# Definition:Homotopy

## Contents

## Definition

### Free Homotopy

Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.

Then $f$ and $g$ are **(freely) homotopic** if and only if there exists a continuous mapping:

- $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$

such that, for all $x \in X$:

- $H \left({x, 0}\right) = f \left({x}\right)$

and:

- $H \left({x, 1}\right) = g \left({x}\right)$

$H$ is called a **(free) homotopy between $f$ and $g$**.

### Relative Homotopy

Let $K \subseteq X$ be a subset of $X$.

We say that $f$ and $g$ are **homotopic relative to $K$** if there exists a free homotopy $H$ between $f$ and $g$, and:

- $(1): \quad \forall x \in K: f \left({x}\right) = g \left({x}\right)$
- $(2): \quad \forall x \in K, t \in \left[{0 \,.\,.\, 1}\right]: H \left({x, t}\right) = f \left({x}\right)$

### Path-Homotopy

Let $X$ be a topological space.

Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be paths.

We say that $f$ and $g$ are **path-homotopic** if they are homotopic relative to $\left\{ {0, 1}\right\}$.

## Also known as

Often, the term **homotopy** is used for free homotopy.

In particular, this happens when there is no danger of confusion with relative homotopy.