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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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.