Submersion with contractible fibers

Question

Let $f: M\to N$ be a surjective submersion of manifolds with contractible fibers. Is it true that $M$ and $N$ are homotopy equivalent?

Answer 1

Surprisingly, your question has positive answer:

Theorem 1. Suppose that $f: M\to N$ is a surjective submersion with contractible fibers. Then $f$ is a fibration (in the sense of Serre). In particular, $f$ is a homotopy-equivalence.

See Corollary 13 in

G. Meigniez, Submersions, fibrations and bundles. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771–3787.

In fact, this result is proven under a weaker assumption, $f$ is only required to be a "homotopy submersion".

Remark. In the same paper it is proven (Corollary 31)

Theorem 2. Suppose that $f: M\to N$ is a surjective submersion with fibers diffeomorphic to $R^p$ for some $p$. Then $f$ is a locally trivial (in $C^\infty$ category) fiber bundle.

However, you do not need this stronger conclusion.