1  The Fundamental Group

1.1  (§51) Homotopy of paths

• $\mathrm{★}$ Def. $f,g:X\to Y$ are homotopic, $f\simeq g$, if there is a map $F:X\times I\to Y$ with $F(x,0)=f(x)$, $F(x,1)=g(x)$. $F$ is a homotopy.

• Notation. $f_t(x)=F(x,t)$; the rule $t\mapsto f_t$ is a path $I\to 𝒞(X,Y)$ from $f$ to $g$.

• Def. A path in $Y$ from $y_0$ to $y_1$: a map $f:I\to Y$ with $f(0)=y_0$, $f(1)=y_1$.

• $\mathrm{★}$ Def. Paths $f,g$ from $y_0$ to $y_1$ are path homotopic, $f{\simeq }_{p}g$, if there is $F:I\times I\to Y$ with $F(s,0)=f(s),F(s,1)=g(s),F(0,t)=y_0,F(1,t)=y_1.$ The endpoints stay fixed throughout.

• Lem. $\simeq$ and ${\simeq }_p$ are equivalence relations.

  Proof. Reflexive: constant homotopy. Symmetric: $\overline{F}(x,t)=F(x,1-t)$. Transitive: concatenate, $(F\star G)(x,t)=F(x,2t)$ for $t\le \frac{1}{2}$, $G(x,2t-1)$ for $t\ge \frac{1}{2}$; continuous by the pasting lemma since $F(x,1)=g(x)=G(x,0)$.

• Notation. $[f]$ = homotopy class; for paths, $[f]$ = path homotopy class.

• $\mathrm{★}$ Ex. Straight-line homotopy. For $f,g:X\to {ℝ}^2$ (or into any convex $Y\subset {ℝ}^n$):   $F(x,t)=(1-t)f(x)+tg(x)$. So any two maps into a convex set are homotopic; and if $f,g$ are paths with the same endpoints, this is a path homotopy.

• $\mathrm{★}$ Ex. In the punctured plane $Y={ℝ}^2-\{0\}$, with $f(s)=(\cos \pi s,\sin \pi s)$, $g(s)=(\cos \pi s,2\sin \pi s)$, $h(s)=(\cos \pi s,-\sin \pi s)$: $f{\simeq }_{p}g$ (the straight-line homotopy stays in $Y$), but the straight-line homotopy from $f$ to $h$ passes through the origin. In fact $f{\simeq ̸}_{p}h$ — proved later.

  Note. $f$ and $h$ are homotopic as maps if endpoints are released. That is why path homotopy, not homotopy, is the right relation here.

• $\mathrm{★}$ Def. For $f$ a path $x_0\to x_1$ and $g$ a path $x_1\to x_2$, the product

  Continuous by the pasting lemma.

• Lem. $f_0{\simeq }_p{f}_1$ and $g_0{\simeq }_p{g}_1$ $\Rightarrow f_0\ast g_0{\simeq }_p{f}_1\ast g_1$. Hence $[f]\ast [g]=[f\ast g]$ is well defined.

• $\mathrm{★}$ Thm. $\ast$ on path homotopy classes satisfies:

  1. Associativity: $([f]\ast [g])\ast [h]=[f]\ast ([g]\ast [h])$;

  2. Units: with $e_x(s)\equiv x$ the constant path, $[e_{x_0}]\ast [f]=[f]=[f]\ast [e_{x_1}]$;

  3. Inverses: with $\overline{f}(s)=f(1-s)$, $[f]\ast [\overline{f}]=[e_{x_0}]$ and $[\overline{f}]\ast [f]=[e_{x_1}]$.

  Proof. All three are reparametrisation homotopies. For (1) let the breakpoints slide: traverse $f$ on $0\le s\le (1+t)/4$, $g$ on $(1+t)/4\le s\le (2+t)/4$, $h$ thereafter. Draw the square with the two subdivision patterns and interpolate.

• Caution. $\ast$ is not a group operation on paths — only on path homotopy classes, and only when endpoints match.

1.2  (§52) The fundamental group

• Def. A loop at $x_0$: a path from $x_0$ to $x_0$.

• $\mathrm{★}$ Def. ${\pi }_1(X,x_0)$ = set of path homotopy classes of loops at $x_0$, with $\ast$. By the theorem above it is a group: unit $e=[e_{x_0}]$, inverse ${[f]}^{-1}=[\overline{f}]$. The fundamental group of $X$ based at $x_0$.

• Ex. $A\subset {ℝ}^n$ convex, $x_0\in A$ $\Rightarrow {\pi }_1(A,x_0)=\{e\}$.

