Let $f: M\to N$ be a smooth map of finite-dimensional manifolds, and let $E\to N$ be a flat vector bundle over $N$. Consider the pullback bundle $f^*E\to M$ over $M$ and consider the twisted de Rham-cohomology groups $$ H^k_{\mathrm{dR}}(N,E)$$
as they are defined in Differential Forms in Algebraic Topology by R. Bott and L. W. Tu. I was wondering whether $f$ induces a homomorphism $$ f^*: H^k_{\mathrm{dR}}(N,E)\to H^k_{\mathrm{dR}}(M,f^*E).$$ I tried to take an element $u_p\otimes e_p\in\bigwedge^k T_p^*N\otimes E_p$ and to map it to $f^*u_p\otimes e_p$. The problem is that $e_p$ is not necessarily an element of $f^*E$. So, my questions are:
- Is $f^*$ well-defined with the definition above?
- If it isn't, is there another way to obtain some kind of "functoriality"; possibly using a different vector bundle over $M$ or over $N$?