The strange adventure of the Universal Coefficient Theorem in the night

If you trawl the internet for more exotic Universal Coefficient Theorems, then you’ll come across comments to the effect that there isn’t a UCT for cohomology with local coefficients in general (for instance, here, for group cohomology, which is ordinary cohomology of a K(G,1), or the Groupprops wiki page, which only gives the trivial coefficients version). I had reason to want a UCT for group cohomology with coefficients a nontrivial module, to try to clear some hypotheses using formal properties of group homology (it’s better behaviour with respect to filtered colimits, in particular). The only reference I could find for a suitable UCT was an exercise in Spanier’s venerable book:

Page 283 of Spanier’s book Algebraic Topology

I found a reference in a 2018 paper, that said “there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited”. Up until this point, I had not actually seen the statement written out anywhere! In particular, it’s not clear what Spanier’s assumptions are (it might be he is assuming R is a PID throughout this section, but I couldn’t see it on a quick search), and in particular, something must break for this to be of “limited” application.

So I asked on math.stackexchange, and Oscar Randal-Williams gave a great answer, pulling out something from the even more venerable source of Cartan and Eilenberg’s Homological Algebra. So I will copy the statement here (under the CC-By-SA 4.0 license)

If R is left hereditary (eg a PID) and either G is an injective R-module (unlikely) or else \Delta(X, A;\Gamma) is a complex of projective R-modules (which holds iff the \Gamma(x) are projective R-modules), then page 114 of Cartan-Eilenberg gives a standard-looking UCT, of the form

0 \to Ext^1_R(H_{i-1}(X,A;\Gamma), G) \to H^i(X,A; Hom_R(\Gamma, G)) \to Hom_R(H_i(X,A;\Gamma), G) \to 0.

Oscar Randal-Williams, math.stackexchange answer

The case that I was looking at was H^1, working over R=\mathbb{Z}, so it is a PID, and my local system \Gamma has each \Gamma(x)\simeq \mathbb{Z}^n (technically, it’s H^2(BG_0,\mathbb{Z})/tors for G_0 a connected Lie group), a free and hence projective module. Since the \Gamma(x)s are finitely generated free abelian groups, they are reflexive, and so taking G=\mathbb{Z} in the above, I can switch the position of the dual Hom_\mathbb{Z}(\Gamma,\mathbb{Z}) from cohomology to the outer terms (this relies on knowing the standard double dual map is natural, so the whole dual local system is what is being swapped here, not just something on fibres).

What I was interested in is knowing when my cohomology group—of a space of the form BK where K is a discrete group—was torsion, based on the group-theoretic properties of \pi_1(BK)=K. Having a hypothesis involving cohomology using arbitrary local systems, even if of a special restricted form, was a bit too generic for my liking. However, with the above UCT, I can reduce the hypothesis on H^1(BK,\Gamma) for arbitrary local coefficient systems \Gamma with fibre \mathbb{Z}^n, to one on H_1(BK,\Gamma). This might not seem like an improvement, but I’m looking as a special case at groups K that are locally finite: the filtered colimit of their finite subgroups, and this means I can say something meaningful about torsion in the homology of BK, and hence in the cohomology.

My thanks to Maxime Ramzi for discussing this and related matters with me via Twitter chat, helping me past all kinds of silly misunderstandings, and to Oscar Randal-Williams for his answer on stackexchange.

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