Let $X$ be a manifold or a CW-complex.

Let

$\pi: \tilde X\longrightarrow X$

be a covering map.

Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an orthogonal representation.

Define the $\rho$-twisted chain complex of $\tilde X$ by

$C_*(\tilde X,\rho)=C_*(\tilde X)\otimes_{\pi_1(X)} \mathbb{R}^n$

where $\pi_1(X)$ acts on $C_*(\tilde X)$ from the right by deck transformations and acts on $\mathbb{R}^n$ from the left by orthogonal transformations.

In the book: Lecture Notes in Algebraic Topology by James F. Davis and Paul Kirk, Chapter 5, the homology with local coefficients is defined as the homology of the $\rho$-twisted chain complex

$H_*(\tilde X,\rho)=H_*(C_*(\tilde X,\rho))$.

Question.

Can we add some additional hypothesis on $X$, the covering space $\tilde X$, and the covering map $\pi:\tilde X\longrightarrow X$ such that for such $X$ and $\tilde X$, we can always find an $n\geq 2$ and a $\rho$ satisfying that $H_*(\tilde X,\rho)$ is trivial?

Thanks for guidance.

universalcover of $X$. But maybe you're coming at this from a different angle than me. $\endgroup$