Definition:Homotopy
Jump to navigation
Jump to search
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 $g$ and we write:
- $f \simeq g$
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
When relative homotopy is not under consideration, free homotopy is usually referred to as simply homotopy.
Also see
- Results about homotopy theory can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): homotopy