If $F:\pi_1(X)\rightarrow Ab$ is a system of local coefficiens, on the topological space $X$, then we can define the homology of $X$ with coefficients $F$ by taking the homology of the chain complex $C_p(X,F)=\bigoplus F(\sigma(e_1))$ where the sum is taken over all continuous $\sigma:\Delta^p \rightarrow X $, and the differentials are obvious (or more or less, care must be taken when computing $d_0\sigma$).

An alternative definition, in the case $X$ admits a universal cover $p:(\tilde X,\tilde x) \rightarrow (X,x)$, is to consider $S(\tilde X)$ as a left $\mathbb Z[\pi_1(X,x)]$ module ($\pi_1(X,X)$ is isomorphic to $Aut(p)$ , by taking $\gamma$ to the unique deck transformation that sends $\tilde x$ to $\tilde x \gamma$), and given $F$ a right $\mathbb Z[\pi_1(X,x)]$ module , we take the tensor product $F\otimes S(X)$ over $\mathbb Z[\pi_1(X,x)]$, and compute the homology of this complex.

If $F:\pi_1(X)\rightarrow Ab$, by restricting the functor to the automorphism group of $x$, which is ismorphic to the opposite of the fundamental group $\pi_1(X,x)$, we get a right $\mathbb Z[\pi_1(X,x)]$ module. Computing thus the homology in this two different ways, do we get the same? Are this definitions ok? Any reference where this is shown?

I could define a morphism from the first chain complex to the second, but I'm having trouble on defining one from the second chain complex to the first (hoping to eventually get an equivalence).

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    $\begingroup$ I cannot parse your 3rd paragraph. Restricting what functor to what automorphism group of $x$? What are the two different ways you mention? In your 4th paaragraph: which are the first complex? the one you described in your 1st paragraph? $\endgroup$ – Mariano Suárez-Álvarez Aug 25 '14 at 13:16
  • $\begingroup$ Restricting the functor $F$ (the one after "If"), to the automorphism group of $x$ (an object of the fundamental grupoid of $X$). Given $F:\pi_1(X)\rightarrow Ab$, we can compute certain homology groups (as defined in the first paragraph), and some others (as defined in the second paragraph, by consider the right $\mathbb Z[\pi_1(X)]$ module induced by F). The first complex is the one referenced to in the first paragraph. Maybe this is too specific. Thanks anyway. $\endgroup$ – Bill Aug 26 '14 at 4:33

Silly me! If I'm not mistaken this is theorem 24.1 of Eilenberg "Homology of Spaces with Operators".


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