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\begin{document}

\title{On spaces with periodic cohomology}

\author{Alejandro Adem}

\address{Mathematics Department,
University of Wisconsin, Madison, Wisconsin 53706}
\email{adem@math.wisc.edu}

\thanks{Both authors were partially supported by grants from the
NSF}

\author{Jeff H. Smith}
\address{Mathematics Department,
Purdue University, West Lafayette, Indiana 47907} 
\email{jhs@math.purdue.edu}

\commby{Dave J. Benson}

\subjclass[2000]{Primary 57S30; Secondary 20J06}
\keywords{Group cohomology, periodic complex}

\date{October 27, 1999}
\begin{abstract}
We define a generalized notion of cohomological periodicity for a 
connected CW-complex $X$,
and show that it is equivalent to the existence of an oriented 
spherical fibration over $X$
with total space homotopy equivalent to a finite dimensional complex. 
As applications we
characterize discrete groups which can act freely and properly on some 
$\mathbb R^n\times
\mathbb S^m$, show that every rank two $p$-group acts freely on a 
homotopy product of two
spheres and construct exotic free actions of many simple groups on such 
spaces.
\end{abstract}

\maketitle

\section{Introduction}

A classical result in group cohomology (proved by R. G. Swan \cite{Sw}) 
is the fact that a
finite group has periodic cohomology if and only if it acts freely on a 
finite complex with the
homotopy type of a sphere. Since then there have been numerous attempts 
to extend this type of
result to other classes of groups, including (1) infinite groups with 
periodic cohomology and
(2) finite groups with non-periodic cohomology.

In this note we describe a homotopy-theoretic characterization of 
\textit{cohomological
periodicity}, a notion which we now make precise.


\begin{definition}\label{defperiodiccohomology}
The cohomology of a CW-complex $X$ is said to be \emph{periodic} if 
there is an integral
cohomology class $\mya\in H^*(X,\mathbb Z)$ and an integer $d\ge 0$ 
such that the cup product
with $\mya$ gives an isomorphism
\[
\mya\cup-\,\colon H^m_{loc}(X,\mathcal B)\to H^{m+
|\mya|}_{loc}(X,\mathcal B)
\]
for every local coefficient system $\mathcal B$ and every integer $m\ge 
d$.
\end{definition}

Our main result is the following

\begin{theorem}
Let $X$ denote a connected CW-complex. The cohomology of $X$ is 
periodic if and only if
there is an orientable spherical fibration $E\ra X$ such that the total 
space $E$ is homotopy
equivalent to a finite dimensional CW-complex.
\end{theorem}

Our result has a number of consequences, which we now list.

\begin{theorem}
A discrete group $\Gamma$ acts freely and properly on $\mathbb 
R^n\times\mathbb S^m$ for some
$m,n>0$ if and only if $\Gamma$ is a countable group with periodic 
cohomology.
\end{theorem}

This represents the definitive extension and verification of a question 
first raised by Wall
\cite{W2} for groups of finite virtual cohomological dimension (and 
settled in that special
case in \cite{CP}).

Specializing to finite groups we show

\begin{theorem}
Let $X$ denote a finite dimensional $G$--CW complex, $G$ a finite 
group, such that every
abelian subgroup of the isotropy subgroups is cyclic. Then for some 
large integer $N>0$ there
exists a finite dimensional $G$--CW complex $Y$ with a free $G$-action 
such that $Y\simeq
\bbS^N\times X$. If $X$ is finitely dominated and simply connected, then 
$Y$ can be taken to be
a finite complex.
\end{theorem}

For finite groups with non-periodic cohomology a key problem has been 
to construct free actions
on 
\textit{products} of spheres. In fact given any finite group $G$, if we 
define its rank $r(G)$
as the dimension of its largest elementary abelian subgroup, then it 
has been conjectured that
Swan's theorem admits the following generalization: if $k$ is the 
smallest integer such that a
finite group $G$ will act freely on a finite complex $Y\simeq\bbS^{n_1}
\times\dots\times\bbS^{n_k}$, then $k=r(G)$. We obtain a complete 
answer for the rank two
situation in the case of finite $p$-groups, representing the first 
substantial result beyond
spherical space forms:

\begin{theorem}
Let $G$ denote a finite $p$-group. Then $G$ acts freely on a finite 
complex $Y\simeq
\bbS^n\times\bbS^m$ if and only if $G$ does not contain a subgroup 
isomorphic to
$\bbZ/p\times\bbZ/p\times\bbZ/p$.
\end{theorem}

