Stallings exact sequence

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Definition

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:

Homomorphism Description
\alpha: H_2(G;\mathbb{Z}) \to H_2(Q;\mathbb{Z}) We apply the second homology group functor (which is covariant) to the quotient map G \to Q.
\beta: H_2(Q;\mathbb{Z}) \to N/[G,N] We can think of this as the special case of the map described in the central extension setting (see below), once we replace the original short exact sequence by the new short exact sequence 1 \to N/[G,N] \to G/[G,N] \to Q \to 1. More explicitly, we have a canonical map Q \wedge Q \to G/[G,N] via the commutator map, and we know that this must send the subgroup M(Q) = H_2(Q;\mathbb{Z}) (the kernel of the natural map Q \wedge Q \to Q) to N/[G,N]. Performing that restriction gives us the map we want.
\sigma: N/[G,N] \to H_1(G;\mathbb{Z}) Note that H_1(G;\mathbb{Z}) = G/[G,G] = G^{\operatorname{ab}}. The injective map N \to G that is part of the original short exact sequence sends the subgroup [G,N] of N inside [G,G], and thus induces a map N/[G,N] \to G/[G,G] = H_1(G;\mathbb{Z}).
\tau: H_1(G;\mathbb{Z}) \to H_1(Q;\mathbb{Z}) We apply the first homology group functor, aka the abelianization, to the quotient map G \to Q

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:

Homomorphism Description
\alpha: H_2(G;\mathbb{Z}) \to H_2(Q;\mathbb{Z}) We apply the second homology group functor (which is covariant) to the quotient map G \to Q.
\beta: H_2(Q;\mathbb{Z}) \to N This is the same as the element of \operatorname{Hom}(H_2(Q;\mathbb{Z}),N) defined at commutator map in central extension defines homomorphism from Schur multiplier of quotient group to central subgroup (note that that page uses the letter G for what we call Q here and the letter A for what we call N here). This map also appears, albeit more indirectly, in the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization.
\sigma: N \to H_1(G;\mathbb{Z}) Note that H_1(G;\mathbb{Z}) = G/[G,G] = G^{\operatorname{ab}}. The map we get is just the composite of the injective map N \to G and the quotient map G \to G/[G,G].
\tau: H_1(G;\mathbb{Z}) \to H_1(Q;\mathbb{Z}) We apply the first homology group functor, aka the abelianization, to the quotient map G \to Q.

Related constructions

References

Journal references