Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$.

**Assumptions**

- The induced map $D\rightarrow C$ is a trivial fibration
- The map $f:B\rightarrow A$ has weakly contractible fibers i.e., for any $a\in A$ we have $f^{-1}(a)\simeq \ast$
- The induced map $D\rightarrow B$ induces an isomorphism in homology.
- The map $C\rightarrow A$ induces a surjective map in homology.
- We can assume that $A$ is simply connected.

**Question**
What can we say about the map $B\rightarrow A$? Does it induce an isomorphism in homology?

**Edit:** The spaces $C,B$ are connected.