Definition:Homotopy
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:
- $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$
such that, for all $x \in X$:
- $H \left({x, 0}\right) = f \left({x}\right)$
and:
- $H \left({x, 1}\right) = g \left({x}\right)$
$H$ 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 there exists a free homotopy $H$ between $f$ and $g$, and:
- $(1): \quad \forall x \in K: f \left({x}\right) = g \left({x}\right)$
- $(2): \quad \forall x \in K, t \in \left[{0 \,.\,.\, 1}\right]: H \left({x, t}\right) = f \left({x}\right)$
Path-Homotopy
Let $X$ be a topological space.
Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be paths.
We say that $f$ and $g$ are path-homotopic if they are homotopic relative to $\left\{ {0, 1}\right\}$.
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): Entry: homotopy