Homotopy

Maps $f, g : X\to Y$ are homotopic, $f\simeq g$, if there is a continuous $H : X\times I\to Y$ with $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$. Write $f_t = H(\cdot, t)$: a continuous one-parameter family of maps from $f$ to $g$. If $A\subseteq X$ and $H(a, t) = f(a)$ for all $a\in A$, $t\in I$, the homotopy is relative to $A$ ($f\simeq g$ rel $A$). Proposition. $\simeq$ (and $\simeq$ rel $A$) is an equivalence relation on maps $X\to Y$, and it respects composition: if $f\simeq f'$ and $g\simeq g'$ then $g\circ f\simeq g'\circ f'$.

If $C\subseteq\mathbb{R}^n$ is convex, then any two maps $f, g : X\to C$ are homotopic via the straight-line homotopy $H(x, t) = (1 - t)f(x) + tg(x)$. If $f = g$ on $A$, the homotopy is rel $A$. More generally the same holds if $C$ is star-shaped about $c$ (for each map, first contract it to the constant $c$). In particular, any two paths in $C$ with the same endpoints are homotopic rel $\{0, 1\}$.

A map $f : X\to Y$ is a homotopy equivalence if there is $g : Y\to X$ with $g\circ f\simeq\mathrm{id}_X$ and $f\circ g\simeq\mathrm{id}_Y$; then $X\simeq Y$ (same homotopy type). Homotopy equivalence is an equivalence relation on spaces, and homeomorphic spaces are homotopy equivalent. $X$ is contractible if $X\simeq\{\mathrm{pt}\}$, equivalently $\mathrm{id}_X$ is homotopic to a constant map. Examples. Convex and star-shaped subsets of $\mathbb{R}^n$ are contractible. $\mathbb{R}^n\setminus\{0\}\simeq S^{n-1}$. The Möbius band, the annulus and the solid torus are all $\simeq S^1$.

$A\subseteq X$ is a retract of $X$ if there is a map $r : X\to A$ with $r|_A = \mathrm{id}_A$ (a retraction). It is a deformation retract if moreover $i\circ r\simeq\mathrm{id}_X$ rel $A$, where $i : A\hookrightarrow X$; then $A\simeq X$. Examples. $S^{n-1}$ is a deformation retract of $\mathbb{R}^n\setminus0$ and of $D^n\setminus0$; the central circle is a deformation retract of the Möbius band; a point is a retract of any space. The punctured torus deformation retracts onto $S^1\vee S^1$ (§6). Big question: is $S^1$ a retract of $D^2$? (No: see Theorem 4.1.)