Homotopy equivalence

Spaces $X$ and $Y$ are homotopy equivalent when there are continuous mapssuch that $g\circ f$ is homotopic to ${\mathrm{id}}_X$ and $f\circ g$ is homotopic to ${\mathrm{id}}_Y$.

Two spaces have the same homotopy type when there is a homotopy equivalence between them. Their homology and homotopy groups are then isomorphic. In Morse theory, attaching a cell records a change in the homotopy type of a sublevel set.

If $g\circ f\simeq {\mathrm{id}}_X$ and $f\circ g\simeq {\mathrm{id}}_Y$, then $g$ is a homotopy inverse of $f$ and conversely.