# Homology with local coefficients

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!

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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 9 '10 at 18:05

## 1 Answer

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.

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Do you know a reference for this fact on Euler characteristic? I'd love to add one to math.stackexchange.com/a/369762/274 ! – Mariano Suárez-Alvarez 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 3:33