Homotopy - Knowlpedia

A homotopy from continuous maps $f,g:X\to Y$ is a continuous map

H:X\times[0,1]\to Y,\qquad H(x,0)=f(x),\quad H(x,1)=g(x),

where the domain has the product topology. If one exists, $f$ and $g$ are homotopic, written $f\simeq g$ .

Fixing a subset

For $A\subseteq X$ , a homotopy relative to $A$ additionally satisfies $H(a,t)=f(a)=g(a)$ for every $a\in A$ and $t\in[0,1]$ . In the fundamental group, homotopies of paths fix both endpoints.

Example

For maps into a convex subset of $\mathbb R^n$ , $H(x,t)=(1-t)f(x)+tg(x)$ is a homotopy.

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002, §0 and §1.1. Author-hosted text.