# Interpretation of universal coefficients theorem for group cohomology

Suppose $G$ and $A$ are abelian groups (I'm setting $G$ abelian to keep the discussion simple, though there are analogues for non-abelian $G$) with $G$ acting trivially on $A$. By the universal coefficients theorem, we have short exact sequences for all positive integers $n$:

$$0 \to \operatorname{Ext}^1(H_{n-1}(G;\mathbb{Z});A) \to H^n(G;A) \to \operatorname{Hom}(H_n(G;\mathbb{Z}),A) \to 0$$

In the case $n = 2$, this becomes:

$$0 \to \operatorname{Ext}^1(G;A) \to H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$$

The surjection $H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ has a nice interpretation:

$H_2(G;\mathbb{Z})$ is the Schur multiplier of $G$, which (since $G$ is abelian) is the exterior square of $G$, so $\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ is the alternating bilinear maps from $G$ to $A$. The mapping $H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A)$ is the "skew" mapping; it sends a 2-cocycle $f:G \times G \to A$ to the function:

$$(x,y) \mapsto f(x,y) - f(y,x)$$

Even without prior knowledge of the universal coefficients theorem, this short exact sequence makes sense: one can check that the skew of a 2-cocycle for an abelian group is alternating and bilinear, and the mapping descends to cohomology classes because any 2-coboundary is symmetric. The kernel of the mapping corresponds to symmetric cohomology classes, which corresponding to the abelian extension groups, given precisely a $\operatorname{Ext}^1(G;A)$. In group extension terms, the image of a given element of $H^2(G;A)$ under the skew map describes the commutator map of the extension group.

My question is: what's an analogous concrete interpretation for higher $n$?

I'm listing below some references with generalizations of the above to non-abelian $G$ and varietal generalizations:

John Burns and Graham Ellis. On the nilpotent multipliers of a group. Math. Zeitschr., Volume 226, Page 405 - 428(Year 1997) (Official gated copy (PDF))

C. R. Leedham-Green and Susan McKay. Baer-invariants, isologism, varietal laws, and homology. PDF online. Pages 37-41 (135-139 in the original) describe a generalization of universal coefficients to varietal laws (where the usual universal coefficients is with respect to abelian groups inside groups)

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Have you tried assuming that $G$ is finite cyclic, so that cohomology is $2$-periodic? The isomorphism $H^q\cong H^{q+2}$ is given by an explicit cup-product so you might deduce something...but that's only a guess. –  Filippo Alberto Edoardo Jun 28 '12 at 10:56
Noting here: the original version of the question had the wrong statement of the universal coefficients theorem for general n (correct for n = 2 though) -- I fixed it today morning. I had said "Ext^{n-1}(G^{ab},A)" and the correct term is "Ext^1(H_{n-1}(G;Z),A)" –  Vipul Naik Jun 29 '12 at 3:12

So in homology, $H_1$ is abelianization, $H_2$ is Schur multiplier, and $H_{n\ge 3}$ is ???
Likewise in cohomology, $H^1(G;A)$ is split extensions. $H^2(G,A)$ corresponds to group extensions $\mathcal{E}(G,A)$ of $G$ by $A$, and this makes its appearance in your UCT (assuming for simplicity that $G$ is abelian and $A$ is a trivial $G$-module); the map $H^2(G;A)\to Hom(H_2(G),A)$ tells us that every alternating map comes from a 2-cocycle, and we note in general that $Ext^1_R(M,N)$ is the set of $R$-module extensions of $M$ by $N$.
As we see, this all ties in to knowing $H^2(G;A)$... if we look at $H^3(G;A)$, we get crossed module extensions $0\to A\to N\to E\to G\to 0$, and these get rather cumbersome. We have no nice interpretation for $H^n(G;A)$ for $n>3$, except more crazy looking exact sequences.