• $\mathrm{★}$ Def. (Functoriality.) $h:(X,x_0)\to (Y,y_0)$ a based map. Define

  $$h_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0),h_{\ast }[f]=[h\circ f].$$

• Lem. $h_{\ast }$ is a group homomorphism.

• $\mathrm{★}$ Thm. ${(k\circ h)}_{\ast }=k_{\ast }\circ h_{\ast }$ and ${(\mathrm{id})}_{\ast }=\mathrm{id}$.

• $\mathrm{★}$ Cor. $h$ a homeomorphism $\Rightarrow h_{\ast }$ an isomorphism. So ${\pi }_1$ is a topological invariant of based spaces.

• Def. $\alpha$ a path from $x_0$ to $x_1$. Define $\widehat{\alpha }:{\pi }_1(X,x_0)\to {\pi }_1(X,x_1),\widehat{\alpha }[f]=[\overline{\alpha }\ast f\ast \alpha ].$

• Thm. $\widehat{\alpha }$ is an isomorphism; and $\alpha {\simeq }_p\beta \Rightarrow \widehat{\alpha }=\widehat{\beta }$.

• Cor. $X$ path connected $\Rightarrow {\pi }_1(X,x_0)\cong {\pi }_1(X,x_1)$ for all $x_0,x_1$.

• Caution. The isomorphism depends on the path: different paths give conjugate isomorphisms, which differ when ${\pi }_1$ is non-abelian. So “${\pi }_1(X)$” without a base point is well defined only up to isomorphism.

• Rmk. $x_0\in P\subset X$ with $P$ the path component $\Rightarrow {\pi }_1(P,x_0)\cong {\pi }_1(X,x_0)$. ${\pi }_1$ sees only one path component.

• Def. $X$ is simply connected if it is path connected and ${\pi }_1(X,x_0)$ is trivial. Then any two paths with the same endpoints are path homotopic.

• $\mathrm{★}$ Thm. (Brouwer.) Every map $f:B^2\to B^2$ has a fixed point. [Mk §55]

1.3  (§53) Covering spaces

• $\mathrm{★}$ Def. $p:E\to B$ a surjective map. An open $U\subset B$ is evenly covered if $p^{-1}(U)={\coprod }_{\alpha }{V}_{\alpha }$ with each $V_{\alpha }\subset E$ open and each $p|V_{\alpha }:V_{\alpha }\to U$ a homeomorphism. The $V_{\alpha }$ are the sheets (slices).

• $\mathrm{★}$ Def. $p$ is a covering map (and $E$ a covering space of $B$) if every point of $B$ has an evenly covered neighborhood.

• Ex. Trivial covering: $F$ discrete, $p:B\times F\to B$; sheets $B\times \{\alpha \}$.

• Lem. Each fibre $p^{-1}(x)$ is discrete.

• $\mathrm{★}$ Lem. Every covering map is an open surjection, hence a quotient map.

  Proof. For $A\subset E$ open and $x\in p(A)$: pick $y\in A$ over $x$, let $V_{\beta }$ be its sheet; $p(A\cap V_{\beta })$ is open in $U$ and contains $x$.

• $\mathrm{★}$ Thm. $p:ℝ\to S^1$,   $p(t)=(\cos 2\pi t,\sin 2\pi t)$, is a covering map.

  Proof. For $U=\{(x,y)\in S^1\mid x>0\}$: $p^{-1}(U)={\coprod }_{n\in \mathbb{Z}}(n-\frac{1}{4},n+\frac{1}{4})$, with inverse $(x,y)\mapsto n+\frac{1}{2\pi }\arcsin y$. Four such $U$ cover $S^1$. Each interval $[n,n+1]$ wraps once around.

• Def. $p$ is a local homeomorphism if each $y\in E$ has a neighborhood $V$ with $p(V)$ open and $p|V:V\to p(V)$ a homeomorphism.

• Lem. Covering map $\Rightarrow$ local homeomorphism.

• $\mathrm{★}$ Caution. Converse fails: $q=p|(0,\mathrm{\infty }):(0,\mathrm{\infty })\to S^1$ is a local homeomorphism but not a covering map — no neighborhood of $(1,0)$ is evenly covered.

• Ex. $p_n:S^1\to S^1$, $z\mapsto z^n$, i.e. $(\cos \theta ,\sin \theta )\mapsto (\cos n\theta ,\sin n\theta )$. A covering map for each $n$. (E.g. $p_2(x,y)=(x^2-y^2,2xy)$.)

• Prop. (Restriction.) $p$ a covering map, $B_0\subset B$, $E_0=p^{-1}(B_0)$ $\Rightarrow$ $p|E_0:E_0\to B_0$ is a covering map.