Constructing interesting free actions of non-abelian finite simple 
groups is a particularly
difficult problem. Note that they all contain a copy of $\mathbb 
Z/2\times \mathbb Z/2$, hence
cannot act freely on a single sphere. Using the classification of 
finite simple groups
\cite{Go}, character theory and local methods in homotopy theory we prove

\begin{theorem}
Let $G$ denote a rank two simple group different from $PSL_3(\bbF_p)$, 
$p$ an odd prime. Then
$G$ acts freely on a finite complex $Y\simeq\bbS^n\times\bbS^m$. 
Furthermore any such action
must be exotic, i.e. it cannot be a product action.
\end{theorem}

In the following section we outline the proof of our main theorem and 
its consequences.%
\footnote {In a recent preprint \cite{MT}, Mislin and Talleli have used 
different methods to
prove a result similar to 1.3 for a large class of discrete groups, 
namely those which are
hierarchically decomposable, with a bound on the orders of their finite 
subgroups.} Complete
details will appear elsewhere.


\section{Proof of the main theorem}

The proof follows from a series of lemmas which we now describe. We 
begin by recalling some
basic notions. Cohomology will be assumed with trivial $\mathbb Z$ 
coefficients unless
specified otherwise. A spherical fibration is a Serre fibration $E\to 
X$ such that the fiber is
homotopy equivalent to a sphere $\bbS^{n-1}$.  If the spherical 
fibration is oriented, its
Euler class is the cohomology class in $H^n(X)$ that is the 
trangression in the Serre spectral
sequence of the fundamental class of the fiber.

We denote the $k$th Postnikov section of the $p$-sphere by 
$P_k\bbS^p$; recall that the
induced map $H^p(P_k\bbS^p) \to H^p(\bbS^p)$ of integral cohomology 
groups is an isomorphism if
$k\ge p$.

\begin{lemma}\label{lempowers}
Let $X$ be a connected CW-complex.  For every integral cohomology class 
$\mya\in H^*(X)$ of
positive degree and every integer $k\ge 0$ there is an integer $q\ge1$ 
and a fibration sequence
$P_{k+q|\mya|-1}\bbS^{q|\mya|-1}\to E\to X$ such that a generator of 
the integral cohomology group
$H^{q|\mya|-1}(P_{k+q|\mya|-1}\bbS^{q|\mya|-1})$ trangresses to 
$\mya^q\in H^{q|\mya|}(X)$ in the Serre
spectral sequence of the fibration.
\end{lemma}


\begin{lemma}\label{lemsphericalfibration}
Let $P_{k+|\mya|-1}\bbS^{|\mya|-1}\to E \to X$ be a fibration sequence such 
that a generator of
$H^{|\mya|-1}(P_{k+|\mya|-1}\bbS^{|\mya|-1})$ trangressses to $\mya\in H^*(X)$ 
in the Serre spectral
sequence. If $k\ge d-2$ and the map $\mya\cup-\,\colon H^m(X)\to H^{m+
|\mya|}(X)$ is an isomorphism
in integral cohomology for $m\ge d$, then there is an orientable 
spherical fibration $\overline
E\to X$ with Euler class $\mya$.  If in addition the map $\mya\cup-\,\colon 
H^m_{loc}(X,\mathcal
B)\to H^{m+|\mya|}_{loc}(X,\mathcal B)$ is an isomorphism for every local 
coefficient system
$\mathcal B$ and every integer $m\ge d$, then $\overline{E}$ is homotopy 
equivalent to a
CW-complex of dimension less than $d+|\mya|$.
\end{lemma}

\begin{lemma}\label{lemproductfibration}
Let $E \to X$ be an orientable spherical fibration, where $X$ is a 
CW-complex with
$k$-skeleton $X^{(k)}$. For every integer $k\ge0$ there is an integer $q$ 
such that the fibration
$E_k\to X^{(k)}$ in the cartesian square
\[
\begin{CD}
E_k @>>> *^q_XE\\ @VVV @VVV\\ X^{(k)} @>>> X
\end{CD}
\]
is a product fibration, where $*^q_XE$ denotes the $q$-fold iterated 
fibered join of the bundle
$E\to X$.
\end{lemma}

