Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand that $f$ is a fibration (or at least a quasi-fibration). However, how about the weaker condition of not demanding that this holds for all points, but only asking that there exists a point $x$ in $X$ such that the natural map $$Y_x \to hofib_x\left(f\right)$$ is a weak homotopy equivalence? I have encountered some non-trivial examples of such maps and was wondering what is known about them, and if there are some easy to check conditions (besides the obvious ones of actually being a fibration of quasi-fibration) that guarantee such a property.

If it helps, in the situation in which I am most interested, the spaces involved are locally contractible. (Also note that if $X$ is contractible, this condition is equivalent to asking for there to be a point such that the inclusion of the fiber $Y_x$ into $Y$ to be a weak homotopy equivalence).

Thanks a lot!

  • $\begingroup$ Do you want connected or path-connected? $\endgroup$ – David Roberts Sep 15 '15 at 5:34
  • $\begingroup$ Given a map $f:X\to Y$ on can always take the mapping cylinder of $\mathrm{hfiber}_y \to X$ over the mapping cylinder of $y \in Y$; for "lots" of finite-dimensional $X$s, only over the cylinder part will the set-fiber be the homotopy fiber. Does that help at all? $\endgroup$ – Jesse C. McKeown Sep 23 '15 at 23:29

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