Is there a discussion in the literature of Künneth-type theorems for (co)homology with local coefficients? The sources I know of that discuss local coefficients (Whitehead's Elements of Homotopy Theory, Davis and Kirk's Lecture Notes in Algebraic Topology, Hather's Algebraic Topology) don't discuss this topic. A result along these lines, for homology, is mentioned in Section 1.4 of the article Semi-Simplicial Spaces by Ebert and Randal-Williams.
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1$\begingroup$ Check Bredon's "Sheaf theory" book, it is almost certainly there. $\endgroup$– Moishe KohanCommented Mar 24, 2023 at 2:27
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$\begingroup$ @MoisheKohan Thanks. I think Bredon does indeed have what I'm looking for. One also needs a comparison between sheaf cohomology and cohomology with local coefficients, which I prefer to think of as coming from a functor out of the fundamental groupoid. I think this should be straightforward once one knows how to go back and forth between local systems and sheaves, and probably it is discussed somewhere in the references collected at mathoverflow.net/questions/17786/… $\endgroup$– Dan RamrasCommented Mar 25, 2023 at 18:42
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$\begingroup$ This is very straightforward (locally constant sheaves). $\endgroup$– Moishe KohanCommented Mar 25, 2023 at 19:54
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1$\begingroup$ @MoisheKohan Right, the correspondence is between local systems and locally constant sheaves. I think the most general Künneth formula in Bredon is on p. 231, with extensions in the exercises on p. 277. It's a little strange to me that he works with an arbitrary pair $(X, A)$ and takes the product with a locally compact Hausdorff space $Y$. The cohomological analogue of the statement in Ebert and Randal-Williams paper would refer to an arbirary space $X$ and its product with a pair $(Y, A)$, where $Y$ is compact Hausdorff. Maybe this works similarly. If I sort it out, I'll write an answer. $\endgroup$– Dan RamrasCommented Mar 26, 2023 at 6:43
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1 Answer
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There's some sort of discussion of this in Section 1 of Greenblatt, "Homology with local coefficients and characteristic classes", Homology, Homotopy and Applications, vol. 8(2), 2006, 91–103.
answered Mar 23, 2023 at 21:06
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2$\begingroup$ Apparently Robert Greenblatt was a student of Bill Massey back in the sixties, and left mathematics shortly thereafter. His thesis was finally published in HHA after Larry Smith pointed out its relevance to the integral cohomology of $BO(n)$ and $BSO(n)$. $\endgroup$ Commented Mar 23, 2023 at 21:35
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2$\begingroup$ Thanks! More about Greenblatt: en.wikipedia.org/wiki/Robert_Greenblatt_(anti_war_activist) $\endgroup$ Commented Mar 24, 2023 at 4:18