• Prop. (Products.) $p\times p^{\prime }:E\times E^{\prime }\to B\times B^{\prime }$ is a covering map.

• $\mathrm{★}$ Ex. $p\times p:{ℝ}^2\to S^1\times S^1=T^2$ covers the torus; each unit square $[n,n+1]\times [m,m+1]$ wraps once around.

• $\mathrm{★}$ Ex. Figure eight. $B_0=(S^1\times \{s_0\})\cup (\{s_0\}\times S^1)\subset T^2$; then $E_0={(p\times p)}^{-1}(B_0)=(ℝ\times \mathbb{Z})\cup (\mathbb{Z}\times ℝ)$, the infinite grid, covers $B_0$.

• Ex. $\exp :ℂ\to ℂ-\{0\}$, in real coordinates $p(x,y)=(e^x\cos y,e^x\sin y)$, is a covering map: it factors through $ℝ\times S^1\cong {ℝ}^2-\{0\}$ as a product of coverings.

1.4  (§54–§55) The fundamental group of the circle [Mk §54–55]

• $\mathrm{★}$ Thm. ${\pi }_1(S^1,s_0)\cong \mathbb{Z}$, generated by the loop that goes once around; the isomorphism sends a loop to its winding number, i.e. the endpoint of its lift along $p:ℝ\to S^1$.

• Rmk. Machinery needed: path lifting, homotopy lifting, the lifting correspondence. Work these through from Munkres §54; then §55 gives no-retraction of $B^2$ onto $S^1$, and Brouwer’s theorem.

1.5  (§58) Deformation retracts and homotopy type

• Def. $j:A\to X$ an inclusion. A retraction is a map $r:X\to A$ with $r\circ j={\mathrm{id}}_A$.

• $\mathrm{★}$ Def. A deformation retraction of $X$ onto $A$: a homotopy $H:X\times I\to X$ from ${\mathrm{id}}_X$ to $j\circ r$, with $H(a,t)=a$ for all $a\in A$, $t\in I$.

• $\mathrm{★}$ Ex. $A=S^n\subset X={ℝ}^{n+1}-\{0\}$, $r(x)=x/\Vert x\Vert$, and $H(x,t)=(1-t)x+tx/\Vert x\Vert .$

• Lem. $H$ a homotopy from $h$ to $k$ with $H(x_0,t)=y_0$ for all $t$ (a homotopy relative to $x_0$) $\Rightarrow$ $h_{\ast }=k_{\ast }$ on ${\pi }_1$.

  Proof. For a loop $f$ at $x_0$, the composite $H\circ (f\times \mathrm{id}):I\times I\to Y$ is a path homotopy from $hf$ to $kf$.

• $\mathrm{★}$ Thm. $A$ a deformation retract of $X$ $\Rightarrow$ $j_{\ast }:{\pi }_1(A,a_0)\stackrel{\cong }{\to }{\pi }_1(X,a_0).$

  Proof. $rj={\mathrm{id}}_A$ gives $r_{\ast }{j}_{\ast }=\mathrm{id}$. And $H$ is a homotopy rel $a_0$ from ${\mathrm{id}}_X$ to $jr$, so $j_{\ast }{r}_{\ast }=\mathrm{id}$ by the lemma.

• $\mathrm{★}$ Ex. ${\pi }_1(ℂ-\{0\})\cong {\pi }_1(S^1)\cong \mathbb{Z}$, infinite cyclic.

• $\mathrm{★}$ Ex. Doubly punctured plane $X=ℂ-\{+i,-i\}$ deformation retracts onto the figure eight $8=(i+S^1)\cup (-i+S^1)=S^1\vee S^1$, meeting at $0$. So ${\pi }_1(8,0)\cong {\pi }_1(X,0)$.

• Rmk. With $x=[f]$, $y=[g]$ the classes of the two circle loops: ${\pi }_1(8,0)=F_2=⟨x,y⟩$, the free group on two generators — every element a word in $x^{\pm 1},y^{\pm 1}$, with no relations beyond the group axioms. We prove less:

• $\mathrm{★}$ Prop. ${\pi }_1(8,0)$ is not abelian.

  Proof. Use the path-connected threefold cover of $8$ which is onefold over one circle and twofold over the other. Starting at the right vertex, the lifts of $f\ast g$ and $g\ast f$ end at different points, so $[f]\ast [g]\ne [g]\ast [f]$. Draw the covering graph.

1.6  (§59) The fundamental group of $S^n$

• $\mathrm{★}$ Thm. $X=U\cup V$ with $U,V$ open and simply connected and $U\cap V$ path connected $\Rightarrow$ $X$ is simply connected.

