Homotopy and the fundamental group

MotivationH

Loops that cannot be undone

A goat tethered by a long rope wanders around a field in the hills above Arusha and, by evening, has walked twice around a tree. The rope is now wrapped twice around the trunk. However the goat wanders, it cannot unwrap the rope without walking back around the tree. Had there been no tree, every tangle could have been pulled straight. The obstruction is the hole in the field where the trunk stands, and the number of turns is a whole number that no continuous wandering can change.

That simple picture leads to deep theorems. Lay a paper map of Santiago flat on a table in Santiago: some point on the map lies exactly above the point of the city it represents. Stir a cup of tea without splashing (and think of the surface as a disc): at every moment some point of the surface is back where it started. These are instances of Brouwer's fixed-point theorem. Its proof in two dimensions is a statement about loops around a hole.

The previous lesson, metric spaces and point-set topology, could tell $\R$ from $\R^2$ by removing a point and counting the pieces. It could not tell $\R^2$ from $\R^3$ : both stay connected when a point is removed. What differs is the loops. In $\R^2 \setminus \{0\}$ a loop around the origin cannot be shrunk to a point; in $\R^3 \setminus \{0\}$ every loop can (slip it over the "top"). Henri Poincaré turned this into algebra in 1895. To each space with a basepoint he attached a group, the fundamental group $\pi_1$ , whose elements are loops up to continuous deformation. This lesson builds that group, computes it for the circle, and uses it.

IntuitionH

Deforming loops

Take the plane with a hole at the origin, and loops that start and end at a basepoint $x_0 = (1, 0)$ . Two loops are homotopic if one can be continuously deformed into the other, keeping the basepoint fixed and never crossing the hole. A loop that does not go around the hole can be shrunk to the constant loop. A loop that goes around once anticlockwise cannot — but any two such loops can be deformed into each other, however much they wiggle. What survives deformation is the winding number: how many times the loop goes around the hole, counted with sign.

Loops can be composed: run $\gamma$ , then $\delta$ . Winding numbers add. Every loop has an inverse, the same loop run backwards, and $\gamma$ followed by its reverse can be shrunk away. So the deformation classes of loops form a group, and for the punctured plane — or for the circle — it should be $\Z$ . Proving that the winding number is really well defined, and that it classifies loops completely, is the heart of the lesson. The tool is a covering map: the real line wound round and round above the circle like a spiral staircase. A loop in the circle is followed upstairs, and how many floors it climbs is its winding number.

The covering map p : ℝ → S¹, drawn as a helix above a circle; p projects straight down. The points above the basepoint (dots) form a copy of ℤ. A loop in the circle lifts to a path on the helix, which ends on the floor where its winding number says.

DefinitionsH

Homotopy, paths and the fundamental group

Throughout, $I = [0, 1]$ and all maps are continuous.

Definition— homotopy

A homotopy between maps $f, g : X \to Y$ is a map $F : X \times I \to Y$ with $F(x, 0) = f(x)$ and $F(x, 1) = g(x)$ ; then $f \simeq g$ . A map is null-homotopic if it is homotopic to a constant map. A homotopy equivalence is a map $f : X \to Y$ with a map $g : Y \to X$ such that $g \circ f \simeq \mathrm{id}_X$ and $f \circ g \simeq \mathrm{id}_Y$ . A space is contractible if it is homotopy equivalent to a point.

Definition— paths and path homotopy

A path from $x_0$ to $x_1$ is a map $\gamma : I \to X$ with $\gamma(0) = x_0$ and $\gamma(1) = x_1$ ; a loop at $x_0$ is a path from $x_0$ to $x_0$ . Two paths $\gamma, \delta$ from $x_0$ to $x_1$ are path-homotopic, $\gamma \simeq \delta$ , if there is a homotopy $F : I \times I \to X$ with

F(s, 0) = \gamma(s), \quad F(s, 1) = \delta(s), \quad F(0, t) = x_0, \quad F(1, t) = x_1 .

So the endpoints stay fixed throughout. Write $[\gamma]$ for the path-homotopy class of $\gamma$ .

