Stallings-Stammbach sequence

Definition

The Stallings-Stammbach sequence is a slight generalization of the Stallings exact sequence. Suppose we have a short exact sequence of groups:

$1\to N\to G\to Q\to 1$

Suppose that $B$ is a $Q$-module, i.e., an abelian group equipped with a homomorphism of groups:

$Q\to \operatorname {Aut} (B)$

The Stallings-Stammbach sequence is the following short exact sequence of groups:

$H_{2}(G;B){\stackrel {\alpha _{B}}{\to }}H_{2}(Q;B){\stackrel {\beta _{B}}{\to }}N^{\operatorname {ab} }\otimes _{Q}B{\stackrel {\sigma _{B}}{\to }}H_{1}(G;B){\stackrel {\tau _{B}}{\to }}H_{1}(Q;B)$

In the case that $B=\mathbb {Z}$ with trivial action of $Q$ on it, the Stallings-Stammback sequence becomes the Stallings exact sequence.

References

• On the homology theory of central group extensions: I—The commutator map and stem extensions by Beno Eckmann, Peter J. Hilton and Urs Stammbach, Commentarii Mathematici Helvetici, Volume 47,Number 1, Page 102 - 122(Year 1972): ^{Official copy (gated)More info}, (1.2) (first page of the paper)