Homotopy Equivalence

Entry conditions

Use homotopy equivalence only when you can provide maps and homotopies showing they are inverses up to homotopy.

Definitions

A map $f:X\to Y$ is a homotopy equivalence if there exists $g:Y\to X$ such that:

$$g\circ f\simeq {\mathrm{id}}_X,f\circ g\simeq {\mathrm{id}}_Y.$$

Vocabulary (plain language)

• Inverse up to homotopy: not strictly inverse, but equivalent by deformation.

Symbols used

• $\simeq$: homotopic.
• ${\mathrm{id}}_X$: identity map on $X$.

Intuition

Homotopy equivalence says two spaces are the same “shape” from the perspective of homotopy, even if they are not homeomorphic.

Worked example

A solid disk is homotopy equivalent to a point. A circle is not homotopy equivalent to a point because it has a nontrivial loop.

How to recognize the structure

• You can provide $f$ and $g$ explicitly.
• You can provide homotopies for $g\circ f$ and $f\circ g$.

Common mistakes

• Confusing homotopy equivalence with homeomorphism.