subgroup is indeed an upper bound for the exponent of ss*(BG^p).
It can also be shown that if G is a 2-perfect group containing the dihedral
group D2n as a Sylow 2-subgroup, then 2n does not bound the torsion in ss*(BG^2)
but 2n+1 does. This is possibly due to the fact that for such a group G the uni-
versal cover of BG^2is the 2-completed classifying space of the 2-universal cen*
*tral
extension of G, which contains the generalized quaternion group Q2n+1as a Sylow
2-subgroup.
Recall the example of BE(p)^p, discussed above, which splits in terms of
loop spaces on Moore spaces and fibres of degree p self maps on spheres. Thus an
exponent in unstable homotopy is obtained for those examples as well by the kno*
*wn
results of Moore, Neisendorfer and the first author. Furthermore, notice that i*
*f the
group theory could be used to inform on the homotopy theory, then there would
be an alternative proof that a Moore space has an exponent.
Also Theorem 6.2 gives that the image of
En*: ss*(BG^p) ____- ss*(nnBG^p);
where En is the n-fold Freudenthal suspension map has an exponent.
The Moore finite exponent conjecture is that if X is a 1-connected finite co*
*m-
plex then the torsion in ss*(X) has an exponent if and only if X is elliptic, n*
*amely
if its rational homotopy is a finite dimensional graded vector space. If X is *
*p-
complete then the same conjecture can be stated replacing the rationals by the
p-adic rationals Qp. Above we exhibited fibrations of the form
L^p____- (L=G)^p____- BG^p
where L is a compact Lie group. Obviously both L^pand (L=G)^phave finite dimen-
sional Qp homotopy. Thus if the Moore conjecture were true it would imply the
existence of unstable homotopy exponents for BG^p. On the other hand a general
exponents theorem for BG^pwill automatically produce a large family of positive
examples to the Moore conjecture, namely the spaces (L=G)^p.
With this in mind we are now ready to state our first conjecture. Recall that
a group G is said to be p-superperfect if BG^pis 2-connected.
Conjecture 7.1. Let G be a finite p-superperfect group. Then the order of
the Sylow p-subgroup of G is an exponent for ss*(BG^p).
22 F. R. COHEN AND R. LEVI
In all the examples, which we could work out, the pr-th power map on 2BG^p,
for G finite and p-superperfect, is null-homotopic. Here pr is, as above the or*
*der of
the Sylow p-subgroup of G. Thus we state an even stronger conjecture
Conjecture 7.2. Let G be a finite p-superperfect group. Then the order of
the Sylow p-subgroup of G is an H-space exponent for 2BG^p.
7.2. Stable and unstable splitting. Next we turn to the problem of ex-
pressing the homotopy type of BG^p, possibly after looping frequently enough, in
terms of more familiar spaces. General information concerning the stable struct*
*ure
of BG^pis given in works of Martino and Priddy [25], Benson and Feshbach [3] and
others.
We have seen that BG^pis a retract of the loop space on a finite complex,
for any finite p-perfect group G. We have also demonstrated an example of a case
where BG^psplits completely in terms of loop spaces on Moore spaces and fibres
on degree p maps on spheres. Thus one gets the (possibly superficial) impression
that BG^pis doing its best to be a finite complex but fails to do so for techni*
*cal
reasons. Let G be a finite p-perfect group. Consider first the stable type of B*
*G^p.
Some of our evidence suggests the following
Conjecture 7.3. Let G be a finite p-perfect group. Then BG^pstably splits
as a wedge of finite complexes. Furthermore, an iterated (finite) suspension of
BG^psplits.
Except for the few examples we know, one can consider Theorem 2.2 above. It
gives a property, the existence of exponents in loop space homology, that BG^p
has in common with finite complexes and homological properties are preserved by
suspensions. Also the existence of stable homotopy exponents supports our last
conjecture as it is true for any finite torsion complex that its stable homotop*
*y has
an exponent.
From an unstable point of view, it is not clear at all that BG^por an iterat*
*ed
loop space splits in terms of familiar spaces. In the examples we know it does,*
* but
we do not believe this is a general phenomenon.
7.3. Homological problems. Next we recapitulate on some problems con-
cerning loop space homology. Consider H*(BG^p; Fp). In all the known examples
this is a finitely generated algebra. It is sometimes commutative but most of t*
*he
time not so (as the homology ring contains a tensor algebra on two generators).
