Lyndon–Hochschild–Serre spectral sequence

The spectral sequence can be used to compute the homology of the Heisenberg group G with integral entries, i.e., matrices of the form

${\displaystyle \left({\begin{array}{ccc}1&a&b\\0&1&c\\0&0&1\end{array}}\right),\ a,b,c\in \mathbb {Z} .}$

This group is a central extension

${\displaystyle 0\to \mathbb {Z} \to G\to \mathbb {Z} \oplus \mathbb {Z} \to 0}$

with center ${\displaystyle \mathbb {Z} }$ corresponding to the subgroup with a=c=0. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that[1]

${\displaystyle H_{i}(G,\mathbb {Z} )=\left\{{\begin{array}{cc}\mathbb {Z} &i=0,3\\\mathbb {Z} \oplus \mathbb {Z} &i=1,2\\0&i>3.\end{array}}\right.}$

For a group G, the wreath product is an extension

${\displaystyle 1\to G^{p}\to G\wr \mathbb {Z} /p\to \mathbb {Z} /p\to 1.}$

The resulting spectral sequence of group cohomology with coefficients in a field k,

${\displaystyle H^{r}(\mathbb {Z} /p,H^{s}(G^{p},k))\Rightarrow H^{r+s}(G\wr \mathbb {Z} /p,k),}$

is known to degenerate at the ${\displaystyle E_{2}}$-page.[2]