Path homotopy

Definition

Let $X$ be a topological space and let $I = [0, 1]$ be the unit interval. A path in $X$ is a continuous map $\gamma \colon I \to X$ , from $\gamma(0)$ to $\gamma(1)$ . Let $\gamma_0, \gamma_1 \colon I \to X$ be paths with $\gamma_0(0) = \gamma_1(0) = x_0$ and $\gamma_0(1) = \gamma_1(1) = x_1$ . A path homotopy from $\gamma_0$ to $\gamma_1$ is a continuous map $H \colon I \times I \to X$ such that:

$H(s, 0) = \gamma_0(s)$ and $H(s, 1) = \gamma_1(s)$ for all $s \in I$ ;
$H(0, t) = x_0$ and $H(1, t) = x_1$ for all $t \in I$ .

The paths are path homotopic, written $\gamma_0 \simeq_p \gamma_1$ , if a path homotopy exists.

Remarks

A path homotopy is the same as a homotopy relative to $\set{0, 1}$ . Path homotopy is an equivalence relation on paths from $x_0$ to $x_1$ . Example: in a convex set $C \subseteq \R^n$ , the map $H(s, t) = (1 - t)\gamma_0(s) + t \gamma_1(s)$ is a path homotopy, since $H(0, t) = x_0$ and $H(1, t) = x_1$ . Non-example: in $S^1$ , the loop $s \mapsto e^{2\pi i s}$ and the constant loop at $1$ are not path homotopic; the fundamental group of the circle tells them apart.

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Interval concept
Relative homotopy concept
Path homotopy definition

Tag	`CHP0QWHFWM`
URI	`m2://CHP0QWHFWM`
Kind	definition
Writing	W0 stub
MSC	55Q05, 57M05
Defines	Path homotopy
Uses	Relative homotopy, Interval
Changed	2 November 2026

Path homotopy

Definition

Remarks

Reading path

Further reading

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