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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 $x \in X$:

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


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

$H$ 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:

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


Let $X$ be a topological space.

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


$f$ and $g$ are path-homotopic

if and only if:

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


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


$f$ is null-homotopic

if and only if:

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