# Definition:Homotopy

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## 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:

- $H: X \times \closedint 0 1 \to Y$

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

- $\map H {x, 0} = \map f x$

and:

- $\map H {x, 1} = \map g x$

$H$ 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 and only if there exists a free homotopy $H$ between $f$ and $g$, and:

- $(1): \quad \forall x \in K: \map f x = \map g x$
- $(2): \quad \forall x \in K, t \in \closedint 0 1: \map H {x, t} = \map f x$

### Path-Homotopy

Let $X$ be a topological space.

Let $f, g: \closedint 0 1 \to X$ be paths.

Then:

- $f$ and $g$ are
**path-homotopic**

- $f$ and $g$ are homotopic relative to $\set {0, 1}$.

### Null-Homotopy

Let $f: X \to Y$ be a continuous mapping.

Then:

- $f$ is
**null-homotopic**

- there exists a constant mapping $g: X \to Y$ such that $f$ and $g$ are homotopic.

## 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.

## Also see

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**homotopy**