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