Homotopy Equivalence & Deformation Retracts

The same shape up to deformation: retracts, contractible spaces, and homotopy type.

The idea

Homotopy equivalence is the standard notion of two spaces having the same shape up to continuous deformation. It replaces homeomorphism, which is too strict for the purpose: the plane and a single point should count as the same shape, yet no bijection between them exists at all.

Definition (Homotopy equivalence).

A map $f \colon X \to Y$ is a homotopy equivalence if there is a map $g \colon Y \to X$ with $fg \simeq \mathrm{id}_{Y}$ and $gf \simeq \mathrm{id}_{X}$. Then $X$ and $Y$ are homotopy equivalent, written $X \simeq Y$, and are said to have the same homotopy type.

The composites are homotopic to the identity maps rather than equal to them. The plane and a point are homotopy equivalent, because the straight-line motion sliding each point of the plane to the origin is a homotopy from the identity map to the constant map.

In practice most homotopy equivalences come from a deformation retraction: a continuous motion that slides $X$ onto a subspace $A$ while holding every point of $A$ fixed. When one exists, the inclusion of $A$ into $X$ is a homotopy equivalence.

Homotopy type is the invariant the rest of the subject computes with. It forgets size and thickness but keeps the holes: it identifies an annulus, a punctured plane and a circle, and it distinguishes each of them from a point.

Ways to work on it

• Walkthrough. Why homeomorphism is too rigid, what a deformation retraction demands, and the homotopy types of the alphabet.
• Proof. The mapping cylinder slides onto its target, so every map is an inclusion followed by a homotopy equivalence.
• Practice. Classify thin letters, name what a space deformation retracts onto, and compare two spaces.
• Hardest. Compute the homotopy types of small graphs and decide how deformation retraction relates them.

Not sure where to start? Take the ten-question placement test.