  Proof. Base point $x_0\in U\cap V$; loop $f$. The cover $\{f^{-1}(U),f^{-1}(V)\}$ of the compact metric space $I$ has a Lebesgue number, giving with each $f([s_{i-1},s_i])$ inside $U$ or $V$. Delete any $s_i$ with $f(s_i)\notin U\cap V$; then all $f(s_i)\in U\cap V$. Pick a loop $g$ in $U\cap V$ agreeing with $f$ at each $s_i$; patch the piecewise path homotopies. Finally $[g]$ dies already in ${\pi }_1(U,x_0)$.

• $\mathrm{★}$ Thm. $S^2$ is simply connected.

  Proof. $U=S^2-\{S\}$, $V=S^2-\{N\}$; stereographic projection $h(x_1,x_2,x_3)=\frac{1}{1+x_3}(x_1,x_2)$ gives $U\cong {ℝ}^2$, likewise $V$; and $U\cap V\cong {ℝ}^2-\{0\}$ is path connected.

• Rmk. Same argument: $S^n$ is simply connected for all $n\ge 2$.

1.7  (§60) Fundamental groups of some surfaces

• Def. Product group $G\times H$: $(g,h)(g^{\prime },h^{\prime })=(g{g}^{\prime },h{h}^{\prime })$, unit $(e,e)$, inverse $(g^{-1},h^{-1})$.

• $\mathrm{★}$ Thm. With $p={\mathrm{pr}}_X$, $q={\mathrm{pr}}_Y$, the map $\mathrm{\Phi }=(p_{\ast },q_{\ast }):{\pi }_1(X\times Y,(x_0,y_0))⟶{\pi }_1(X,x_0)\times {\pi }_1(Y,y_0)$ is a group isomorphism.

  Proof. Surjective: given loops $f,g$ set $h(s)=(f(s),g(s))$. Injective: path homotopies $F,G$ assemble to $H=(F,G)$.

• $\mathrm{★}$ Ex. ${\pi }_1(T^2)={\pi }_1(S^1\times S^1)\cong \mathbb{Z}\times \mathbb{Z}={\mathbb{Z}}^2,$ free abelian on two generators.

• Rmk. The inclusion $8=S^1\vee S^1\subset S^1\times S^1=T^2$ induces the canonical surjection $F_2\to {\mathbb{Z}}^2$ (abelianisation). The commutator $xy{x}^{-1}{y}^{-1}$ maps to $0$: in the square model of $T^2$, the loop $f\ast g\ast \overline{f}\ast \overline{g}$ is the boundary of $I^2$, which is nullhomotopic in $I^2$.

• $\mathrm{★}$ Def. Double torus $T^2\mathrm{\#}{T}^2$: the quotient of an octagon with boundary word $ab{a}^{-1}{b}^{-1}cd{c}^{-1}{d}^{-1}.$ Cutting along the diagonal gives two tori each with a disc removed; regluing along that circle is the connected sum.

• $\mathrm{★}$ Thm. ${\pi }_1(T^2\mathrm{\#}{T}^2)$ is not abelian.

  Proof. Let $A\subset X=T^2\mathrm{\#}{T}^2$ be the edges labelled $a$ and $c$; then $A\cong 8$. There is a retraction $r:X\to A$: collapse the cutting circle to get $T^2\vee T^2$, then retract each torus onto a circle. So $j_{\ast }:{\pi }_1(8)\to {\pi }_1(X)$ is injective, and ${\pi }_1(X)$ has a non-abelian subgroup.

• $\mathrm{★}$ Thm. $S^2$, $T^2$ and $T^2\mathrm{\#}{T}^2$ are pairwise non-homeomorphic.

  Proof. ${\pi }_1(S^2)=1$ and ${\pi }_1(T^2)={\mathbb{Z}}^2$ are abelian and non-isomorphic; ${\pi }_1(T^2\mathrm{\#}{T}^2)$ is not abelian. Fundamental groups distinguish all three.

• Rmk. Closing remark (3-manifolds). A compact 3-manifold splits into finitely many compact connected ones; a connected one is irreducible if every embedded $S^2$ bounds an embedded $B^3$, and reducible ones simplify by cutting along such a sphere and capping off. By Thurston’s geometrization conjecture, proved by Perelman, an irreducible 3-manifold decomposes along canonical tori into geometric pieces (eight model geometries). Combining the classifications — Waldhausen for Haken manifolds, Seifert for six of the geometries, Mostow rigidity for the hyperbolic one — the fundamental group turns out to be a complete invariant for irreducible 3-manifolds, apart from lens spaces.

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