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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!

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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.2010-10-09

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