Homotopy

Let $X$ and $Y$ be topological spaces, and let $f, g: X \to Y$ be continuous maps.

Definition. A homotopy from $f$ to $g$ is a continuous map $H: X \times [0,1] \to Y$ satisfying $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$. If such an $H$ exists, $f$ and $g$ are homotopic, written $f \simeq g$. For each $t \in [0,1]$, the map $H_t = H(-, t): X \to Y$ is called the stage of the homotopy at time $t$.

Definition. Let $A \subseteq X$. A homotopy $H: X \times [0,1] \to Y$ is relative to $A$ (written $f \simeq g$ rel $A$) if $H(a, t) = f(a) = g(a)$ for all $a \in A$ and all $t \in [0,1]$.

Proposition. Homotopy (respectively, homotopy rel $A$) is an equivalence relation on the set of continuous maps $X \to Y$ (respectively, on maps agreeing on $A$).

Proof sketch. Reflexivity: $H(x,t) = f(x)$. Symmetry: $H'(x,t) = H(x, 1-t)$. Transitivity: given $H: f \simeq g$ and $K: g \simeq h$, define $(H \cdot K)(x,t) = H(x, 2t)$ for $t \leq 1/2$ and $K(x, 2t-1)$ for $t \geq 1/2$; continuity follows from the pasting lemma. $\square$

Proposition. Homotopy is compatible with composition: if $f_0 \simeq f_1: X \to Y$ and $g_0 \simeq g_1: Y \to Z$, then $g_0 \circ f_0 \simeq g_1 \circ f_1: X \to Z$.

Proof sketch. If $H$ is a homotopy from $f_0$ to $f_1$ and $K$ is a homotopy from $g_0$ to $g_1$, then $(x, t) \mapsto K(H(x, t), t)$ is a homotopy from $g_0 \circ f_0$ to $g_1 \circ f_1$. $\square$

Examples.

• A path in $Y$ is a map $\gamma: [0,1] \to Y$, equivalently a homotopy $H: \{*\} \times [0,1] \to Y$ between two points viewed as maps $\{*\} \to Y$.
• A homotopy rel $\{0,1\}$ between two paths is the data underlying the fundamental group and higher homotopy groups.
• Any two maps $f, g: X \to \mathbb{R}^n$ are homotopic via the linear homotopy $H(x,t) = (1-t)f(x) + tg(x)$.
• Two spaces are homotopy equivalent when there exist maps between them whose composites are homotopic to the respective identities.

Remark. In higher category theory, homotopies are the 2-morphisms of the ∞-category of spaces, homotopies between homotopies are 3-morphisms, and so on.