The Lumenary · First-year graduate

Homotopy theory

Name: ______________________

Date: ______________________

You will learn to

Define the homotopy groups $\pi_n(X, x_0)$ and prove, by the Eckmann–Hilton argument, that they are abelian for $n \ge 2$
Define fibrations, derive the long exact sequence of homotopy groups, and compute $\pi_n$ of covering spaces such as $S^1$ and $\R P^n$
Construct the Hopf fibration $S^1 \to S^3 \to S^2$ , prove that its fibres are circles, and deduce $\pi_3(S^2) \cong \Z$
State Whitehead's, Hurewicz's and Freudenthal's theorems, with their hypotheses, and apply them to spheres and CW complexes
Define Eilenberg–MacLane spaces and explain how they represent cohomology

1.
For which $k$ is $\pi_k(S^3) = 0$ ? (Select all that apply.)
- A. $k = 1$
- B. $k = 2$
- C. $k = 3$
- D. $k = 4$
- E. $k = 6$
2.
Which list gives $\pi_1, \pi_2, \pi_3$ of $\R P^3$ ?
- A. $\Z/2,\ 0,\ \Z$
- B. $\Z/2,\ \Z/2,\ 0$
- C. $\Z,\ 0,\ \Z$
- D. $\Z/2,\ 0,\ 0$
3.
The quaternionic Hopf fibration $S^3 \to S^7 \to S^4$ has null-homotopic fibre inclusion (since $\pi_3(S^7) = 0$ ), and this splits its long exact sequence: $\pi_n(S^4) \cong \pi_n(S^7) \oplus \pi_{n-1}(S^3)$ . Given $\pi_6(S^3) \cong \Z/12$ and $\pi_7(S^7) \cong \Z$ , what is the order of the torsion subgroup of $\pi_7(S^4)$ ?
4.
The Eckmann–Hilton argument proves that $\pi_n(X)$ is abelian for $n \ge 2$ . Why does it not apply to $\pi_1(X)$ ?
- A.Loops have only one coordinate, so there is only one way to concatenate them and no second operation to interchange with.
- B.The constant loop is not an identity element for concatenation.
- C.Homotopy classes of loops cannot be concatenated in a well-defined way.
- D.The interchange law fails for loops because the unit interval is not a cube.
5.
Which example shows that the CW hypothesis in Whitehead's theorem cannot be dropped?
- A.The map from the Warsaw circle to a point
- B.The map from $\R$ to a point
- C.The degree-two map $S^1 \to S^1$
- D.The inclusion $S^1 \hookrightarrow D^2$
6.
Let $\Sigma$ be Poincaré's homology sphere and $c : \Sigma \to S^3$ a degree-one map. Which statement is correct?
- A. $c$ induces isomorphisms on all homology groups, but it is not a homotopy equivalence, because $\pi_1(\Sigma) \neq 0$ ; so the homology Whitehead theorem really needs simple connectivity.
- B. $c$ is a homotopy equivalence by the homology Whitehead theorem.
- C. $c$ cannot induce an isomorphism on $H_3$ , since $\Sigma \not\cong S^3$ .
- D. $c$ induces isomorphisms on all homotopy groups.
7.
Let S3 z w abbreviate Complex.normSq z + Complex.normSq w = 1. Which Lean statement says that the fibre of the Hopf map through a point of $S^3$ is exactly the circle through it?
- A.S3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → ∃ l : ℂ, Complex.normSq l = 1 ∧ z' = l * z ∧ w' = l * w
- B.∀ l : ℂ, Complex.normSq l = 1 → hopf (l * z, l * w) = hopf (z, w)
- C.S3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → ∃ t : ℝ, z' = t * z ∧ w' = t * w
- D.S3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → z = z' ∧ w = w'
8.
Write hopf(z, w) returning the triple (x, y, t) of real numbers with $x + iy = 2 z \bar w$ and $t = |z|^2 - |w|^2$ . The tests check that points of $S^3$ go to $S^2$ , that the map is constant on circles $\lambda(z, w)$ , and two special values.
Computational — Practice

Answer key — Homotopy theory

1.
$\pi_1(S^3) = \pi_2(S^3) = 0$ by Theorem 5, while $\pi_3(S^3) \cong \Z$ , $\pi_4(S^3) \cong \Z/2$ and $\pi_6(S^3) \cong \Z/12$ . Unlike homology, the homotopy groups of $S^3$ do not vanish above degree $3$ .
2.
$\pi_1(\R P^3) \cong \Z/2$ (the deck group of the universal cover), and for $n \ge 2$ , $\pi_n(\R P^3) \cong \pi_n(S^3)$ , giving $\pi_2 = 0$ and $\pi_3 \cong \Z$ .
3.
$\pi_7(S^4) \cong \pi_7(S^7) \oplus \pi_6(S^3) \cong \Z \oplus \Z/12$ , so its torsion subgroup has order $12$ . The free summand is generated by the quaternionic Hopf map; it is the $\Z$ that Serre's finiteness theorem allows in $\pi_{2n-1}(S^n)$ for $n$ even.
4.
The proof uses $+_1$ and $+_2$ , concatenation in the first and second coordinates, and the interchange law between them. For $n = 1$ there is only $+_1$ , and indeed $\pi_1(S^1 \vee S^1)$ is a non-abelian free group.
5.
The Warsaw circle $W$ has $\pi_n(W) = 0$ for all $n$ (no path can pass from the oscillating curve onto the limit segment, so the image of any map from a sphere lies in a contractible part of $W$ ), so $W \to \mathrm{pt}$ is a weak equivalence. But $W$ is not contractible, as its Čech cohomology $\check H^1(W) \cong \Z$ shows. $W$ is not a CW complex.
6.
$H_\bullet(\Sigma) \cong H_\bullet(S^3)$ and $c_*$ is an isomorphism on $H_0$ and $H_3$ (degree one), the other groups being zero. But $\pi_1(\Sigma)$ has order $120$ , so $c$ is not a homotopy equivalence. Corollary 10(b) needs simple connectivity.
7.
Statement (a), Lumenary.HomotopyTheory.hopf_fibre, says every point of the fibre is $\lambda \cdot (z, w)$ with $|\lambda| = 1$ . The reverse inclusion is (b), hopf_circle_invariant. Statements (c) and (d) are false.
8.
```
def hopf(z, w):
    v = 2 * z * complex(w).conjugate()
    return (v.real, v.imag, abs(z) ** 2 - abs(w) ** 2)
```
On $S^3$ , $x^2 + y^2 + t^2 = 4|z|^2|w|^2 + (|z|^2 - |w|^2)^2 = (|z|^2 + |w|^2)^2 = 1$ , and multiplying $(z, w)$ by $\lambda$ with $|\lambda| = 1$ multiplies $z \bar w$ by $\lambda \bar \lambda = 1$ .