Stallings exact sequence

General definition
The Stallings exact sequence is a five-term exact sequence associated with any short exact sequence of groups.

Consider a short exact sequence of groups:

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

The Stallings exact sequence corresponding to this short exact sequence is the following five-term exact sequence:

$$H_2(G;\mathbb{Z}) \stackrel{\alpha}{\to} H_2(Q;\mathbb{Z}) \stackrel{\beta}{\to} N/[G,N] \stackrel{\sigma}{\to} H_1(G;\mathbb{Z}) \stackrel{\tau}{\to} H_1(Q;\mathbb{Z})$$

(Note that the letters used for the morphisms are only being used on this page (and in some of the references) and are not standardized).

Here, the groups $$H_2(G;\mathbb{Z}) = M(G)$$ and $$H_2(Q;\mathbb{Z}) = M(Q)$$ are the respective Schur multipliers for the groups $$G$$ and $$Q$$. The groups $$H_1(G;\mathbb{Z}) = G^{\operatorname{ab}}$$ and $$H_1(Q;\mathbb{Z}) = Q^{\operatorname{ab}}$$ are the respective abelianizations of $$G$$ and $$Q$$.

The maps are as follows:

Particular case of a central extension
Consider a short exact sequence of groups:

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

where the image of $$N$$ in $$G$$ is a central subgroup of $$G$$. In this case, the Stallings exact sequence is as follows:

$$H_2(G;\mathbb{Z}) \stackrel{\alpha}{\to} H_2(Q;\mathbb{Z}) \stackrel{\beta}{\to} N \stackrel{\sigma}{\to} H_1(G;\mathbb{Z}) \stackrel{\tau}{\to} H_1(Q;\mathbb{Z})$$

The maps are as follows:

Related constructions

 * Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization: This says a little bit more about the homomorphism $$H_2(Q;\mathbb{Z}) \to N$$ in the special case that the extension is a central extension.
 * Dual universal coefficients theorem for group cohomology
 * Stallings-Stammbach sequence
 * Hopf's formula for Schur multiplier
 * Ganea sequence

Journal references

 * , main theorem, stated on the first page of the paper.
 * , (1.3), listed on the first page. This reference also explains (Theorem 2.2) how this is related to the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization