homotopy equivalence

Definition

Let $X$, $Y$ be topological spaces. We call these spaces homotopy equivalent given there exists two maps $f:X\to Y$ and $g:Y\to X$ with homotopies

$G:f\circ g\sim i{d}_x\text{and}H:g\circ f\sim i{d}_y$

Notes

Homotopy equivalence is weaker than homeomorphism. Homeomorphism implies homotopy, since we can take the homotopies to be constant of the inverse maps.

$X$ and $X\times [0,1]$ are homotopy equivalent, but $X\sim X\times (0,1)$ are also homotopy equivalent, so homotopy equivalence doesn’t preserve compactness.

Any star convex subset $S\subset {\mathbb{R}}^n$ is homotopy equivalent to a point. We can take the homotopies to be the contraction down to a point and the “inverse” contraction that is an inclusion into $S$.