group cohomology (topological definition)

Let G be a topological groupMathworldPlanetmath. Suppose some contractible space X admits a fixed point free action of G, so that the quotient map p:XX/G is a fibre map. Then X/G, denoted BG is called the classifying spacePlanetmathPlanetmath of G. Classifying spaces always exist and are unique up to homotopyMathworldPlanetmathPlanetmath. Further, if G has the structure of a CW- complex, we can choose BG to have one too.

The group (co)homologyMathworldPlanetmathPlanetmath of G is defined to be the (co)homology of BG. From the long-exact sequence associated to the fibre map, p, we know that πn(G)=πn+1(BG) for n0. In particular the fundamental groupMathworldPlanetmathPlanetmath of BG is π0(G), which inherits a group structure as a quotientPlanetmathPlanetmath of G. Let H denote π0(G). Then H acts freely on the cells of BG*, the universal over of BG. Hence the cellular resolution for BG*, denoted, C*(BG*), is a sequence of free ZH- modules and ZH- linear maps. Taking coefficients in some ZH- module A, we have


In particular, when G is discrete, p must be the covering map associated to a universal coverMathworldPlanetmath. Hence X=BG* and C*(BG*) is exact, as X is contractibleMathworldPlanetmath and hence has trivial homology. Note in this case H=G. So for a discrete group G, we have,


Also, as passing to the universal cover preserves πn for n>1, we know that πn(BG)=0 for n>1. BG is always connected and for a discrete group π0(G)=G so we have BG=k(G,1), the Eilenberg - Maclane space.

As an example take G=SU1. Note topologically, SU1=S1=k(Z,1). As πn(G)=πn+1(BG) for n0, we know that BSU1=k(Z,2)=CP.

More explicitly, we may identify SU1 with the unit complex numbers. This acts freely on the infinite complex sphere (which is contractible) leaving a quotient of CP.

Hence Hn(SU1,Z)=Z if 2 divides n and 0 otherwise.

Similiarly BC2=RP and BSU2=HP, as C2 and SU2 are isomorphicPlanetmathPlanetmathPlanetmath to U(R) and U(H) respectively. So Hn(C2,Z2)=Z2 for all n and Hn(SU2,Z)=Z if 4 divides n and 0 otherwise.

Title group cohomologyMathworldPlanetmathPlanetmath (topological definition)
Canonical name GroupCohomologytopologicalDefinition
Date of creation 2013-03-22 14:32:24
Last modified on 2013-03-22 14:32:24
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 18
Author whm22 (2009)
Entry type Definition
Classification msc 55N25
Related topic CohomologyGroupTheorem
Defines group cohomology
Defines classifying spaces