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The Lumenary · Advanced undergraduate

Homotopy and the fundamental group

Name: ______________________

Date: ______________________

You will learn to

Define homotopy of maps and of paths and the fundamental group, and prove that the fundamental group is a group and a homotopy invariant
Compute $\pi_1(S^1) \cong \Z$ using the covering map $\R \to S^1$ and the path and homotopy lifting properties
Deduce the Brouwer fixed-point theorem for the disc and the fundamental theorem of algebra
Use the Seifert–van Kampen theorem to compute fundamental groups of wedges of circles, spheres and surfaces
Relate connected covering spaces to subgroups of the fundamental group

1.
Which of these spaces are simply connected? (Choose all that apply.)
- A.The closed disc $D^2$
- B. $\mathbb{R}^3 \setminus \{0\}$
- C.The sphere $S^2$
- D. $\mathbb{R}^2 \setminus \{0\}$
- E.The torus $S^1 \times S^1$
- F.The annulus $\{1 \le |z| \le 2\}$
2.
Under the isomorphism $\pi_1(S^1, 1) \cong \mathbb{Z}$ of Theorem 6, what integer corresponds to the class of the loop $\gamma(s) = e^{-6\pi i s}$ ?
3.
In the proof that the degree is well defined, we lift a path homotopy $F$ from $\gamma$ to $\gamma'$ to $\tilde F$ , and then claim that $t \mapsto \tilde F(1, t)$ is constant. Why is it constant?
- A.It is a path in the discrete set $p^{-1}(1) = \mathbb{Z}$ , and paths in a discrete space are constant.
- B.Because $\mathbb{R}$ is simply connected.
- C.Because lifts are unique.
- D.Because $I \times I$ is compact.
4.
Brouwer's theorem says every continuous map $D^2 \to D^2$ has a fixed point. For which of these spaces is there a continuous map to itself with no fixed point? (Choose all that apply.)
- A.The annulus $\{1 \le |z| \le 2\}$
- B.The open disc $\{|z| < 1\}$
- C.The sphere $S^2$
- D.The closed disc $D^2$
- E.The interval $[0, 1]$
5.
The closed orientable surface of genus 2 is an octagon with sides glued by the word $a b a^{-1} b^{-1} c d c^{-1} d^{-1}$ . What does the Seifert–van Kampen theorem give for its fundamental group?
- A. $\langle a, b, c, d \mid a b a^{-1} b^{-1} c d c^{-1} d^{-1} \rangle$
- B. $\mathbb{Z}^4$
- C.The free group on $a, b, c, d$
- D. $\mathbb{Z}^2 \times \mathbb{Z}^2$ , one torus for each handle
6.
The lesson's Lean theorem circle_not_retract takes a simply connected space X, maps i : C(AddCircle 1, X) and r : C(X, AddCircle 1), and a hypothesis h, and proves False. Which hypothesis h makes it the no-retraction theorem?
- A.h : ∀ z, r (i z) = z
- B.h : ∀ x, i (r x) = x
- C.h : Function.Injective i
- D.h : Function.Surjective r
7.
Implement path lifting for the angle. Write winding_number(points) for a list of non-zero complex numbers points describing a closed loop around $0$ (the last point equals the first). Add up the changes of angle between consecutive points, each taken in $(-\pi, \pi]$ , and return the total divided by $2\pi$ , rounded to the nearest integer.
Computational — Practice
8.
How many connected 2-sheeted covering spaces does the figure eight $S^1 \vee S^1$ have, up to isomorphism?
9.
In the plane with two holes, let $a$ and $b$ be loops going once around the first and the second hole. True or false: the loop $a b a^{-1} b^{-1}$ , which winds $0$ times around each hole, is null-homotopic.
- A.False: $aba^{-1}b^{-1} \ne 1$ in $\pi_1 \cong F_2$ .
- B.True: its winding number around each hole is $0$ .
- C.True: $\pi_1$ is abelian, so $aba^{-1}b^{-1} = 1$ .

