View source for Equivariant cohomology

A [[Cohomology|cohomology]] that takes the action of some group into account. More precisely, an equivariant cohomology in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e0360902.png" />-spaces (that is, topological spaces on which the continuous action of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e0360903.png" /> is defined) and equivariant mappings is a sequence of contravariant functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e0360904.png" /> (taking values in the category of Abelian groups) and natural transformations

<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/e/e036/e036090/e0360905.png" /></td> </tr></table>

with the following properties: a) equivariantly-homotopic mappings of pairs induce identity homomorphisms of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e0360906.png" />; b) an inclusion of the form

<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/e/e036/e036090/e0360907.png" /></td> </tr></table>

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/e/e036/e036090/e0360908.png" /></td> </tr></table>

and c) for every pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e0360909.png" /> the following cohomology sequence is exact:

<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/e/e036/e036090/e03609010.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609011.png" /> be a universal [[G-fibration|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609013.png" />-fibration]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609014.png" /> be the space associated with the universal fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609015.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609016.png" /> (that is, the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609017.png" /> under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609018.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609019.png" />). Then the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609020.png" /> yield an equivariant cohomology theory; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609021.png" /> is an arbitrary cohomology theory.

For any fixed group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609022.png" /> the collection of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609023.png" /> together with all possible homomorphisms induced by inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609024.png" /> of subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609025.png" /> is usually called the system of coefficients for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609026.png" />-theory. In some cases the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609027.png" /> are uniquely defined by their systems of coefficients (for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609028.png" /> is finite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609030.png" />).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E. Bredon,   "Equivariant cohomology theories" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.Y. Hsiang,   "Cohomology theory of topological transformation groups" , Springer  (1975)</TD></TR></table>



====Comments====
The principal use of equivariant cohomology is in equivariant obstruction theory and in such problems of equivariant (stable) homotopy theory as the solution of the Segal conjecture by G. Carlsson [[#References|[a1]]] (see also [[#References|[a2]]] and [[Homotopy|Homotopy]]).

Quite generally, in many parts of mathematics it is useful to consider also family and equivariant versions of various constructions and results. The various family (relative) and equivariant versions of parts of mathematics are then often also important tools in the non-equivariant and non-family settings. An example of this is the use of equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609031.png" />-theory in proofs of the Atiyah–Singer index and fixed-points theorems, cf., e.g., [[#References|[a3]]].

Thus, many theories, e.g. cohomology theories, have equivariant versions, e.g. equivariant (stable) homotopy theory [[#References|[a1]]], equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609033.png" />-theory [[#References|[a3]]], [[#References|[a4]]], equivariant cobordism [[#References|[a5]]], [[#References|[a7]]]. And there are equivariant versions of many theorems and constructions such as equivariant surgery [[#References|[a1]]], [[#References|[a7]]], equivariant smoothing [[#References|[a6]]] and equivariant transversality [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Carlsson,   "Equivariant stable homotopy and Segal's Burnside ring conjecture"  ''Ann. of Math.'' , '''120'''  (1984)  pp. 189–224</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.G. Lewis,   J.P. May,   M. Steinberger,   "Equivariant stable homotopy theory" , ''Lect. notes in math.'' , '''1213''' , Springer  (1986)  (With contributions by J.E. McClure)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Petrie,   J.D. Randall,   "Transformation groups on manifolds" , M. Dekker  (1984)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. tom Dieck,   "Transformation groups and representation theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Petrie,   "Pseudoequivalences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609034.png" />-manifolds"  R.J. Milgram (ed.) , ''Algebraic and geometric topology'' , '''33.1''' , Amer. Math. Soc.  (1978)  pp. 169–210</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Lashof,   M. Rothenberg,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609035.png" />-smoothing theory"  R.J. Milgram (ed.) , ''Algebraic and geometric topology'' , '''33.1''' , Amer. Math. Soc.  (1978)  pp. 211–266</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Browder,   F. Quinn,   "A surgery theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036090/e03609036.png" />-manifolds and stratified sets" , ''Manifolds (Tokyo)'' , Univ. Tokyo Press  (1973)  pp. 27–36</TD></TR></table>