Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \times_{\pi_1(B)} F $, where $\widetilde{B}$ is the fundamental cover of $B$.
Let's call two $\pi_1(B)$-actions $\phi_1, \phi_2$ on $F$ equivalent if the corresponding bundles $E_{\phi_1} , E_{\phi_2}$ are homotopy equivalent as fibre bundles over $B$. (Meaning there is a homotopy equivalence with respect to fibre preserving homotopies.)
Are the following two statements true?
- If $F$ is Riemannian and $\phi$ acts by isometries, i.e. $\phi: \pi_1(B) \rightarrow Iso(F)$, then $\phi$ is equivalent to an action $\phi'$ that is finite, i.e. factors through a finite subgroup: $\phi': \pi_1(B) \rightarrow H\subset Aut(F)$.
- If the induced actions of $\phi_1, \phi_2$ on all homotopy and (co-)homology groups of $F$ are identical, then $\phi_1$ and $\phi_2$ are equivalent.
