I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.

Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap X_2$ is empty. Let $f: Y \to X $ be a surjective map such that $f: f^{-1}(X_i) \to X_i$ (with fibre $F_i$, I am not assuming $F_1=F_2$) is trivial fibre bundle with $H_*(F_i; \mathbb{Z})=0$, for $i=1,2$. Then my question is whether $f:H_*(Y;\mathbb{Z})\to H_*(X;\mathbb{Z})$ is isomorphism at each level?

Note as $H_*(F_i;\mathbb{Z})=0$ it implies $f: f^{-1}(X_i) \to X_i$ induces isomorphism is homology.

Thanks in advance!!