The strongest conjecture one can make here is that this is always the case.
Conjecture 7.4. The homology algebra H*(BG^p; Fp) is finitely generated
for every finite p-perfect group G.
However, we would not claim that there is enough evidence to make this con-
jecture plausible. Several weaker statements could be easier to verify. First n*
*otice
that H*(BG^p; Fp) is always infinite dimensional. Thus finite generation would
imply that there is always an element of infinite height in H*(BG^p; Fp).
Conjecture 7.5. For every finite p-perfect group G there exists an element
of infinite height in H*(BG^p; Fp).
Another conjecture already mentioned above concerns the nilpotency degree of
the loop space homology Lie algebra.
F. R. COHEN AND R. LEVI 23
Conjecture 7.6. For every positive integer 1 n 1 and every prime p,
there is a finite p-perfect group G with the property that LSH*(G; Fp) is a nil*
*potent
Lie algebra of rank precisely n.
This conjecture is known for n at the extremes namely if n = 1 or 1.
7.4. The p-essential dimension. Recall that the essential dimension of a
group G is defined to be the least positive integer n such that for any CW model
for BG, the natural map
BnG^p____- BG^p
admits a right homotopy inverse after looping. This mysterious invariant of gro*
*ups
might have interesting applications. Indeed some were already discussed above.
However the upper bound suggested by the proof of Theorem 6.1 is not satisfac-
tory. Indeed this bound is one more than the minimal dimension of a faithful
representation of G, or more generally than that of a faithful homotopy represe*
*nta-
tion. For details on the use of homotopy representations in this context the re*
*ader
is referred to [10]. Although the use of those improves on the result we get by*
* using
Lie groups, it should be pointed out, by means of examples that one can do bett*
*er.
One example is the groups E(p) discussed above, where the existing approximation
is edp(E(p)) 9, whereas in fact the decomposition of the previous section gives
that edp(E(p)) 6. Another example, this time out of our direct context is of t*
*he
group SL(3; Z) at the prime 2. There, the upper bound on the 2-essential dimens*
*ion
given by [19] is 6 whereas one can show that the actual 2-essential dimension i*
*s 4
[17]. The question is thus what determines the p-essential dimension of a p-per*
*fect
group G. The following conjecture has been verified in all known examples. For a
graded algebra A, let QA denote the module of indecomposable elements. If A is
finitely generated let tA denote the top dimension in which QA is non-zero.
Conjecture 7.7. Let G be a finite p-perfect group and let A denote its mod-p
cohomology algebra. Then
edp(G) tA :
It is worth pointing out though that for the group SL(3; Z) the bound given
by this conjecture is not best possible. However it is precisely this example w*
*hich
motivates the following
Conjecture 7.8. Let H be a central extension of a finite p-perfect group by
a finite abelian p-group. Then edp(G) edp(H). Thus for a finite p-perfect group
let "Gdenote the p-universal central extension. Let A denote the mod-p cohomolo*
*gy
of "G. Then with the notation above
edp(G) tA :
8. Appendix
A list of examples where the homotopy type of BG^phas been worked out
appears below. Details are omitted. The interested reader is referred to the ap*
*pro-
priate references.
8.1. Periodic p-Sylow subgroup. If G has periodic mod-p cohomology and
is p-perfect then BG^pis homotopy equivalent to S2n-1{pr} for appropriate n and
r [8].
24 F. R. COHEN AND R. LEVI
8.2. Groups of Lie type. Let G(C) be a complex reductive connected Lie
group. Let G(Fq) denote the corresponding finite group of Lie type over the fie*
*ld
of q elements. Let p be a prime not dividing q and suppose that the integral
cohomology of G(C) has no p-torsion. Then BG(Fq)^pis spherically resolvable of
finite length [22]. This applies in particular to the groups SL(n; Fq) and Sp(n*
*; Fq)
at all primes p-different from the characteristic. In those examples the resolu*
*tion is
given by spaces of the form S2n-1{pr} which allows one to conclude that the ord*
*er
of the Sylow p-subgroup is an upper bound for the order of torsion in ss*(BG^p).