Definition— concatenation and reverse

If $\gamma$ ends where $\delta$ begins, the concatenation $\gamma \cdot \delta$ runs $\gamma$ at double speed, then $\delta$ :

(\gamma \cdot \delta)(s) = \begin{cases} \gamma(2s) & 0 \le s \le \tfrac12, \\ \delta(2s - 1) & \tfrac12 \le s \le 1. \end{cases}

It is continuous by the pasting lemma. The reverse of $\gamma$ is $\bar\gamma(s) = \gamma(1 - s)$ , and $c_{x}$ denotes the constant path at $x$ .

Definition— fundamental group

The fundamental group $\pi_1(X, x_0)$ is the set of path-homotopy classes of loops at $x_0$ , with the product $[\gamma][\delta] = [\gamma \cdot \delta]$ . A map $f : X \to Y$ induces $f_* : \pi_1(X, x_0) \to \pi_1(Y, f(x_0))$ , $f_*[\gamma] = [f \circ \gamma]$ . A path-connected space with trivial fundamental group is simply connected.

Definition— covering map

A map $p : E \to B$ is a covering map if every $b \in B$ has an open neighbourhood $U$ that is evenly covered: $p^{-1}(U)$ is a disjoint union of open sets (the sheets over $U$ ), each mapped homeomorphically onto $U$ by $p$ . A lift of a map $f : Z \to B$ is a map $\tilde f : Z \to E$ with $p \circ \tilde f = f$ .

Example— The basic covering

$p : \R \to S^1 \subseteq \C$ , $p(x) = e^{2\pi i x}$ , is a covering map. For $b = e^{2\pi i \theta}$ , the open arc $U$ of points $e^{2\pi i x}$ with $|x - \theta| < 1/2$ is evenly covered: $p^{-1}(U)$ is the disjoint union of the intervals $(\theta + n - \tfrac12, \theta + n + \tfrac12)$ , $n \in \Z$ . So is $z \mapsto z^n$ from $S^1$ to itself, for $n \ne 0$ : every point has $|n|$ preimages, and small arcs are evenly covered.

Worked examplesH

A first gallery

The theorems below justify every entry of this table.

Space	$\pi_1$	Reason
$\R^n$ , a convex set, a disc	trivial	straight-line homotopy (Example below)
circle $S^1$ , punctured plane, annulus	$\Z$	Theorem 6 (and homotopy equivalence)
torus $S^1 \times S^1$	$\Z \times \Z$	product formula; also van Kampen
figure eight $S^1 \vee S^1$	free group $F_2$ on two generators	van Kampen (Theorem 9)
sphere $S^n$ , $n \ge 2$	trivial	van Kampen
real projective plane $\R P^2$	$\Z/2$	van Kampen, or the double cover $S^2 \to \R P^2$
closed orientable surface of genus $g$	$\langle a_1, b_1, \dots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle$	van Kampen

Example— Convex sets are simply connected

If $C \subseteq \R^n$ is convex and $\gamma, \delta$ are paths in $C$ from $x_0$ to $x_1$ , then $F(s, t) = (1 - t)\gamma(s) + t\delta(s)$ stays in $C$ (by convexity), is continuous, equals $\gamma$ at $t = 0$ and $\delta$ at $t = 1$ , and keeps the endpoints fixed because $(1 - t)x_0 + t x_0 = x_0$ . So any two paths with the same endpoints are path-homotopic; in particular every loop is null-homotopic. This straight-line homotopy is formalised in Lean below.

Example— Products

A loop in $X \times Y$ is a pair of loops, and a path homotopy is a pair of path homotopies, so $\pi_1(X \times Y, (x_0, y_0)) \cong \pi_1(X, x_0) \times \pi_1(Y, y_0)$ . For the torus this gives $\Z \times \Z$ : a loop is determined up to homotopy by how many times it goes around "the long way" and "the short way".

Example— Same group, different spaces

$S^2$ and a single point both have trivial fundamental group, but they are not homeomorphic (and not even homotopy equivalent: $S^2$ has a non-trivial second homotopy group). The fundamental group is an invariant, not a complete classification.

