Definition:Homotopy/Relative
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Definition
Let $X$ and $Y$ be topological spaces.
Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.
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:
- $(1): \quad \forall x \in K: \map f x = \map g x$
- $(2): \quad \forall x \in K, t \in \closedint 0 1: \map H {x, t} = \map f x$
Trivially, if $K = \O$, then $(1)$ and $(2)$ are vacuous truths, so relative homotopy generalises free homotopy.
Also known as
A homotopy relative to $K$ can also be referred to as stationary on $K$.
Also see
- Definition:Free Homotopy
- Definition:Homotopy Class
- Definition:Path Homotopy
- Definition:Smooth Homotopy
Sources
- 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy