Definition Suppose that $X$ and $Y$ are topological spaces and $f: X \to Y$ is a continuous map. If there exists a continuous map $g:Y \to X$ such that $f\circ g \simeq id_{Y}$ (i.e. $f\circ g$ is homotopic to the identity mapping on $Y$ , and $g \circ f \simeq id_{X}$ then $f$ is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence.

If there exist a homotopy equivalence between the topological spaces $X$ and $Y$ we say that $X$ and $Y$ are homotopy equivalent, or that $X$ and $Y$ are of the same homotopy type. We then write $X\simeq Y$

Properties

1. Any homeomorphism $f:X\to Y$ is obviously a homotopy equivalence with $g=f^{-1}$
2. For topological spaces, homotopy equivalence is an equivalence relation.
3. A topological space $X$ is (by definition) contractible, if $X$ is homotopy equivalent to a point, i.e., $X\simeq \{x_0\}$

Bibliography

1

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available online.