= Homotopy
{title2=$H:X\times[0,1]\to Y$}
{wiki}
= Homotopic
{synonym}
A <homotopy> between <continuous maps> $f,g:X\to Y$ is a <continuous map> $H:X\times[0,1]\to Y$ satisfying $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Think of $t$ as time: continuity is required jointly in position and time. The target $Y$ specifies where every intermediate point is allowed to go. See https://pi.math.cornell.edu/~hatcher/AT/ATch0.pdf#page=3[Hatcher, Algebraic Topology, Chapter 0, pp. 3–4].
For the filled <disk> $D^2=\{x\in\mathbb R^2:\lVert x\rVert\leq1\}$, the formula
$$
H(x,t)=(1-t)x
$$
is a <contraction of a topological space>: it starts at the <identity map> and ends at the constant map to zero. The <Euclidean norm> obeys $\lVert H(x,t)\rVert=(1-t)\lVert x\rVert\leq1$, so the formula never leaves the <disk>. For $x=(3/5,4/5)$, the midpoint of the deformation is $(3/10,2/5)$, whose <Euclidean norm> is $1/2$.
On the <circle> $S^1=\{x:\lVert x\rVert=1\}$, the same formula fails: at $t=1/2$ every image has <Euclidean norm> $1/2$, so lies outside $S^1$. This checks one failed candidate; the nonzero <winding number> of a once-around <loop (topology)> proves that no contraction within the <circle> exists. The distinction is between allowing the filling and requiring the deformation to stay on its boundary.
\Image[/disk-circle-loop-obstruction.png]
{title=Contraction inside a filled disk, obstruction on its boundary circle, and the winding numbers of circle loops}
Back to article page
