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Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite dimension. Assume that for some (equivalently, any) point $y\in Y$ the cohomology groups $H^i(f^{-1}(y),\mathbb{R})$ are finite dimensional for all $i$.

Let $\underline{\mathbb{R}}_X$ be the constant sheaf on $X$.

Question. Is it true that the sheaves $R^if_*(\underline{\mathbb{R}}_X)$ are locally trivial of finite rank which is equal to $\dim H^i(f^{-1}(y),\mathbb{R})$?

General sufficient conditions are also of interest.

Remark. It is well known (and easy to see) that dimensions of $H^i(f^{-1}(y),\mathbb{R})$ are independent of $y$.

A reference would be helpful.

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    $\begingroup$ Yes, this is the proper base change theorem. This is proved for example in "Sheaves on manifolds" by Kashiwara-Schapira. This is also discussed in detail here: front.math.ucdavis.edu/1404.7630 (their interest is more general, but the discussion is useful even in the "classical" case). $\endgroup$
    – Geordie Williamson
    Commented Mar 14, 2016 at 20:10
  • $\begingroup$ @GeordieWilliamson: Thanks for the references. However my problem is that I do not see how to formally deduce my statement from the proper base change theorem. $\endgroup$
    – asv
    Commented Mar 15, 2016 at 7:15
  • $\begingroup$ I think I was too hasty. Will try to think a bit more... $\endgroup$
    – Geordie Williamson
    Commented Mar 15, 2016 at 16:15

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