Group homology

Group homology is a technical mathematical tool that plays an important role in group theory and algebraic topology .

Definitions

Abstract definition

It is a group . The functor from the category of modules to the category of Abelian groups , of a module the group of co-invariants ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle A_{G}}$

${\displaystyle A_{G}:=A\otimes _{\mathbb {Z} [G]}\mathbb {Z} }$

assigns is very precise . Its n th left derivative is the n th homology group of with coefficients in the module . ${\displaystyle H_{n}(G,A)}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle A}$

Group homology can also be defined using the Tor functor :

${\displaystyle \mathrm {H} _{n}(G,A)=\mathrm {Tor} _{n}^{\mathbb {Z} [G]}(\mathbb {Z} ,A);}$

It is the group ring of and with the trivial provided operation. ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle G}$

From the description using the functor it can be seen that the group homology can be calculated using any projective resolution of the trivial module. That is, one chooses a long exact sequence of modules ${\displaystyle Tor}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Z} [G]}$

${\displaystyle 0\leftarrow A\leftarrow I_{0}\leftarrow I_{1}\leftarrow I_{2}\leftarrow \cdots }$

in which all projective modules are and is then defined as the homology of the chain complex obtained by tensing with the trivial module ${\displaystyle I_{i}}$ ${\displaystyle H_{*}(G,A)}$ ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle 0\leftarrow A\otimes _{\mathbb {Z} [G]}\mathbb {Z} \leftarrow I_{0}\otimes _{\mathbb {Z} [G]}\mathbb {Z} \leftarrow I_{1}\otimes _{\mathbb {Z} [G]}\mathbb {Z} \leftarrow I_{2}\otimes _{\mathbb {Z} [G]}\mathbb {Z} \leftarrow \cdots }$.

From the fundamental lemma of homological algebra it follows that it depends only on the module and not on the selected projective resolution. ${\displaystyle H_{*}(G,A)}$ ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle A}$

Explicit definition

As the projective resolution of the module , one can use the differential ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle A}$ ${\displaystyle (\mathbb {Z} [G^{n+1}],d_{n})}$

${\displaystyle d_{n}(\sigma _{0},\ldots ,\sigma _{n})=\sum _{i=0}^{n-1}(-1)^{i}(\sigma _{0},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n})}$      in which ${\displaystyle (\sigma _{0},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n}):=(\sigma _{0},\ldots ,\sigma _{i-1},\sigma _{i+1},\ldots ,\sigma _{n})}$

and then define it as the homology of the chain complex obtained by tensing with the trivial module . The elements of this complex are called homogeneous chains . ${\displaystyle H_{*}(G,A)}$ ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle \mathbb {Z} }$

The so-called bar resolution provides an equivalent definition . Here one looks with the differential ${\displaystyle \mathbb {Z} [G^{n}]}$

${\displaystyle \partial _{n}(g_{1},\ldots ,g_{n})=g_{1}\cdot (g_{2},\ldots ,g_{n})+\sum _{i=1}^{n-1}(-1)^{i}(g_{1},\ldots ,g_{i}g_{i+1},\ldots ,g_{n})-(g_{1},\ldots ,g_{n-1})}$

and then defined as the homology of the chain complex obtained by tensing with the trivial module . The elements of this complex are called inhomogeneous chains . ${\displaystyle H_{*}(G,A)}$ ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle \mathbb {Z} }$

Topological definition

Equivalent can also be defined as the singular homology with coefficients in the Eilenberg-MacLane space : ${\displaystyle H_{*}(G,A)}$ ${\displaystyle A}$ ${\displaystyle K(G,1)}$

${\displaystyle H_{*}(G,A)=H_{*}(K(G,1),A)}$.

This definition is the only one that can be used for practical calculations.

Low grades of homology

The following applies to the 0 th homology , especially for the trivial module . ${\displaystyle H_{0}(G,A)=A_{G}}$ ${\displaystyle H_{0}(G,\mathbb {Z} )=\mathbb {Z} }$ ${\displaystyle \mathbb {Z} [G]}$ ${\displaystyle \mathbb {Z} }$

For the 1st homology is

${\displaystyle H_{1}(G,\mathbb {Z} )=G/[G,G]}$

the abelization of . ${\displaystyle G}$

The 2nd homology with trivial coefficients can be calculated with the Hopf formula : if there is a finitely presented group with a finitely generated free group , then is ${\displaystyle G=F/R}$ ${\displaystyle F}$

${\displaystyle H_{2}(G,\mathbb {Z} )=(R\cap [F,F])/[F,R]}$.

Examples

• ${\displaystyle H_{i}(\mathbb {Z} ,\mathbb {Z} )=\left\{{\begin{array}{cc}\mathbb {Z} &i=0,1\\0&{\mbox{sonst}}\end{array}}\right\}}$
• ${\displaystyle H_{i}(\mathbb {Z} /n\mathbb {Z} ,\mathbb {Z} )=\left\{{\begin{array}{cc}\mathbb {Z} &i=0\\\mathbb {Z} /n\mathbb {Z} &i{\mbox{ ungerade}}\\0&i\not =0{\mbox{ gerade}}\end{array}}\right\}}$
• ${\displaystyle H_{i}(SL(2,\mathbb {Z} ),\mathbb {Z} )=\left\{{\begin{array}{cc}\mathbb {Z} &i=0\\\mathbb {Z} /12\mathbb {Z} &i{\mbox{ ungerade}}\\0&i\not =0{\mbox{ gerade}}\end{array}}\right\}}$

history

The history of group homology begins with a work published in 1936 by Witold Hurewicz Contributions to the Topology of Deformations. IV. Aspherical spaces , in which it is proven that the homotopy type of an aspherical space depends only on its fundamental group and therefore group homology can be defined as the homology of an aspherical space with a fundamental group. In his 1942 published work fundamental group and Betti second group showed Heinz Hopf that the cokernel of Hurewicz figure in degrees and that of the producers and relations of a presentation can be calculated. After Hopf's publication, the area developed rapidly in the 1940s through the work of Eckmann, Eilenberg-MacLane, Hopf and Freudenthal; Eilenberg and MacLane found the definition in their 1945 work Relations between homology and homotopy groups of spaces through the dissolution of the bar and soon afterwards the general definition in terms of projective resolutions was also given. ${\displaystyle H_{*}(\pi )}$ ${\displaystyle \pi }$ ${\displaystyle H_{2}(\pi )}$ ${\displaystyle 2}$ ${\displaystyle H_{2}(\pi )}$

See also

• Group cohomology

literature

• Kenneth S. Brown : Cohomology of groups (= Graduate Texts in Mathematics 87). Corrected 2nd printing. Springer, New York et al. 1994, ISBN 0-387-90688-6
• DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , chap. 11.2: Homology Groups and Cohomology Groups (without prior knowledge of homological algebra)