ProofH

The fundamental group is a group, and an invariant

Theorem— Lemma 1 (reparametrisation)

If $\varphi : I \to I$ is continuous with $\varphi(0) = 0$ and $\varphi(1) = 1$ , then $\gamma \circ \varphi \simeq \gamma$ for every path $\gamma$ .

Proof

$F(s, t) = \gamma\big((1 - t)\varphi(s) + ts\big)$ is defined because $I$ is convex, it is continuous, it equals $\gamma \circ \varphi$ at $t = 0$ and $\gamma$ at $t = 1$ , and $F(0, t) = \gamma(0)$ , $F(1, t) = \gamma(1)$ .

∎

Theorem— Theorem 2 (π₁ is a group)

Path homotopy is an equivalence relation, concatenation is well defined on classes, and $\pi_1(X, x_0)$ is a group with identity $[c_{x_0}]$ and inverses $[\gamma]^{-1} = [\bar\gamma]$ .

Proof

Equivalence relation. Reflexivity: $F(s, t) = \gamma(s)$ . Symmetry: $F(s, 1 - t)$ . Transitivity: run a homotopy from $\gamma$ to $\delta$ for $t \in [0, \tfrac12]$ and one from $\delta$ to $\varepsilon$ for $t \in [\tfrac12, 1]$ , at double speed.

Well defined. If $F$ is a path homotopy from $\gamma$ to $\gamma'$ and $G$ from $\delta$ to $\delta'$ , then placing $F$ on $[0, \tfrac12] \times I$ and $G$ on $[\tfrac12, 1] \times I$ (with $s$ rescaled) gives a path homotopy from $\gamma \cdot \delta$ to $\gamma' \cdot \delta'$ ; the pieces agree on $s = \tfrac12$ , where both equal the common endpoint.

Associativity. $(\gamma \cdot \delta) \cdot \varepsilon$ runs $\gamma, \delta, \varepsilon$ on $[0, \tfrac14], [\tfrac14, \tfrac12], [\tfrac12, 1]$ , while $\gamma \cdot (\delta \cdot \varepsilon)$ uses $[0, \tfrac12], [\tfrac12, \tfrac34], [\tfrac34, 1]$ . So $(\gamma \cdot \delta) \cdot \varepsilon = (\gamma \cdot (\delta \cdot \varepsilon)) \circ \varphi$ , where $\varphi$ is piecewise linear, mapping $[0, \tfrac14] \to [0, \tfrac12]$ , $[\tfrac14, \tfrac12] \to [\tfrac12, \tfrac34]$ and $[\tfrac12, 1] \to [\tfrac34, 1]$ . Apply Lemma 1.

Identity. $c_{x_0} \cdot \gamma = \gamma \circ \varphi$ with $\varphi(s) = \max(0, 2s - 1)$ , and similarly for $\gamma \cdot c_{x_0}$ . Apply Lemma 1.

Inverses. $F(s, t) = \gamma\big(2s(1 - t)\big)$ for $s \le \tfrac12$ and $F(s, t) = \gamma\big((2 - 2s)(1 - t)\big)$ for $s \ge \tfrac12$ agree at $s = \tfrac12$ , equal $\gamma \cdot \bar\gamma$ at $t = 0$ and the constant loop at $t = 1$ , and fix both ends at $x_0$ . So $\gamma \cdot \bar\gamma \simeq c_{x_0}$ , and likewise $\bar\gamma \cdot \gamma \simeq c_{x_0}$ .

∎

Theorem— Proposition 3 (basepoints)

A path $h$ from $x_0$ to $x_1$ gives an isomorphism $\beta_h : \pi_1(X, x_0) \to \pi_1(X, x_1)$ , $\beta_h[\gamma] = [\bar h \cdot \gamma \cdot h]$ . So for path-connected $X$ the group $\pi_1(X, x_0)$ does not depend on $x_0$ , up to isomorphism.

Proof

$\beta_h[\gamma] \, \beta_h[\delta] = [\bar h \cdot \gamma \cdot h \cdot \bar h \cdot \delta \cdot h] = [\bar h \cdot \gamma \cdot \delta \cdot h] = \beta_h([\gamma][\delta])$ , using $h \cdot \bar h \simeq c_{x_0}$ and the group laws of Theorem 2 (which hold for paths, not only loops, with the same proofs). The inverse of $\beta_h$ is $\beta_{\bar h}$ .