8.3. Clark-Ewing groups. These are groups of the form T o W , where T
is a finite product of cyclic groups of order pr (r fixed) and W is a p-adic ps*
*eudo-
reflection group. The resulting groups have the property that their mod-p coho-
mology is a symmetric algebra. It is shown in [22] that most of those are resol*
*vable
of finite length, where the resolving spaces are of the form S2n-1{pr}. Thus, a*
*s for
groups of Lie type, there is a homotopy exponent for those examples as well.
8.4. Finite simple groups of 2-rank two. Let G be a finite simple group
of 2-rank two. Then one of the following holds
1. G has dihedral Sylow 2-subgroups and G is isomorphic to either A7 or
P SL(2; Fq) for q odd,
2. G has semi-dihedral Sylow 2-subgroup and so G is isomorphic to either
P SL(3; Fq) for q 3(mod 4); U(3; Fq) for q 1(mod 4), or M11,
3. G has wreathed Sylow 2-subgroup and G is isomorphic to either P SL(3; Fq)
for q 1(mod 4), or U(3; Fq) for q 3(mod 4); or
4. is isomorphic to U(3; F4).
In case 1 it is shown in [22] that the 1-connected cover of BG^2is homotopy
equivalent to S3{2r} for the appropriate r. If G has as semi-dihedral Sylow 2-
subgroup then a combination of results from [22] and [14] gives that BG^2is
spherically resolvable. In fact BG^2fibres over S5{2} with fibre S3{2r} for a
suitable r. It is worth pointing out though that work of Martino and Priddy
shows that the 2-completed classifying spaces of SL(3; F3) and M11 are homotopy
equivalent even before looping [26]. The existence of spherical resolutions of *
*length
4 in case 3 follows from [22]. The type of resolutions is similar to case 2. *
*The
only homotopy type in this family that has not been decided is case 4, namely
G = U(3; F4). Here one might guess that it behaves like one of the "exponential*
*ly
growing" examples discussed above.
8.5. The sporadic simple group J1. Using an unpublished theorem of J.
Harper which characterizes the 2-local homotopy type of the Lie group G2 cohomo-
logically, it is shown in [1, pp. 281-282] that the homotopy fibre of a map (BJ*
*1)^2
to (BG2)^2is (G2)^2. Thus (BJ1)^2is the homotopy fibre of a self map of (G2)^2.
8.6. The groups D(p) and E(p). The groups D(p) and E(p) are defined
above. The homotopy type of BD(p)^phasn't been worked out completely. It
is known though that P 16(p) splits off as a retract of the first. A complete
decomposition for E(p)^pwas given in section 5.
8.7. Amalgamated products of finite groups. Amalgamated products of
finite p-perfect groups are also p-perfect. They are generally infinite but if *
*the amal-
gam is finite then the resulting groups have finite virtual cohomological dimen*
*sion
[5]. Thus by [19] for such groups G, the loop space BG^pbehaves very much as
F. R. COHEN AND R. LEVI 25
if G were finite. In particular they have finite p-essential dimension. The p-l*
*ocal
homology of such groups is entirely torsion and in fact has an exponent bounded
above by the highest exponent for the factors. Thus having a finite p-essential
dimension implies, in conjunction with [20], that BG^phas an integral loop space
homology exponent.
There are interesting examples in the literature of finite groups which have*
* the
cohomology of an amalgamated product of some of their subgroups. One example
of this sort is given by the finite simple group M12 [1].
8.8. The group SL(3; Z). The special linear group of rank 3 over the integers
is in fact mildly out of our context being infinite. However, it can be describ*
*ed as a
generalized amalgamated product of finite subgroups and thus exhibits a somewhat
similar behavior as one expects from a finite group. For instance its integral *
*coho-
mology is entirely torsion (2 and 3 torsion). Thus the space BSL(3; Z)^pis only
interesting at 2 and 3. At the prime 3 the homotopy type has been worked out in
[8] by the first author. More recently the second author worked out the homotopy
type at p = 2 [17]. In both cases the answer is given in terms of Moore spaces *
*and
fibres of degree pr maps on spheres.
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Department of Mathematics, University of Rochester, Rochester, NY 14627 U.S.*
*A.
E-mail address: cohf@db1.cc.rochester.edu
Department of mathematics, Northwestern University, 2033 Sheridan Rd., Evans*
*ton,
IL 60208, U.S.A.
E-mail address: ran@math.nwu.edu