Why group cohomology and not group homology?

In algebraic topology, one studies the homology and cohomology of spaces. However, when we study group homology/cohomology, we almost exclusively talk about cohomology. Why is this? Is there an aesthetic reason to prefer cohomology in this context? Does group cohomology just happen to be easier to compute?

I understand that the cup product gives cohomology more structure than homology, but this alone doesn't seem to justify the lopsided amount of attention that group cohomology gets.

Solution 1:

If you read, say, Brown's book, it becomes quite clear that cohomology is better than homology. Firstly, as Don Alejo noted, cohomology comes equipped with the cup-product, but also in regard to various finiteness conditions for groups; Brown has the entire chapter (chapter 8) about this. One first defines finiteness conditions in terms of projective resolutions but then identifies it in cohomological terms; at the same time, there is no clear-cut homological interpretation. For instance, a group $G$ has cohomological dimension $cd(G)$ (over ${\mathbb Z}$) at most $n$ if there exists a finite resolution by projective ${\mathbb Z} G$-modules $$ 0\to P_n \to ... \to P_0 \to {\mathbb Z}\to 0. $$ (Why this is the "right" definition is not immediate, read Brown's book to see an answer.) Equivalently, $G$ has cohomological dimension over ${\mathbb Z}$ at most $n$ if $H^i(G, M)=0$ for all $i>n$ and all ${\mathbb Z} G$-modules $M$. There is the "dual" notion of homological dimension defined via injective resolutions and group-homology but it is not as nicely-behaved. For instance Stallings proved that $cd(G)=1$ if and only if $G$ is a nontrivial free group. On the other hand, other locally free groups also have homological dimension $1$, like the group $G={\mathbb Q}$. In higher dimensions, $n\ge 3$, cohomological dimension of a group turns out to be equal to its geometric dimension: A group $G$ is said to have geometric dimension $\le n$ if there exists an $(n-1)$-connected complex $X$ and a (properly discontinuous) free cellular action $G\times X\to X$. In contrast, the best one gets (for $n\ge 3$) with the homological dimension $hd(G)$ is that if $hd(G)=n$ then the geometric of $G$ is $\le n+1$ (not the equality).

Solution 2:

Just some random thoughts on homology vs. cohomology.

• I think in algebraic topology, cohomology often comes together with homology, as historically, cohomology is usually introduced by explicitly dualizing the homological singular complex $C_\bullet (X)$ (cellular complex, etc.): $$C^\bullet (X;A) = \operatorname{Hom}_\mathbb{Z} (C_\bullet (X), A).$$ In group cohomology, to calculate $$H^n (G, M) = R^n (-)^G (M) \quad\text{and}\quad H_n (G,M) = L_n (-)_G (M)$$ (do you find invariants $(-)^G$ to be more natural and interesting than coinvariants $(-)_G$?), one also usually works with some specific complex (the bar-resolution) and dualizes it to get cohomology, so we have the same "duality" (the universal coefficient theorem; when the action of $G$ is trivial): $$ 0\to \operatorname{Ext}_\mathbb{Z}^1 (H_{n-1} (G,\mathbb{Z}),M) \to H^n (G,M) \to \operatorname{Hom} (H_n (G,\mathbb{Z}), M)\to 0.$$ Thus far, it's not much different from topology.

• In general, cohomological theorems and calculations have their homological counterparts, and many texts discuss both, e.g. Chapter 6 of Weibel's book and Brown's "Cohomology of groups".

• Probably (?) one exception is interpretation of lower-dimensional (co)homology groups. For instance, $H^1 (G,M)$ often appears in explicit terms, as some $1$-cocycles modulo $1$-coboundaries (crossed homomorphisms modulo principal crossed homomorphisms). Something similar should exist for $H_1 (G,M)$, but the result is not quite as nice, in my opinion.

• In some settings homology and cohomology appear together, for instance in Tate cohomology $\hat{H}^n (G,M)$, when $G$ is finite. Tate cohomology puts together homology and cohomology, it also has cup-products defined for all $n\in \mathbb{Z}$, etc.

• As in topology, we also have cap-products between cohomology and homology $H^p (G,M) \otimes H_q (G,N) \to H_{q-p} (G, M\otimes N)$.

So I think that the situation with (co)homology of topological spaces is not that different, and it's not quite true that nobody cares about group homology.