Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $X$ and $Y$ are homotopy equivalent.

PS-1:By Whitehead's Theorem it will be enough to show that $f$ induces an isomorphism between all homotopy groups.

PS-2:In question Equivariant Cohomology for actions with finite stabilizers there are some discussion regarding the above question but in terms of homology. If anybody thinks that my question can be a consequence of this discussion please explain the connection.