Homology fibration

A map of topological spaces $f:X\to Y$ is a homology fibration if, for every $y\in Y$, the natural map $f^{-1}(y)\to P{f}_y$ from the fiber over $y$ to the homotopy fiber over $y$ induces an isomorphism on homology groups. By a theorem of McDuff and Segal, this is implied, for instance, by the condition that for sufficiently small neighborhoods $U$ of $y$, the inclusion $f^{-1}(y)\to f^{-1}(U)$ induces an isomorphism on homology.