TITLE: Pullback and homology
QUESTION [14 upvotes]: Suppose I have two maps of topological spaces, $f:X\rightarrow B$ and $g:Y\rightarrow B$, such that $f$ induces a homology isomorphism and $g$ is a fibration and $B$ is connected. Is it true that the natural map $X\times_{B}Y\rightarrow Y$ induces an isomorphism in homology?

REPLY [11 votes]: Here is a positive answer to a slightly different question.
Call a map $X\to B$ "acyclic" if it induces an isomorphism in homology for every coefficient system on $B$. (If $B$ is simply connected then this is equivalent to your hypothesis.) Note that a map $F\to \ast$ is acyclic if and only if the space $F$ is acyclic in the sense that it has trivial integral homology.
Using Serre spectral sequences one can prove the following:
If $X\to B$ is acyclic then for any fibration $Y\to B$ the map $X\times_BY\to Y$ is acyclic.
A fibration $X\to B$ is acyclic if and only if for every point $b\in B$ the fiber over $B$ is acyclic.