∎

The isomorphism $\beta_h$ depends on the homotopy class of $h$ : changing $h$ composes $\beta_h$ with conjugation by a loop. So when $\pi_1$ is non-abelian, "the" fundamental group of a path-connected space is defined only up to non-canonical isomorphism.

Theorem— Proposition 4 (functoriality and homotopy invariance)

(a) $f_*$ is a homomorphism, $(g \circ f)_* = g_* \circ f_*$ and $(\mathrm{id}_X)_* = \mathrm{id}$ . (b) If $F : X \times I \to Y$ is a homotopy from $f$ to $g$ with $F(x_0, t) = y_0$ for all $t$ , then $f_* = g_*$ . (c) A homotopy equivalence $f : X \to Y$ induces an isomorphism $\pi_1(X, x_0) \cong \pi_1(Y, f(x_0))$ .

Proof

(a) $f \circ (\gamma \cdot \delta) = (f \circ \gamma) \cdot (f \circ \delta)$ , and $f$ carries path homotopies to path homotopies. The other two equations hold already on loops.

(b) For a loop $\gamma$ at $x_0$ , $(s, t) \mapsto F(\gamma(s), t)$ is a path homotopy from $f \circ \gamma$ to $g \circ \gamma$ ; its ends stay at $F(x_0, t) = y_0$ .

(c) When a homotopy $F$ from $f$ to $g$ does not fix the basepoint, one shows instead that $g_* = \beta_h \circ f_*$ , where $h(t) = F(x_0, t)$ is the track of the basepoint; see Hatcher (2002), Lemma 1.19. Now let $g$ be a homotopy inverse of $f$ . Applying this to $g \circ f \simeq \mathrm{id}_X$ shows that $g_* \circ f_*$ is a change-of-basepoint isomorphism, so $g_* : \pi_1(Y, f(x_0)) \to \pi_1(X, g f(x_0))$ is surjective. Applying it to $f \circ g \simeq \mathrm{id}_Y$ shows that $f_* \circ g_*$ is an isomorphism, so the same $g_*$ is injective. Hence $g_*$ is an isomorphism, and so is $f_* = g_*^{-1} \circ (g_* \circ f_*)$ .

∎

In particular contractible spaces are simply connected, and the punctured plane, the annulus and the solid torus all have the same fundamental group as the circle — each deformation retracts onto a circle.

ProofH

Lifting, and the fundamental group of the circle

Theorem— Theorem 5 (path and homotopy lifting)

Let $p : E \to B$ be a covering map. (a) For every path $\gamma$ in $B$ and every $e \in p^{-1}(\gamma(0))$ there is a unique lift $\tilde\gamma$ with $\tilde\gamma(0) = e$ . (b) For every map $F : I \times I \to B$ and every $e \in p^{-1}(F(0, 0))$ there is a unique lift $\tilde F$ with $\tilde F(0, 0) = e$ . If $F$ is a path homotopy, so is $\tilde F$ .

Proof

(a) Existence. The sets $\gamma^{-1}(U)$ , for $U$ evenly covered, form an open cover of the compact metric space $I$ . By the Lebesgue number lemma there is a subdivision $0 = s_0 < s_1 < \dots < s_m = 1$ such that each $\gamma([s_{k-1}, s_k])$ lies in one evenly covered set $U_k$ . Suppose $\tilde\gamma$ is defined on $[0, s_{k-1}]$ . Let $\tilde U$ be the sheet over $U_k$ that contains $\tilde\gamma(s_{k-1})$ , and set $\tilde\gamma = (p|_{\tilde U})^{-1} \circ \gamma$ on $[s_{k-1}, s_k]$ . The two definitions agree at $s_{k-1}$ , so the pasting lemma keeps $\tilde\gamma$ continuous. After $m$ steps, $\tilde\gamma$ is defined on $I$ .