The \textit{fiber join} of a fiber bundle $F\to E\to X$ with itself 
produces a new fiber bundle
over $X$ with fiber the join $F*F$. In particular a spherical fibration 
over $X$ will again
yield a spherical fibration over $X$. To prove the three lemmas above 
we use induction combined
with a systematic application of fiber joins. Roughly stated the 
process of fiber joins has the
effect of multiplying certain relevant obstructions by an integer; on 
the other hand these
obstructions lie in homotopy groups of spheres which have 
\textit{finite exponent}. Hence by
taking suitable fiber joins to begin with, we can ensure the vanishing 
of the obstruction and
hence complete our inductive arguments. One sees that a proper notion 
of periodicity must
contemplate twisted coefficients, as otherwise we cannot ensure finite 
dimensionality of the
total space $E$. Note that for $X=BG$, $G$ finite, it is well known 
that periodicity with
trivial coefficients implies our more general notion of periodicity. 
This should be considered
an `accidental' occurrence, as our approach makes explicit how the full 
notion of periodicity
is required.

We now have
\begin{theorem}\label{thmfinitedimensionalE}
Let $X$ denote a connected CW-complex. The cohomology of $X$ is 
periodic if and only if
there is an orientable spherical fibration $E \to X$ such that the 
total space $E$ is homotopy
equivalent to a finite dimensional CW-complex.
\end{theorem}

\begin{proof}
If $E\to X$ is an oriented spherical fibration and the total space is 
homotopy equivalent to a
finite dimensional complex, then the Gysin sequence with local 
coefficients shows that $X$ has
periodic cohomology.

Conversely, assume that $X$ has periodic cohomology. Let $\mya\in H^*(X)$ 
be an integral
cohomology class and $d\ge0$ be an integer such that $\mya\cup-\,\colon 
H^m(X)\to H^{m+|\mya|}(X)$
is an isomorphism for $m\ge d$. By Lemma~\ref{lempowers}, there is an 
integer $q\ge1$ and a
fibration sequence
\[
P_{d+q|\mya|-1}\bbS^{q|\mya|-1}\to E\to X
\]
such that a generator of $H^{q|\mya|-1}(P_{d+q|\mya|-1}\-\bbS^{q|\mya|-1})$ 
trangresses to $\mya^q\in
H^{q|\mya|}(X)$ in the Serre spectral sequence of the fibration. By
Lemma~\ref{lemsphericalfibration}, since $d\ge d-2$, there is an 
orientable spherical
fibration with Euler class $\mya^q$, and the total space has the homotopy 
type of a finite
dimensional complex thanks to our generalized periodicity assumption.
\end{proof}

Let $\Gamma$ denote a discrete group; its classifying space has the 
homotopy type of a
CW-complex, whence we obtain

\begin{corollary}
For a discrete group $\Gamma$, $B\Gamma$ has periodic cohomology if and 
only if $\Gamma$ acts
freely on a finite dimensional complex homotopy equivalent to a sphere.
\end{corollary}

If a countable discrete group $\Gamma$ acts freely on a finite 
dimensional complex $Y$ homotopy
equivalent to a sphere, then it acts freely and properly on $\mathbb 
R^n\times\mathbb S^m$ for
some $m,n>0$. Indeed $Y/\Gamma$ has countable homotopy groups, hence is 
homotopic to a
countable complex which in turn is homotopic to an open submanifold $V$ 
in some Euclidean
space; applying the h-cobordism theorem we can infer that for 
sufficiently large $q$ we have a
diffeomorphism $\widetilde V\times\mathbb R^q\cong \mathbb 
R^n\times\mathbb S^m$ for some $m,n
\ge 0$ (see \cite{CP}). Hence we obtain

\begin{theorem}
A discrete group $\Gamma$ acts freely and properly on $\mathbb 
R^n\times \mathbb S^m$ for some
$m,n>0$ if and only if $\Gamma$ is countable and has periodic cohomology.
\end{theorem}

By a result of Wall \cite{W1}, a finitely dominated CW-complex $X$ is 
homotopy equivalent to a
finite CW-complex if and only if its finiteness obstruction vanishes; 
the finiteness
obstruction is an element in $\widetilde K_0(\bbZ\pi_1(X))$.

\begin{theorem}\label{thmfiniteE}
Let $X$ be a connected CW-complex of finite type such that the reduced 
projective class group
$\widetilde K_0(\bbZ\pi_1(X))$ is a torsion group. The cohomology of 
$X$ is periodic if and
only if there is a spherical fibration $E \to X$ such that the total 
space $E$ is homotopy
equivalent to a finite complex.
\end{theorem}

\begin{corollary}
Let $\Gamma$ denote a discrete group such that $B\Gamma$ is of finite 
type and $\widetilde
K_0(\mathbb Z\Gamma)$ is torsion. Then $B\Gamma$ has periodic 
cohomology if and only if it acts
freely and co-compactly on a CW-complex $Y$ homotopy equivalent to 
a sphere.
\end{corollary}

Again these results rely on the fact that fiber joins multiply 
obstructions by an integer---the
hypothesis of finite exponent is used to dispose of the finiteness 
obstruction by successive
fiber joins. This argument was used by Swan, based on the fact that 
$\widetilde K_0(\mathbb Z
G)$ is a finite abelian group for any finite group $G$.

