Smith theory of group actions

A collection of techniques and results first obtained by P.A. Smith around 1940 (see [a5], [a6], [a7]) in the area of finite transformation groups. Smith theory is now (2000) best understood via cohomological methods, following an approach introduced by A. Borel (see [a2], [a3]).

The main goal of Smith theory is to study actions of finite $p$-groups on familiar and accessible spaces such as polyhedra or manifolds (cf. also Action of a group on a manifold; $p$-group). However, it can easily be adapted to a very large class of spaces, the so-called finitistic spaces. These are spaces such that every covering has a finite-dimensional refinement (see [a4], p. 133, for details; see also General topology; Covering (of a set)). The most important examples are compact spaces and finite-dimensional spaces. The spaces occurring below are assumed to be of this type.

Let $X$ be such a finitistic space and let $P$ be a finite $p$-group acting on it (here $p$ is a fixed prime number). Let $X ^ { P }$ be the fixed-point set of the action, that is,

\begin{equation*} X ^ { P } = \{ x \in X : g x = x , \forall g \in P \}. \end{equation*}

The two basic theorems of Smith theory are as follows:

a) If $X$ has the $\operatorname{mod} p$ homology of a point (cf. also Homology), then the fixed-point set $X ^ { P }$ also has the $\operatorname{mod} p$ homology of a point; in particular, it is non-empty.

b) If $X$ has the $\operatorname{mod} p$ homology of a sphere, then the fixed-point set $X ^ { P }$ (possibly empty) also has the $\operatorname{mod} p$ homology of a sphere.

Of course, the main examples here are when $X \cong D ^ { n}$, the $n$-dimensional disc, and when $X \cong S ^ { m }$, the $m$-dimensional sphere. However, the $\operatorname{mod} p$ homological nature of the results are important, as they can fail at other prime numbers.

Homological methods building on Smith's original approach can be used to verify very general restrictions associated to actions of finite $p$-groups. For example, if $X$ satisfies the additional hypothesis that its total $\operatorname{mod} p$ cohomology is finite, then there is an inequality arising from an action of a finite $p$-group $P$ on $X$:

\begin{equation*} \sum _ { i = 0 } \operatorname { dim } H ^ { i } ( X , \mathbf{Z} / p ) \geq \sum _ { i = 0 } \operatorname { dim } H ^ { i } ( X ^ { P } , \mathbf{Z} / p ). \end{equation*}

Note that this implies that the fixed-point set $X ^ { P }$ has finitely many components and that each of them has finite $\operatorname{mod} p$ cohomology. The two previous results can be derived from this inequality.

Another important result which follows from Smith theory is the fact that if $G$ is a finite group acting on a space $X$ which is finitistic and acyclic (i.e. has the integral homology of a point), then the orbit space $X / G$ is also acyclic.

Smith theory can be considered a precursor to the general cohomological theory of transformation groups (cf. also Transformation group). Given a finite group $G$ acting on a space $X$, one constructs a space, called the Borel construction on $X$, as follows: $X \times_ { G } E G = ( X \times E G ) / G$, where $E G$ is a free, contractible $G$-space. The projection induces a bundle mapping $X \times_{G} EG \rightarrow B G$, where $B G = E G / G$ is the so-called classifying space of $G$, an Eilenberg–MacLane space of type $K ( G , 1 )$. The analysis of this bundle and related constructions is the basic tool in this area. In particular, the main results from Smith theory follow from considering the case $G = \mathbf{Z} / p$; if $X$ is an $n$-dimensional complex with a $G$-action, then the inclusion $X ^ { G } \hookrightarrow X$ induces an isomorphism

\begin{equation*} H ^ { j } ( X \times _ { G } E G , \mathbf{Z} / p ) \rightarrow H ^ { j } ( X ^ { G } \times B G , \mathbf Z / p ) \end{equation*}

provided $j > n$. This fact, combined with the spectral sequence in $\operatorname{mod} p$ cohomology associated to the fibration

\begin{equation*} X \times _ { G } E G \rightarrow B G, \end{equation*}

are the two main elements used in this reformulation of Smith theory.

See [a1], [a4] and [a8] for excellent references regarding Smith theory and transformation groups.

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