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Is there any relation between the homology of a space with local coefficients (in $\mathbb Q$ vector space) and the homology with coefficients in $\mathbb Q$? Thanks!

Raeder
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  • Is $Q$ the rational numbers? In general these two notions are the same if and only if the local coefficient system is un-twisted. See for example the local-coeffiecients section in Hatcher's Algebraic Topology text. – Ryan Budney Oct 09 '10 at 18:05

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If you compute homology with twisted coefficients, where the coefficient system involves vector spaces of dimension $d$, then the Euler characteristic of the resulting homology spaces (i.e. the alternating sum of the $H_i$ with twisted coefficients) is equal to $d$ times the Euler characteristic of the space computed via homology with trivial coefficients.

Matt E
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  • Do you know a reference for this fact on Euler characteristic? I'd love to add one to http://math.stackexchange.com/a/369762/274 ! – Mariano Suárez-Álvarez Apr 22 '13 at 21:49
  • @MarianoSuárez-Alvarez: Dear Mariano, I'm not sure of a reference in general. I started to write down an argument here, but it was getting a bit long and meandering. I'll think a bit about how to make an efficient argument in at least some degree of generality, and if I succeed, add it to my answer. Cheers, – Matt E Apr 23 '13 at 03:33
  • @MarianoSuárez-Álvarez in the case of finite CW complexes, this is stated in Dimca's Sheaves in topology, Prop. 2.5.4(ii), with proofs deferred to the standard works of Spanier and Whitehead. But I suppose that case is somehow trivial as you get a complex $C_\bullet(X,\mathscr L)$ whose $i^{\text{th}}$ term has dimension given by the number of $i$-cells times $\operatorname{rk} \mathscr L$ and whose cohomology computes $H^i(X,\mathscr L)$. – Remy Apr 10 '22 at 22:32
  • On a compact manifold (maybe even compact manifold with corners?), one can take a finite cover by contractible opens and use the alternating Čech resolution. The terms of this resolution agree for $\mathscr L$ and $\underline{\mathbf Q}^r$ (but the maps differ), so additivity of Euler characteristic gives the result. This leaves me kind of curious how general this result is: does it hold on paracompact Hausdorff spaces (with finite-dimensional total cohomology)? – Remy Apr 10 '22 at 23:34