$\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}$ The key idea of homotopy theory is that one can construct a group out of loops. > [!definition] The ==**fundamental group**== of a space $X$ based at $x_0$ as the group of homotopy classes of loops based at $x_0$. This is denoted $\pi_1(X, x_0)$. In particular: > > > > - The identity element is the loop $p(t)=x_0$. > > - The inverse of a loop $p$ is the loop $q(t)=p(1-t)$. > > - The product of two loops $p$ and $q$ is the loop $ (p\star q) = \begin{cases} p(2t) & \text{if } t\leq \frac{1}{2} \\ q(2t-1) & \text{if } t>\frac{1}{2} \end{cases} $ > > To distinguish the equivalence classes from the paths, we denote the group element corresponding to path $p$ as $[p]$. Observe some trivialities: - Suppose $x_0$ and $x_1$ are path-connected by $\gamma$. Then, via conjugation by $\gamma$, $\pi_1(X, x_0)\cong \pi_1(X, x_1)$. - A space $X$ is ==**simply connected**== if $\pi_1(X, x_0)=1$ for all $x_0 \in X$. This is *stronger* than being path-connected; you also can't have any holes. > [!idea] Originally, my intuition was to \"drop all the basepoints,\" but it seems like they encode a certain *discrete* amount of information which is useful for algebraic topology. Just bear with me; they're useful, and you need to pay attention to them. Now, how do these groups interact with maps? > [!theorem] Continuous Maps yield Homomorphisms Let $f: X\to Y$ be a continuous map between two spaces. Then $f$ induces the natural map on loops $ p\mapsto f\circ p $ This in turn generates a homomorphism, which we will forever denote $ f_* : \pi_1(X, x_0)\to \pi_1(Y, f(x_0)) $ via $ [p]\mapsto [f\circ p] $ > [!problem] Show that the above map $p\mapsto f\circ p$ generates a well-defined homomorphism $\pi_1(X, x_0)\to \pi_1(Y, f(x_0))$. > [!solution]- Observe that our function acts on *equivalence classes* of loops. To show that this is well-defined as a function, we need to show that if $\gamma \simeq \delta$ in $X$, then $f\circ \gamma \simeq f\circ \delta$ in $Y$. This is obvious though; see a previous question. This function now obviously preserves identities, inverses, and composition. > [!idea] There is an additional nice property we might want this function to have: if $p$ in $X$ gets sent to $f\circ p$, and I have another loop $q'$ in $\tilde{X}$ which is homotopic to $f\circ p$, then there should exist some loop $q$ in $X$, homotopic to $p$, such that $f\circ q = q'$. In other words, you might want $ f([p]) = [f\circ p] $ *as sets*. Unfortunately, this is cap; let $Y$ be the interval, and let $f$ be the constant map to $0$. The good thing about *covering maps* (which we'll get to in a bit!) is that they *do* have this property. > [!theorem] Homotopic Maps and Homomorphism behave identically Suppose $f,g: (X,x_0)\to (Y, y_0)$ such that $f\simeq g$. Then $f_*=g_*$. Suppose $h : (Y, y_0)\to (Z, z_0)$. Then, $h_*\circ f_* = (h\circ f)_*$. In particular, if $f$ is a homotopy equivalence, then $f_*$ is an isomorphism. > [!proof] For any loop $\gamma$ at $x_0$, $f^*([\gamma]) = g^*([\gamma])$ iff $f(\gamma) \simeq g(\gamma)$, but this is just a composition of homotopies. The second statement is by definition; for any curve $\gamma$, $ (h_*\circ f_*)([\gamma]) = h_*([f\circ \gamma]) = [h\circ f\circ \gamma] = (h\circ f)_*([\gamma]) $ If $f:(X,x_0)\to (Y, y_0)$ is a homotopy equivalence, then there exists a $g: (Y, y_0)\to (X, x_0)$ such that $g\circ f \simeq \id_X$ and $f\circ g \simeq \id_Y$. Then, $f_*\circ g_* = (\id_X)_* = \id_{\pi_1(X, x_0)}$ and $g_*\circ f_* = (\id_Y)_* = \id_{\pi_1(Y, y_0)}$. So $f_*$ is an isomorphism (with inverse $g_*$). Suppose $X$ deformation retracts onto $A\subset X$. Just like any other continuous map, the inclusion map $i:A\to X$ induces a homomorphism $\pi_1(A,x_0)\to \pi_1(X,x_0)$ for $x_0\in A$. For any loop $\gamma \simeq 1$ in $\pi_1(X,x_0)$, there exists the required homotopy $\gamma_t: I\to X$; but then $i\circ \gamma_t$ induces a homotopy $\gamma\simeq 1$ in $\pi_1(A, x_0)$. Thus $i^*$ is an *injective* homomorphism. As a corollary, $D^2$ does not retract onto its boundary, for instance. ## A word on graphs > [!example] Modding out a Tree For a tree $T$ that is a subgraph of a connected graph $G$, the projection $G\to G/T$ is homotopy equivalence.