Homotopy equivalence

Summary

A homotopy equivalence is a continuous map $f \colon X \to Y$ that has an inverse up to homotopy. This is a map $g \colon Y \to X$ with $g \circ f$ homotopic to $\id_X$ and $f \circ g$ homotopic to $\id_Y$. It is a weaker notion than a homeomorphism. Homotopy invariants, such as the fundamental group and homology, cannot distinguish spaces that are related by a homotopy equivalence.