# group cohomology (topological definition)

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

The group (co)homology^{} 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 ${\pi}_{n}(G)={\pi}_{n+1}(BG)$ for $n\ge 0$. In particular the fundamental group^{} of $BG$ is ${\pi}_{0}(G)$, which inherits a group structure as a quotient^{} of $G$. Let $H$ denote ${\pi}_{0}(G)$. Then $H$ acts freely on the cells of $B{G}^{*}$, the universal over of $BG$. Hence the cellular resolution for $B{G}^{*}$, denoted, ${C}_{*}(B{G}^{*})$, is a sequence of free $ZH$- modules and $ZH$- linear maps. Taking coefficients in some $ZH$- module $A$, we have

$${H}^{n}(G;A)={H}^{n}({C}_{*}(B{G}^{*});A)\mathrm{and}{\mathrm{H}}_{\mathrm{n}}(\mathrm{G};\mathrm{A})={\mathrm{H}}_{\mathrm{n}}({\mathrm{C}}_{*}({\mathrm{BG}}^{*});\mathrm{A})$$ |

In particular, when $G$ is discrete, $p$ must be the covering map associated to a universal cover^{}. Hence $X=B{G}^{*}$ and ${C}_{*}(B{G}^{*})$ is exact, as $X$ is contractible^{} and hence has trivial homology. Note in this case $H=G$. So for a discrete group $G$, we have,

$${H}^{n}(G;A)=Ex{t}_{ZG}^{n}(Z,A)\mathrm{and}{\mathrm{H}}_{\mathrm{n}}(\mathrm{G};\mathrm{A})={\mathrm{Tor}}_{\mathrm{ZG}}^{\mathrm{n}}(\mathrm{Z},\mathrm{A})$$ |

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

As an example take $G=S{U}_{1}$. Note topologically, $S{U}_{1}={S}^{1}=k(Z,1)$. As ${\pi}_{n}(G)={\pi}_{n+1}(BG)$ for $n\ge 0$, we know that $BS{U}_{1}=k(Z,2)=C{P}^{\mathrm{\infty}}$.

More explicitly, we may identify $S{U}_{1}$ with the unit complex numbers. This acts freely on the infinite complex sphere (which is contractible) leaving a quotient of $C{P}^{\mathrm{\infty}}$.

Hence ${H}^{n}(S{U}_{1},Z)=Z$ if $2$ divides $n$ and $0$ otherwise.

Similiarly $B{C}_{2}=R{P}^{\mathrm{\infty}}$ and $BS{U}_{2}=H{P}^{\mathrm{\infty}}$, as ${C}_{2}$ and $S{U}_{2}$ are isomorphic^{} to U(R) and U(H) respectively. So ${H}^{n}({C}_{2},{Z}_{2})={Z}_{2}$ for all $n$ and ${H}^{n}(S{U}_{2},Z)=Z$ if $4$ divides $n$ and $0$ otherwise.

Title | group cohomology^{} (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 |