\section{Finite group actions}

Let $G$ denote a finite group; to begin we will be interested in 
actions of $G$ on spaces such
that the isotropy subgroups have periodic cohomology. The key fact is 
given by

\begin{proposition}
Let $X$ denote a finite dimensional $G$--CW complex, where $G$ is a 
finite group. Then the
cohomology of $X\times_GEG$ is periodic if and only if all the isotropy 
subgroups have periodic
cohomology.
\end{proposition}

This allows us to describe important specific situations where the 
results from the previous
section will apply. In particular, the following theorem is a direct 
consequence of
Theorem~\ref{thmfinitedimensionalE} and 
Lemma~\ref{lemproductfibration}:

\begin{theorem}\label{thmrankoneisotropy}
Let $X$ denote a finite dimensional $G$--CW complex ($G$ a finite 
group) such that all of its
isotropy subgroups have periodic cohomology. Then there exists a finite 
dimensional CW-complex
$Y$ with a free $G$-action such that $Y\simeq \bbS^N\times X$. If $X$ 
is simply connected and
finitely dominated, then we can assume that $Y$ is a finite complex.
\end{theorem}

One can show that given any finite $p$-group $P$ and an element $x$ of 
order $p$ in its
center, there exists a complex representation $V$ for $G$ such that the 
subgroup $\langle
x\rangle$ acts freely on the associated sphere $S(V)$. In particular if 
$P$ is a rank two
$p$-group, such a sphere will have periodic isotropy and so we can use 
the result above to
construct a free action on a product of two spheres up to homotopy, 
from which we infer 1.5.

Turning finally to the case of simple groups our strategy is again to 
construct an action on a
sphere such that the isotropy is periodic. Unfortunately this cannot 
always be done, hence
instead we construct a homotopy action such that the associated Borel 
construction has periodic
cohomology. This can be done one prime at a time (using methods from 
local homotopy theory) and
furthermore we can use local decompositions of the classifying spaces 
for the groups involved.
Making use of the classification of finite simple groups (i.e. a case 
by case analysis) we show

\begin{theorem}
Let $G$ denote a rank two finite simple group other than 
$PSL_3(\bbF_p)$, $p$ an odd prime.
Then $G$ acts freely on a finite complex $Y\simeq \bbS^n\times\bbS^m$. 
Furthermore any such
action must be exotic, i.e. it cannot be a product action.
\end{theorem}

The last statement follows from the observation that any rank two 
simple group contains a copy
of the alternating group $A_4$ and the fact that by a result due to 
Oliver \cite{O} this group
must act exotically.

We conclude with examples illustrating the theorem above.
\begin{example}
Both the alternating group $A_5$ and the linear group $SL_3(\mathbb 
F_2)$ have complex
representations of dimension three such that the action on the 
associated sphere has isotropy
with periodic cohomology. Hence both of these groups will act freely on 
finite complexes of the
form $\mathbb S^N\times \mathbb S^5$. On the other hand, for $G=M_{11}$, 
the first Mathieu
group, we must use homotopy representations at the primes $p=2, 3$ to 
construct an action on
$\mathbb S^{383}$ giving rise to a Borel construction with periodic 
cohomology. Hence $M_{11}$
acts freely on a finite complex $Y\simeq \bbS^N\times\bbS^{383}$.
\end{example}

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\bibitem{Go} Gorenstein, D., \emph{The Classification of Finite
Simple Groups}, Plenum Press (1983).
\MR{86i:20024} 
\bibitem{MT} Mislin, G. and Talelli, O.,
\emph{On Groups which Act Freely and Properly on Finite Dimensional 
Homotopy Spheres}, preprint
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\bibitem{O} Oliver, R., \emph{Free Compact Group Actions on Products
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\bibitem{Sw} Swan, R.G., \emph{Periodic Resolutions for Finite Groups},
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\bibitem{W1} Wall, C.T.C., \emph{Finiteness Conditions for 
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\bibitem{W2} Wall, C.T.C., \emph{Periodic Projective Resolutions},
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\end{thebibliography}

\end{document}