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

Statement
This provides an interpretation of the Baer correspondence (and somewhat more generally, of the LCS-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 (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 $$\operatorname{Hom}(\bigotimes^2G,A)$$, 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:

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