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Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions (in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not necessarily submersions).

Let $\mathcal{E}$ be a locally free locally constant sheaf of finite rank on $X$.

Is it true that the push-forwards in the derived category of sheaves $Rf_*(\mathcal{E}),Rg_*(\mathcal{E})$ are isomorphic to each other?

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    $\begingroup$ Yes, it is true: locally constant sheaves only depend on the homotopy type of $X$, functorially with respect to pullbacks (i.e. the pullback functor along any map $f$ restricted to locally constant sheaves only depends on the homotopy class of $f$). When $f$ is a proper submersion, the functor $Rf_*$ is a right adjoint of the pullback and thus only depends on the homotopy type of $f$. $\endgroup$
    – D.-C. Cisinski
    Commented Jan 31, 2023 at 22:21
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    $\begingroup$ The last statement of the preceding comment is not clear to me. If the adjointness property depends on $f$ being a submersion, why does it depend only on the homotopy type of $f$ when the homotopies can be through non-submersions? $\endgroup$
    – Greg Friedman
    Commented Feb 1, 2023 at 1:02
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    $\begingroup$ @GregFriedman The pullback $f^*$ is defined for all maps $f$. Now, if two left adjoints are isomorphic, so are their right adjoints. Hence the right adjoints, whenever they exist, only depend on the homotopy type. In fact, all this depends on the weak homotopy type only, and up to a weak equivalence, any map is a Serre fibration, which implies that the right adjoint always exists. the point of being a proper submersion is that this right adjoint can be computed as the ordinary (derived) sheaf-theoretic push forward. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 1, 2023 at 12:02
  • $\begingroup$ @D.-C.Cisinski: It sounds like a final answer. Isn’t it? $\endgroup$
    – asv
    Commented Feb 2, 2023 at 19:31
  • $\begingroup$ @asv This is an answer indeed. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 2, 2023 at 22:55

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