Homotopy of Maps

Homotopy is the central idea of algebraic topology: two continuous maps are homotopic if one can be continuously deformed into the other. This notion captures the intuition that topology studies properties invariant under "rubber sheet" deformations.

Definition

Definition9.1Homotopy

Let $X$ and $Y$ be topological spaces and $f, g: X \to Y$ continuous maps. A homotopy from $f$ to $g$ is a continuous map $H: X \times [0, 1] \to Y$ such that: $H(x, 0) = f(x) \quad \text{and} \quad H(x, 1) = g(x) \quad \text{for all } x \in X.$

If such an $H$ exists, we say $f$ and $g$ are homotopic and write $f \simeq g$ . The map $H$ is also written $H_t(x) = H(x, t)$ , giving a one-parameter family of continuous maps $H_t: X \to Y$ with $H_0 = f$ and $H_1 = g$ .

RemarkHomotopy as Deformation

Think of the parameter $t \in [0, 1]$ as "time." At time $t = 0$ , the map is $f$ ; at time $t = 1$ , the map is $g$ . The homotopy $H$ provides a continuous deformation from $f$ to $g$ through the intermediate maps $H_t$ .

Basic Properties

Theorem9.1Homotopy is an Equivalence Relation

Homotopy $\simeq$ is an equivalence relation on the set of continuous maps $f: X \to Y$ .

Proof

Reflexivity: $f \simeq f$ via the constant homotopy $H(x, t) = f(x)$ .

Symmetry: If $H: f \simeq g$ , then $\bar{H}(x, t) = H(x, 1 - t)$ gives $g \simeq f$ .

Transitivity: If $H: f \simeq g$ and $K: g \simeq h$ , define: $L(x, t) = \begin{cases} H(x, 2t) & \text{if } 0 \leq t \leq 1/2, \\ K(x, 2t - 1) & \text{if } 1/2 \leq t \leq 1. \end{cases}$ By the pasting lemma ( $[0, 1/2]$ and $[1/2, 1]$ are closed, $H(x, 1) = g(x) = K(x, 0)$ ), $L$ is continuous. So $f \simeq h$ .

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Definition9.2Homotopy Classes

The equivalence class $[f]$ of a continuous map $f: X \to Y$ under homotopy is called its homotopy class. The set of all homotopy classes is denoted $[X, Y]$ .

Examples

ExampleBasic Homotopies

Null-homotopy in $\mathbb{R}^n$ : Any continuous $f: X \to \mathbb{R}^n$ is homotopic to any other $g: X \to \mathbb{R}^n$ via the straight-line homotopy $H(x, t) = (1 - t)f(x) + tg(x)$ . In particular, every map to $\mathbb{R}^n$ is null-homotopic (homotopic to a constant map).
Identity and constant maps on $S^n$ : The identity $\operatorname{id}: S^n \to S^n$ is not null-homotopic (not homotopic to a constant) for $n \geq 1$ . This is a deep fact, proved using the fundamental group (for $n = 1$ ) or higher homotopy groups.
Maps between spheres: Two maps $f, g: S^1 \to S^1$ are homotopic if and only if they have the same degree: $\deg(f) = \deg(g)$ . So $[S^1, S^1] \cong \mathbb{Z}$ .

Homotopy Equivalence

Definition9.3Homotopy Equivalence

Two spaces $X$ and $Y$ are homotopy equivalent (or have the same homotopy type) if there exist continuous maps $f: X \to Y$ and $g: Y \to X$ such that $g \circ f \simeq \operatorname{id}_X$ and $f \circ g \simeq \operatorname{id}_Y$ .

We write $X \simeq Y$ and call $f$ a homotopy equivalence with homotopy inverse $g$ .

Definition9.4Contractible Space

A space $X$ is contractible if it is homotopy equivalent to a point. Equivalently, $\operatorname{id}_X$ is null-homotopic: there exists a homotopy $H: X \times [0, 1] \to X$ with $H_0 = \operatorname{id}_X$ and $H_1 \equiv x_0$ for some $x_0 \in X$ .

ExampleHomotopy Equivalences

$\mathbb{R}^n$ is contractible: $H(x, t) = (1 - t)x$ contracts $\mathbb{R}^n$ to the origin.
$\mathbb{R}^n \setminus \{0\} \simeq S^{n-1}$ : The inclusion $\iota: S^{n-1} \hookrightarrow \mathbb{R}^n \setminus \{0\}$ and the retraction $r(x) = x/\|x\|$ are homotopy inverses.
M"obius band $\simeq S^1$ : The M"obius band deformation retracts to its central circle.
Figure-eight $\simeq$ punctured torus: The torus with a point removed is homotopy equivalent to the wedge of two circles.

Relative Homotopy and Path Homotopy

Definition9.5Relative Homotopy

Let $A \subseteq X$ . Two maps $f, g: X \to Y$ are homotopic relative to $A$ (written $f \simeq g \; (\text{rel } A)$ ) if there exists a homotopy $H: X \times [0, 1] \to Y$ with $H_0 = f$ , $H_1 = g$ , and $H(a, t) = f(a) = g(a)$ for all $a \in A$ and $t \in [0, 1]$ .

Definition9.6Path Homotopy

Two paths $\gamma_0, \gamma_1: [0, 1] \to X$ with the same endpoints ( $\gamma_0(0) = \gamma_1(0) = x_0$ and $\gamma_0(1) = \gamma_1(1) = x_1$ ) are path-homotopic if $\gamma_0 \simeq \gamma_1 \; (\text{rel } \{0, 1\})$ .

RemarkHomotopy vs. Homeomorphism

Homotopy equivalence is much weaker than homeomorphism. For example, $\mathbb{R}^n \simeq \{0\}$ (a single point) for all $n$ , while $\mathbb{R}^m \cong \mathbb{R}^n$ only when $m = n$ . Algebraic topology primarily studies homotopy-invariant properties (fundamental group, homology, cohomology), which cannot distinguish homotopy equivalent spaces.