Uniqueness. Let $\tilde\gamma, \hat\gamma$ be lifts with the same start, and $A = \{s \mid \tilde\gamma(s) = \hat\gamma(s)\}$ , which contains $0$ . Given $s$ , choose an evenly covered $U \ni \gamma(s)$ and the sheets $\tilde U \ni \tilde\gamma(s)$ , $\hat U \ni \hat\gamma(s)$ . On a small interval $J$ around $s$ , both lifts stay in their sheets. If $\tilde U = \hat U$ , both equal $(p|_{\tilde U})^{-1} \circ \gamma$ on $J$ , so $J \subseteq A$ . If $\tilde U \ne \hat U$ the sheets are disjoint, so $J \cap A = \varnothing$ . Hence $A$ and its complement are open; $I$ is connected, so $A = I$ .

(b) The same argument on the square: subdivide $I \times I$ into small squares, each mapped into an evenly covered set, and lift them one at a time, row by row; each new square meets the part already lifted in a connected set (an edge, or two edges), so the lift agrees there. Uniqueness follows from (a) applied to the segments $s \mapsto (s, t)$ and $t \mapsto (0, t)$ . If $F$ is a path homotopy, then $t \mapsto F(0, t)$ and $t \mapsto F(1, t)$ are constant, and by uniqueness so are their lifts. Details: Hatcher (2002), Proposition 1.30.

∎

Theorem— Theorem 6 (Poincaré: the fundamental group of the circle)

$\pi_1(S^1, 1) \cong \Z$ . The isomorphism sends the class of a loop $\gamma$ to its degree: $\tilde\gamma(1)$ , where $\tilde\gamma$ is the lift of $\gamma$ to $\R$ with $\tilde\gamma(0) = 0$ under $p(x) = e^{2\pi i x}$ .

Proof

The lift ends in $p^{-1}(1) = \Z$ , so $\deg \gamma = \tilde\gamma(1)$ is an integer.

Well defined. If $F$ is a path homotopy from $\gamma$ to $\gamma'$ , lift it with $\tilde F(0, 0) = 0$ (Theorem 5b). By uniqueness, $\tilde F(\cdot, 0) = \tilde\gamma$ and $\tilde F(\cdot, 1) = \tilde\gamma'$ . The path $t \mapsto \tilde F(1, t)$ lifts the constant path at $1$ , so it lies in the discrete set $\Z$ ; a path in a discrete space is constant. Hence $\tilde\gamma(1) = \tilde\gamma'(1)$ .

Homomorphism. Let $\tilde\gamma$ lift $\gamma$ from $0$ to $m$ and $\tilde\delta$ lift $\delta$ from $0$ to $n$ . The translate $\tilde\delta + m$ also lifts $\delta$ (as $p(x + m) = p(x)$ ), from $m$ to $m + n$ . So $\tilde\gamma \cdot (\tilde\delta + m)$ lifts $\gamma \cdot \delta$ from $0$ to $m + n$ : $\deg(\gamma \cdot \delta) = \deg \gamma + \deg \delta$ .

Surjective. The loop $\omega_n(s) = e^{2\pi i n s}$ lifts to $s \mapsto ns$ , so it has degree $n$ .

Injective. If $\deg \gamma = 0$ , then $\tilde\gamma$ is a loop at $0$ in $\R$ , and $\tilde F(s, t) = (1 - t)\tilde\gamma(s)$ is a path homotopy from $\tilde\gamma$ to the constant loop. Then $p \circ \tilde F$ is a path homotopy from $\gamma$ to the constant loop.

∎

The same proof, with the covering $\C \to \C \setminus \{0\}$ , $z \mapsto e^z$ , shows that $\pi_1(\C \setminus \{0\}) \cong \Z$ , with the winding number as the isomorphism.

ProofH

Brouwer and the fundamental theorem of algebra

Theorem— Corollary 7 (no retraction; Brouwer's fixed-point theorem)

(a) There is no map $r : D^2 \to S^1$ with $r(x) = x$ for all $x \in S^1$ . (b) Every map $f : D^2 \to D^2$ of the closed disc to itself has a fixed point.

