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I have a problem in understanding Prop.1.10 in https://pages.uoregon.edu/sadofsky/691/sseq-local-coefficients.pdf My interpretation is as follows:

Let $X$ be a smooth n-manifold and $U\to X$ be its universal cover. Notice that $U$ is a $\pi_1 X$ principal bundle. Let $\rho: \pi_1 X\to Aut(S) $ be a an action of $\pi_1 X$ on the group $S$. Then we can form the associated bundle $\tilde{X}=U\times_\rho S $. The proposition 1.10 says that $$H_*(X,S) \simeq H_*(\tilde{X},\mathbb{Z})$$ where the LHS is the homology with twisted coefficients in $S$.

Problem: If $S=V$ is a vector space then $\tilde{X}=U\times_{\rho} V$ is the flat bundle associated to the flat connection induced by $\rho.$ This is a vector bundle over $X$ hence it retract over $X$ therefore $$H_*(X,V) \simeq H_*(\tilde{X},\mathbb{Z})\simeq H_*(X, \mathbb{Z}).$$ Which is not true for any flat connection.

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  • $\begingroup$ The highlighted proposition 1.10 is false as stated. $\endgroup$ – Moishe Kohan Jan 9 '20 at 18:16
  • $\begingroup$ @MoisheKohan, am I interpreting it wrongly? Or is the proposition in the document false as well? $\endgroup$ – Warlock of Firetop Mountain Jan 10 '20 at 9:27
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Your general statement is simply false. For instance, if the action is trivial, your statement would say that the homology of any space agrees with the homology of its universal cover, which is ridiculous.

What is true is that if $A = \Bbb Z[\pi_1(X)/H]$ as a $\pi_1(X)$-module, and $p: X' \to X$ is the covering space associated to the subgroup $H \subset \pi_1(X)$, then $\mathcal A = \tilde X \times_{\pi_1 X} A$ gives a local coefficient system over $X$ (if you like to think of a local coefficient system as being a bundle of abelian groups over $X$). Then we have $$H_*(X;\mathcal A) \cong H_*(X'; \Bbb Z).$$

What this says is that a very specific local coefficient system gives the homology of this covering space. Not that you can compute the homology of general coefficient systems by passing to a covering space.

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  • $\begingroup$ Thank you Jacque, do you have also a good reference to recommend? $\endgroup$ – Warlock of Firetop Mountain Jan 11 '20 at 16:15
  • $\begingroup$ @WarlockofFiretopMountain Sorry, I don't, I picked up most of this by brute-force computation. It might be a nice exercise for you to compute the homology of the circle equipped with a local coefficient system with fiber $\Bbb C^n$, and $1 \in \pi_1 S^1$ acting by $A \in GL_n(\Bbb C)$. Also, if you use the definition of the local-coefficient chain complex from bundle of groups, I suggest you show explicitly that we have an isomorphism of chain complexes $C_*(X;\mathcal A) \cong C_*(X';\Bbb Z)$, it's not too hard and is revealing. $\endgroup$ – Jacque Lemure Jan 11 '20 at 18:22

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