If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is an isomorphism.

My question is as follows:

Given a surjective submersion (I am not considering it to be proper) $f:X \to Y$ (connected smooth manifolds) such that all the fibers have trivial integral homology does this mean the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is isomorphism?

I will be grateful to see some counterexamples or proof.

Thanks in advance!!