Proof

(a) Let $i : S^1 \to D^2$ be the inclusion. If $r \circ i = \mathrm{id}_{S^1}$ , then $r_* \circ i_* = \mathrm{id}$ on $\pi_1(S^1) \cong \Z$ (Proposition 4a). But $i_*$ lands in $\pi_1(D^2)$ , which is trivial (the disc is convex). So the identity of $\Z$ would factor through the trivial group, which is absurd.

(b) Suppose $f(x) \ne x$ for every $x$ . Let $r(x)$ be the point where the ray from $f(x)$ through $x$ leaves the disc. Explicitly, with $u = x - f(x) \ne 0$ , $r(x) = x + \tau u$ , where $\tau \ge 0$ solves $|x + \tau u|^2 = 1$ :

\tau = \frac{-\langle x, u\rangle + \sqrt{\langle x, u\rangle^2 + |u|^2 (1 - |x|^2)}}{|u|^2}.

The square root is of a non-negative continuous function (as $|x| \le 1$ ), so $r$ is continuous. If $|x| = 1$ , then $\langle x, u\rangle = 1 - \langle x, f(x)\rangle \ge 0$ (Cauchy–Schwarz and $|f(x)| \le 1$ ), so $\tau = 0$ and $r(x) = x$ . Thus $r$ is a retraction of $D^2$ onto $S^1$ , contradicting (a).

∎

The same theorem holds for the closed ball $D^n$ in every dimension, with $\pi_1$ replaced by higher homotopy or homology groups; in dimension one it is just the intermediate value theorem (formalised below as brouwer_interval). The hypotheses matter: a rotation of an annulus, the antipodal map of a sphere and the map $x \mapsto (x + e_1)/2$ of the open disc have no fixed points.

Theorem— Corollary 8 (fundamental theorem of algebra)

Every non-constant polynomial with complex coefficients has a complex root.

Proof

We may take $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ with $n \ge 1$ . Suppose $p$ has no root. For $r \ge 0$ define the loop

f_r(s) = \frac{p(r e^{2\pi i s}) / |p(r e^{2\pi i s})|}{p(r) / |p(r)|}

in $S^1$ at $1$ . As $r$ varies, $(s, r) \mapsto f_r(s)$ is a path homotopy between any two of these loops, and $f_0$ is constant. So $[f_r] = 0$ in $\pi_1(S^1)$ for all $r$ .

Now fix $R > \max(1, |a_0| + \dots + |a_{n-1}|)$ . For $|z| = R$ ,

|a_{n-1} z^{n-1} + \dots + a_0| \le (|a_{n-1}| + \dots + |a_0|) R^{n-1} < R^n = |z^n|,

so $p_t(z) = z^n + t(a_{n-1} z^{n-1} + \dots + a_0)$ has no zero on $|z| = R$ for $t \in [0, 1]$ . Replacing $p$ by $p_t$ in the formula for $f_R$ gives a path homotopy from $f_R$ (at $t = 1$ ) to $s \mapsto e^{2\pi i n s}$ (at $t = 0$ ), whose degree is $n \ne 0$ . This contradicts $[f_R] = 0$ .

∎

ProofH

Gluing spaces: van Kampen, and covering spaces

To compute $\pi_1$ of a space built from simpler pieces, one glues the groups.

Theorem— Theorem 9 (Seifert–van Kampen; stated, proof referenced)

Let $X = U \cup V$ with $U, V$ open and path-connected, $U \cap V$ path-connected, and $x_0 \in U \cap V$ . Then the inclusions induce an isomorphism

\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0),

the free product of $\pi_1(U)$ and $\pi_1(V)$ modulo the relations $i_U(\omega) = i_V(\omega)$ for $\omega \in \pi_1(U \cap V)$ .

