Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product

(i) $E \cong \widetilde{B} \times_{\pi_{1}} F$,

where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.

Now, how nice can we assume this $\pi_1$-action to be? In particular, is it very restricting to only look at flat fiber bundles of the form

(ii) $E \cong \widetilde{B} \times_{\pi_{1}} |L|$,

where $L$ is a simplicial $\pi_{1}$-complex? What, if we further assume the action on $L$ to be regular?

I can't think of a flat bundle that is not of form (ii). Do you know counterexamples that clarify what (ii) cannot describe? (possibly with relaxed conditions on $E$, $F$, $B$.) And are there theorems that specify exact conditions on a flat bundle to be of form (ii)?

I am particularly interested in the case $B$, $F$ compact, $\pi_{1}$ infinite.

Edit: After helpful comments, I now know that I am concerned with flat fiber bundles up to fiber preserving homotopy.

Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product

(i) $E \cong \widetilde{B} \times_{\pi_{1}} F$,

where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.

Now, how nice can we assume this $\pi_1$-action to be? I would like to use additional structure in my problem without giving up too much generality. In particular, is it very restricting to only look at flat fiber bundles of the form

(ii) $E \cong \widetilde{B} \times_{\pi_{1}} |L|$,

where $L$ is a simplicial $\pi_{1}$-complex? What, if we further assume the action on $L$ to be regular?

I can't think of a flat bundle that is not of form (ii). Do you know counterexamples that clarify what (ii) cannot describe? (possibly with relaxed conditions on $E$, $F$, $B$.) And are there theorems that specify exact conditions on a flat bundle to be of form (ii)?

I am particularly interested in the case $B$, $F$ compact, $\pi_{1}$ infinite.

Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product

(i) $E \cong \widetilde{B} \times_{\pi_{1}} F$,

where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.

Now, how nice can we assume this $\pi_1$-action to be? In particular, is it a reasonable restriction to only look at flat fiber bundles of the form

(ii) $E \cong \widetilde{B} \times_{\pi_{1}} |L|$,

where $L$ is a simplicial $\pi_{1}$-complex? Can we further assume the action on $L$ to be regular?

This seems to be sufficient for 'reasonable' bundles. Is there a theorem that specifies exact conditions on a flat bundle to be of form (ii)? And do you know counterexamples that clarify what (ii) cannot describe? (possibly with relaxed conditions on $E$, $F$, $B$.)

I am particularly interested in the case $B$, $F$ compact, $\pi_{1}$ infinite.

Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product

(i) $E \cong \widetilde{B} \times_{\pi_{1}} F$,

where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.

Now, how nice can we assume this $\pi_1$-action to be? In particular, is it very restricting to only look at flat fiber bundles of the form

(ii) $E \cong \widetilde{B} \times_{\pi_{1}} |L|$,

where $L$ is a simplicial $\pi_{1}$-complex? What, if we further assume the action on $L$ to be regular?

I can't think of a flat bundle that is not of form (ii). Do you know counterexamples that clarify what (ii) cannot describe? (possibly with relaxed conditions on $E$, $F$, $B$.) And are there theorems that specify exact conditions on a flat bundle to be of form (ii)?

I am particularly interested in the case $B$, $F$ compact, $\pi_{1}$ infinite.