$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ By replacing homomorphisms with [[homotopic maps|homotopies]], one can greatly generalize the notion of a space. The identity function $X\to X$ is now expanded to an entire homotopy class. In this sense, the notion of an inverse must be broadened. We say that $f:X\to Y$ is a ==**homotopy equivalence**== (read: \"invertible in the homotopic sense\") if there exists $g:Y\to X$ such that $g\circ f\simeq \id_X$ and $f\circ g\simeq \id_Y$. If any such homotopic equivalence exist between two spaces $f:X\to Y$, then we say that $X$ and $Y$ have the same ==**homotopy type**==, and write $X\simeq Y$. > [!idea] Homotopy Theory cannot discern between spaces that are homotopy equivalent. > [!idea] As is standard in these category-theoretic type situations, the objects are second-class citizens, with their properties governed by the morphisms between them. > [!problem] Show that $gf \simeq \id_X$ and $fh \simeq \id_Y$ implies $g\simeq h$; this is pure symbol-pushing. > Thus $f$ is a homotopy equivalence if and only if there exist $g,h$ such that $g\circ f\simeq \id_X$ and $f\circ h\simeq \id_Y$. > >One last exercise: $f:X\to Y$ is a homotopy equivalence iff there exist $g:W\to X$, $h:Y\to Z$ such that $gf$ and $fh$ are homotopy equivalences. > >Push the symbols around until you're comfortable with them; they're literally just arrows. > [!problem] Show that homotopy type is an equivalence relation. > [!idea] Homotopy equivalence of spaces is a much coarser classification than homeomorphism of spaces (clearly homeomorphisms are homotopy equivalences). > [!example] Contractible Spaces that are homotopy equivalent to a point are called ==**contractible**==. This occurs iff, for some point $*\in X$, there exists a map $f:[0,1]\times X\to X$ such that $f(0,x) = x$ and $f(1,x) = *$ for all $x\in X$. >[!example] Examples of Contractible Spaces >Examples include: > - $\RR^n$ for any $n$. > - Any convex subset $X\subset \RR^n$. > - Any [[topological cone]] > [!example] Circle The Mobius band and cylinder are homotopy equivalent to a circle. We show this for a cylinder $[0,1]\times \CC^1$; the Mobius band is similar. Indeed, let $f: [0,1]\times S^1\to S^1$ via $(r,\theta)\to \theta$, and let $g$ send $\theta \to (0, \theta)$. Then, $f\circ g$ sends $(r,\theta)\to (0,\theta)$, which is homotopy equivalent to the identity via $\gamma_t(r,\theta) = (rt,\theta)$ and $g\circ f$ is literally the identity. > [!example] Punctured Torus The once-punctured surface of genus $g$ is homotopy equivalent to a wedge of $2g$ circles.