View source for Smith theory of group actions

A collection of techniques and results first obtained by P.A. Smith around 1940 (see [[#References|[a5]]], [[#References|[a6]]], [[#References|[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 [[#References|[a2]]], [[#References|[a3]]]).

The main goal of Smith theory is to study actions of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304001.png" />-groups on familiar and accessible spaces such as polyhedra or manifolds (cf. also [[Action of a group on a manifold|Action of a group on a manifold]]; [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304002.png" />-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 [[#References|[a4]]], p. 133, for details; see also [[General topology|General topology]]; [[Covering (of a set)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304003.png" /> be such a finitistic space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304004.png" /> be a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304005.png" />-group acting on it (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304006.png" /> is a fixed prime number). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304007.png" /> be the fixed-point set of the action, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304008.png" /></td> </tr></table>

The two basic theorems of Smith theory are as follows:

a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s1304009.png" /> has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040010.png" /> homology of a point (cf. also [[Homology|Homology]]), then the fixed-point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040011.png" /> also has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040012.png" /> homology of a point; in particular, it is non-empty.

b) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040013.png" /> has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040014.png" /> homology of a sphere, then the fixed-point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040015.png" /> (possibly empty) also has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040016.png" /> homology of a sphere.

Of course, the main examples here are when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040017.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040018.png" />-dimensional disc, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040019.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040020.png" />-dimensional sphere. However, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040022.png" />-groups. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040023.png" /> satisfies the additional hypothesis that its total <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040024.png" /> [[Cohomology|cohomology]] is finite, then there is an inequality arising from an action of a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040025.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040027.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040028.png" /></td> </tr></table>

Note that this implies that the fixed-point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040029.png" /> has finitely many components and that each of them has finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040030.png" /> cohomology. The two previous results can be derived from this inequality.

Another important result which follows from Smith theory is the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040031.png" /> is a [[Finite group|finite group]] acting on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040032.png" /> which is finitistic and acyclic (i.e. has the integral [[Homology|homology]] of a point), then the orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040033.png" /> is also acyclic.

Smith theory can be considered a precursor to the general cohomological theory of transformation groups (cf. also [[Transformation group|Transformation group]]). Given a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040034.png" /> acting on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040035.png" />, one constructs a space, called the Borel construction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040036.png" />, as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040038.png" /> is a free, contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040039.png" />-space. The projection induces a bundle mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040041.png" /> is the so-called [[Classifying space|classifying space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040042.png" />, an [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040043.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040044.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040045.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040046.png" />-dimensional complex with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040047.png" />-action, then the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040048.png" /> induces an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040049.png" /></td> </tr></table>

provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040050.png" />. This fact, combined with the [[Spectral sequence|spectral sequence]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040051.png" /> cohomology associated to the fibration

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130400/s13040052.png" /></td> </tr></table>

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Allday,   V. Puppe,   "Cohomological methods in transformation groups" , ''Studies Adv. Math.'' , '''32''' , Cambridge Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Borel,   "Nouvelle démonstration d'un théorème de P.A. Smith"  ''Comment. Math. Helv.'' , '''29'''  (1955)  pp. 27–39</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Borel,   "Seminar on transformation groups" , ''Ann. of Math. Stud.'' , '''46''' , Princeton Univ. Press  (1960)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.E. Bredon,   "Introduction to compact transformation groups" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.A. Smith,   "Transformations of finite period"  ''Ann. of Math.'' , '''39'''  (1938)  pp. 127–164</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.A. Smith,   "Transformations of finite period II"  ''Ann. of Math.'' , '''49'''  (1939)  pp. 690–711</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.A. Smith,   "Fixed point theorems for periodic transformations"  ''Amer. J. Math.'' , '''63'''  (1941)  pp. 1–8</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  T. tom-Dieck,   "Transformation groups" , ''Studies in Math.'' , '''8''' , de Gruyter  (1987)</TD></TR></table>