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

if and only if:

$f$ 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

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


Sources