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


**Assumptions**


1. The induced map $D\rightarrow C$ is a trivial fibration
2. The map $f:B\rightarrow A$ has weakly contractible fibers i.e., for any $a\in A$ we have $f^{-1}(a)\simeq \ast$
3. The induced map $D\rightarrow B$ induces an isomorphism in homology.
4. The map $C\rightarrow A$ induces a surjective map in homology.
5. 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. 



fiber, homotopy fiber of spaces
Counterexample:


Let $A=[0,1]$ with the usual topology, $B=C=[0,1]^{\delta}$ (this means the discrete topology). The maps $f:B \to A$ and $g:C \to A$ are the identity. Then $f^{-1} (t) = \*$ for all $t \in A$, but $f$ is not a homology isomorphism, and $g$ is surjective in homology. Furthermore, the pullback $D$ is the diagonal in $B \times C$, hence it is also $[0,1]^{\delta}$. The two maps $D \to B$ and $D \to C$ are homeomorphisms, hence acyclic fibrations and homology equivalences.