The proof subdivides a loop, or a homotopy, into pieces lying in $U$ or in $V$ by the Lebesgue number lemma; see Hatcher (2002), Theorem 1.20. In practice it is used like this:

Wedge of circles. For $S^1 \vee S^1$ , take $U$ and $V$ to be each circle together with a short open arc of the other at the wedge point. Each deformation retracts to a circle and $U \cap V$ is contractible, so $\pi_1(S^1 \vee S^1) \cong \Z * \Z = F_2$ , the free group on two generators $a, b$ : its elements are reduced words such as $a b^2 a^{-1} b$ . In particular $ab \ne ba$ , so fundamental groups need not be abelian.
Spheres. For $S^n$ , $n \ge 2$ , take $U$ and $V$ to be the complements of the north and south poles. Each is homeomorphic to $\R^n$ , and $U \cap V \cong S^{n-1} \times \R$ is path-connected when $n \ge 2$ . So $\pi_1(S^n)$ is trivial. (For $n = 1$ the intersection is two arcs and the theorem does not apply — rightly so.)
Surfaces. The torus is a square with sides glued by the word $a b a^{-1} b^{-1}$ . Take $U$ to be the open square and $V$ a neighbourhood of the glued boundary, which deformation retracts onto the figure eight $a \cup b$ . Then $U \cap V$ is an annulus, whose generator goes once around the boundary of the square, i.e. maps to $a b a^{-1} b^{-1}$ in $\pi_1(V)$ . So $\pi_1(T^2) = \langle a, b \mid a b a^{-1} b^{-1} \rangle \cong \Z^2$ . The genus- $g$ surface, a $4g$ -gon glued by $[a_1, b_1] \cdots [a_g, b_g]$ , has $\pi_1 = \langle a_1, \dots, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle$ , whose abelianisation is $\Z^{2g}$ . Different genera give different groups, so the surfaces are pairwise non-homeomorphic. The projective plane, a disc with antipodal boundary points glued (word $aa$ ), has $\pi_1 = \langle a \mid a^2 \rangle \cong \Z/2$ .

Covering spaces turn $\pi_1$ into a Galois-like correspondence.

Theorem— Theorem 10 (classification of covering spaces; stated, proof referenced)

Let $B$ be path-connected, locally path-connected and semilocally simply connected, with basepoint $b_0$ . Then $(E, e_0) \mapsto p_*\pi_1(E, e_0)$ is a bijection from isomorphism classes of connected based covering spaces $p : (E, e_0) \to (B, b_0)$ onto subgroups of $\pi_1(B, b_0)$ . Forgetting basepoints, connected coverings correspond to conjugacy classes of subgroups. The number of sheets equals the index of the subgroup. The trivial subgroup corresponds to the simply connected universal cover $\tilde B$ , whose group of deck transformations is isomorphic to $\pi_1(B, b_0)$ .

The proof (Hatcher 2002, Theorem 1.38) constructs $\tilde B$ as the space of homotopy classes of paths starting at $b_0$ and obtains the other coverings as its quotients. Examples: the coverings of $S^1$ are $\R \to S^1$ (subgroup $0$ ) and $z \mapsto z^n$ (subgroup $n\Z$ ); the universal cover of the torus is the plane, with deck group $\Z^2$ acting by translations. An $n$ -sheeted covering of the figure eight is described by two permutations of the $n$ sheets (where the lifts of $a$ and $b$ lead), and it is connected when they act transitively — the second experiment below counts them.

Formal mathematicsF

Homotopy in Lean and Mathlib

Mathlib's notions match the lesson's, with one difference of convention: a homotopy $F(t, s)$ takes the time first. Path x y is a bundled continuous map from the unit interval with the right endpoints, Path.Homotopy p q a homotopy rel endpoints, and FundamentalGroup X x the automorphism group of x in the fundamental groupoid (so $\pi_1$ is literally a groupoid's vertex group — see the categories viewpoint).

The straight-line homotopy of the convex-set example, and its consequences. The helper lemma is the identity $(1 - t)v + tv = v$ that keeps endpoints fixed:

smul_combo_selfLean 4Proved here from basic factsFormally verified

/-- The convex combination of a vector with itself is that vector: `(1 - t) • v + t • v = v`.
This is why a straight-line homotopy keeps the endpoints of paths fixed. -/
theorem smul_combo_self {E : Type*} [AddCommGroup E] [Module ℝ E] (t : ℝ) (v : E) :
    (1 - t) • v + t • v = v := by
  rw [← add_smul, sub_add_cancel, one_smul]