Interpretation of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem

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Statement

This provides an interpretation of the Baer correspondence (and various types of generalized Baer correspondence) as providing a natural splitting of the short exact sequence arising from the dual universal coefficients theorem for group cohomology (more specifically, see formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization).

General background

Suppose G and A are abelian groups. Consider the second cohomology group for trivial group action \! H^2(G;A). By formula for second cohomology group for trivial group action in terms of second homology group and abelianization#Case of abelian group (which follows from the dual universal coefficients theorem for group cohomology, we have the following natural short exact sequence:

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

where the image of \operatorname{Ext}^1 is H^2_{sym}(G;A), i.e., the group of cohomology classes represented by symmetric 2-cocycles and corresponding to the abelian group extensions. We also know, again from the general theory, that the short exact sequence above splits, i.e., H^2_{sym}(G;A) has a complement inside H^2. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

Reduction to problem of splitting the short exact sequence at the level of cocycles

Before describing the splitting strategy, we make the Skew map more explicit. First, note that the Skew map is a map from the group of 2-cochains for the trivial group action to itself, i.e.:

C^2(G;A) \stackrel{\operatorname{Skew}}{\to} C^2(G;A)

It turns out that, because skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism, the restriction of this map to the subgroup Z^2(G;A) (of 2-cocycles) has image inside the group of alternating bihomomorphisms from G to A, which can be viewed as \operatorname{Hom}(\bigwedge^2G,A). Thus, we get a map:

Z^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)

It further turns out that the group of 2-coboundaries B^2(G;A) is contained in the kernel of the map, because any 2-coboundary is symmetric (by definition). Thus, the map descends to a map:

H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)

The last map is the map appearing in the short exact sequence.

Our splitting strategy will provide a one-sided inverse to the map:

Z^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)

i.e., it will split the short exact sequence:

0 \to Z^2_{sym}(G;A) \to Z^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

As a corollary, the short exact sequence at the level of cohomology also gets split.

Reduction to a linear problem

We note that, because bihomomorphism to abelian group implies 2-cocycle, we have an injective map:

\operatorname{Hom}(\bigotimes^2G,A) \to Z^2(G;A)

Further, the skew map restricts to \operatorname{Hom}(\bigwedge^2G,A), i.e., we have a restricted skew map:

\operatorname{Hom}(\bigotimes^2G,A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A)

Thus, in order to split the short exact sequence at the level of cocycles, it suffices to invert the above map, i.e., it suffices to split the short exact sequence:

0 \to \operatorname{Hom}(\operatorname{Sym}^2G,A) \to \operatorname{Hom}(\bigotimes^2G,A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

Splitting for the Baer correspondence

Suppose that both G and A are uniquely 2-divisible groups. We want to find a section for the mapping:

f \mapsto \operatorname{Skew}(f)

The section is as follows:

f \mapsto \frac{f}{2}

Division by 2 is permissible because A, the target group, is uniquely 2-divisible. The mapping is a homomorphism and provides the required inverse, splitting the short exact sequence.

Putting things together

The splitting of the original short exact sequence:

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

is done using the following section \operatorname{Hom}(\bigwedge^2G,A) \to H^2(G;A):

  • Start with an element of \operatorname{Hom}(\bigwedge^2G,A).
  • Consider the bihomomorphism obtained by halving this element.
  • Since bihomomorphism to abelian group implies 2-cocycle, the halved bihomomorphism is a 2-cocycle from G to A.
  • Consider the cohomology class of the 2-cocycle. That is the desired image.

This choice of section splits the short exact sequence and we get a canonical isomorphism:

H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \oplus \operatorname{Hom}(\bigwedge^2G,A)

More explicitly, we have an internal direct sum decomposition:

H^2(G;A) = H^2_{sym}(G;A) + J

where J is the group of cohomology classes represented by 2-cocycles that are in fact alternating bihomomorphisms.

Explication of relationship with Baer correspondence

The Baer correspondence starts off with an extension group E viewed as an element of H^2(G;A). The projection on the H^2_{sym}(G;A) coordinate gives the additive group of its corresponding Lie ring. The projection on the other coordinate can be used to recover the Lie bracket structure.

More explictly, if we consider elements in the extension group, with multiplication denoted by concatenation, the element of H^2_{sym}(G;A) corresponds to the addition:

\! (x,y) \mapsto x + y := \frac{xy}{\sqrt{[x,y]}}

and the element of the other direct summand is:

\! (x,y) \mapsto \sqrt{[x,y]}

Related correspondences

Relation with various generalizations of Baer correspondence

The extension group E is a group of class at most two on the group side of the Baer or Baer-like correspondence. We have flexibility in (a) the choice of A and G (isomorphic to E/A), and (b) the way we do the splitting.

We say "partial" splitting type if the splitting applies only to part of the short exact sequence, i.e., we have a way of going back up to G^2(G;A) from a subgroup of \operatorname{Hom}(\bigwedge^2 G, A) rather than from the whole group.

Name of correspondence Key idea What this means in terms of the flexibility for G and A What this means in terms of the choice of splitting Splitting type?
Baer correspondence unique 2-divisibility in whole group A can be any subgroup between the derived subgroup and center Splitting is as above, using halving Canonical and complete
LCS-Baer correspondence unique 2-divisibility in derived subgroup A is the derived subgroup Splitting is as above, using halving Canonical and complete
UCS-Baer correspondence unique 2-divisibility in center A is the center Splitting is as above, using halving Canonical and complete
LUCS-Baer correspondence every element of derived subgroup has unique square root in center A is the center Splitting is as above, using halving, but is partial: it only applies to the subgroup of \operatorname{Hom}(\bigwedge^2 G, A) whose image is inside the derived subgroup Canonical and partial
CS-Baer correspondence unique 2-divisibility in some subgroup intermediate between derived subgroup and center A can be chosen as a subgroup between the derived subgroup and center; but not every subgroup would work Splitting is as above, using halving Canonical and complete
linear halving generalization of Baer correspondence linear half exists for commutator map A can be chosen as the center Splitting is specified by the choice of the linear half, and is not natural. Moreover, it may not apply to the whole of \operatorname{Hom}(\bigwedge^2 G, A) but only to a subgroup. Note that there are cases where this can give a full splitting; see for instance second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4 Partial (sometimes complete) and probably unique
cocycle halving generalization of Baer correspondence cyclicity-preserving cocycle half exists for commutator map A can be chosen as the center Splitting is specified by the choice of the cocycle half, and is not natural. Moreover, it may not apply to the whole of \operatorname{Hom}(\bigwedge^2 G, A) but only to a subgroup. Note that there are cases where this can give a full splitting; see for instance second cohomology group for trivial group action of V4 on Z4 Partial (sometimes complete) and probably unique (cf. Symmetric and cyclicity-preserving 2-cocycle implies 2-coboundary)
cocycle skew reversal generalization of Baer correspondence cyclicity-preserving cocycle skew reversal exists for commutator map A can be chosen as the center Splitting is specified by the choice of the cocycle skew reversal, and is not natural. Moreover, it may not apply to the whole of \operatorname{Hom}(\bigwedge^2 G, A) but only to a subgroup. Note that there are cases where this can give a full splitting; see for instance second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2 Partial (sometimes complete) and probably unique (cf. Symmetric and cyclicity-preserving 2-cocycle implies 2-coboundary)