Group cohomology of free groups

This article gives specific information, namely, group cohomology, about a family of groups, namely: free group.
[[:Category:group cohomology {{{connective}}} group families|View group cohomology {{{connective}}} group families]] | View other specific information about free group

[[Category:group cohomology {{{connective}}} group families]]

We discuss here the group homology and cohomology for the free group $F_{n}$ on a freely generating set of size $n$.

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

$H_{q}(F_{n};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&q=0\\\mathbb {Z} ^{n},&q=1\\0,&q>1\\\end{array}}\right.$

Over an abelian group

The homology groups over an abelian group $M$ for the trivial group action are as follows:

$H_{q}(F_{n};M)=\left\lbrace {\begin{array}{rl}M,&q=0\\M^{n},&q=1\\0,&q>1\\\end{array}}\right.$

Homology groups for trivial group action

FACTS TO CHECK AGAINST (cohomology group for trivial group action):
First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization
In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

Over the integers

$H^{q}(F_{n};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&q=0\\\mathbb {Z} ^{n},&q=1\\0,&q>1\\\end{array}}\right.$

Over an abelian group

The cohomology groups over an abelian group $M$ for the trivial group action are as follows:

$H^{q}(F_{n};M)=\left\lbrace {\begin{array}{rl}M,&q=0\\M^{n},&q=1\\0,&q>1\\\end{array}}\right.$