# 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.