Answer key — Homotopy and the fundamental group

1.
Simply connected: the disc (convex), $S^2$ (van Kampen) and $\mathbb{R}^3 \setminus \{0\}$ (homotopy equivalent to $S^2$ ). Not simply connected: $\mathbb{R}^2 \setminus \{0\}$ and the annulus ( $\pi_1 \cong \mathbb{Z}$ , both homotopy equivalent to $S^1$ ) and the torus ( $\mathbb{Z}^2$ ). Note that $\mathbb{R}^2 \setminus \{0\}$ and $\mathbb{R}^3 \setminus \{0\}$ have different $\pi_1$ , which proves $\mathbb{R}^2 \not\cong \mathbb{R}^3$ .
2.
$\gamma(s) = e^{2\pi i(-3s)}$ , so the lift starting at $0$ is $\tilde\gamma(s) = -3s$ , which ends at $-3$ . The degree is $-3$ : the loop runs three times around the circle clockwise.
3.
Since $F$ fixes the endpoint, $p(\tilde F(1, t)) = F(1, t) = 1$ , so $\tilde F(1, t)$ lies in $p^{-1}(1) = \mathbb{Z}$ for all $t$ . A continuous map from the connected space $I$ to the discrete space $\mathbb{Z}$ is constant. So $\tilde\gamma(1) = \tilde F(1, 0) = \tilde F(1, 1) = \tilde\gamma'(1)$ .
4.
The annulus (rotate it), the open disc (push towards a missing boundary point) and the sphere (antipodal map) have fixed-point-free self-maps. The closed disc and $[0, 1]$ do not, by Brouwer's theorem in dimensions 2 and 1. The examples show that both compactness and the absence of holes matter.
5.
With $U$ the open octagon (contractible), $V$ a neighbourhood of the boundary (homotopy equivalent to a wedge of four circles, $\pi_1 = F(a, b, c, d)$ ) and $U \cap V$ an annulus whose generator maps to the boundary word, van Kampen gives $\pi_1 = \langle a, b, c, d \mid [a, b][c, d] \rangle$ . Its abelianisation is $\mathbb{Z}^4$ , which distinguishes it from the torus ( $\mathbb{Z}^2$ ).
6.
The condition is ∀ z, r (i z) = z, that is $r \circ i = \mathrm{id}_{S^1}$ . The Lean proof then follows the lesson: the loop $\gamma$ winding once is sent by $i$ to a null-homotopic loop in X, and $r$ carries that null-homotopy to one of $r \circ i \circ \gamma = \gamma$ , contradicting winding number $1 \ne 0$ .

import cmath
import math

def winding_number(points):
    total = 0.0
    for z0, z1 in zip(points, points[1:]):
        step = cmath.phase(z1) - cmath.phase(z0)
        step = (step + math.pi) % (2 * math.pi) - math.pi
        total += step
    return round(total / (2 * math.pi))

This is the proof of the path lifting theorem run as an algorithm: each small step of the loop stays in an evenly covered arc, where the lift is determined by choosing the nearby branch of the angle. It is correct as long as consecutive points turn by less than $\pi$ around $0$ .

8.
A 2-sheeted cover is given by two permutations of $\{1, 2\}$ , the monodromies of $a$ and $b$ ; there are $4$ pairs, and the cover is connected unless both are the identity. Relabelling the two sheets conjugates both permutations, which changes nothing in $S_2$ . So there are $3$ connected double covers. Equivalently, $F_2$ has $3$ subgroups of index $2$ : the kernels of the three surjections $F_2 \to \mathbb{Z}/2$ (send $a, b$ to $(1, 0)$ , $(0, 1)$ or $(1, 1)$ ). The lesson's experiment confirms the counts $1, 3, 13, 71, 461$ of subgroups of index $1, \dots, 5$ .
9.
False. $\pi_1(\mathbb{R}^2 \setminus \{p, q\}) \cong F_2$ (the space deformation retracts onto a figure eight, then use van Kampen), and $aba^{-1}b^{-1}$ is a non-empty reduced word, so it is not the identity. One can also see it with the homomorphism to $S_3$ sending $a \mapsto (0\,1)$ , $b \mapsto (1\,2)$ : the commutator maps to a 3-cycle, not the identity. Its winding numbers vanish because winding numbers factor through the abelianisation.