MAP5703 - Introduction to Reidemeister torsion in 3-manifold topology (2020)

Let \( X \) be a finite cell complex and let \( V \) be a finite-dimensional vector space over a field \( \mathbb{F} \).
Suppose that \( \rho:\pi_1(X) \to \mathrm{Aut}(V)=GL(V) \) is a group homomorphism such that the cellular chain complex \( C_\bullet (X, V^\rho) \) is acyclic.
Let \( \rho' \) be a representation conjugate to \( \rho \), i.e., one for which there exists an element \( A\in \mathrm{Aut}(V) \) satisfying
\( \rho'(\alpha) = A\rho(\alpha) A^{-1} \)
for all \( \alpha\in\pi_1(X) \). Prove that the chain complex \( C_\bullet(X,V^{\rho'}) \) is also acyclic and that
\( \mathbb{T}(C_\bullet(X, V^\rho)) = \pm \mathbb{T}(C_\bullet(X, V^{\rho'})). \)