Let $X$ be a path-connected manifold (or a CW complex). 

 Let $\pi_1(X)$ be the fundamental group of $X$. 

Let $\pi: \tilde X\longrightarrow X$ be a covering map. 

For each $m\geq 0$, let $C_m(\tilde X)$ be the real chain group generated by all the $m$-cells of $\tilde X$. 

Then $C_m(\tilde X)$ is a module over the real group algebra $\mathbb{R}(\pi_1(X))$. 

Here $\pi_1(X)$ acts on $C_m(\tilde X)$ be the deck transformation. 

Let $\rho: \pi_1(X)\longrightarrow O(n)$ be an orthogonal representation. 

Define the twisted chain complex by $C_m(\tilde X,\rho)=C_m(\tilde X)\otimes _{\pi_1(X)}\mathbb{R}^n$. 

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Question. Does there always exist a representation $\rho$ such that the homology of the twisted chain complex is trivial?

Thanks for guidance!