# fiber, homotopy fiber of spaces

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.

• Doesn't the isomorphism on homology follow already from assumption 2 thanks to the Vietoris Begle theorem? – Vidit Nanda Oct 14 '15 at 21:50
• No, the Vietoris-Begle theorem only holds for proper maps. – Johannes Ebert Oct 14 '15 at 23:37

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.