Ganea sequence

Definition
The Ganea sequence is a six-term exact sequence associated with any short exact sequence of groups that corresponds to a central extension.

Explicitly, suppose:

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

is a short exact sequence of groups describing a central extension (i.e., the image of $$N$$ in $$G$$ is a central subgroup of $$G$$). The Ganea sequence for this extension is a sequence defined as follows:

$$G^{\operatorname{ab}} \otimes N \stackrel{\chi_0}{\to} H_2(G;\mathbb{Z}) \stackrel{\alpha}{\to} H_2(Q;\mathbb{Z}) \stackrel{\beta}{\to} N/[G,N] \stackrel{\sigma}{\to} H_1(G;\mathbb{Z}) \stackrel{\tau}{\to} H_1(Q;\mathbb{Z})$$

The last five terms (and corresponding four homomorphisms) of the sequence are the Stallings exact sequence. The first homomorphism is the new feature introduced by Ganea using methods from algebraic topology.

Note that the Ganea sequence does not completely subsume the usefulness of the Stallings exact sequence, because the latter is defined for all short exact sequences whereas the Ganea sequence is defined only for the short exact sequences that correspond to central extensions.

The terms Ganea extension or Ganea morphism may be used to put emphasis on the new homomorphism introduced by Ganea.