The Lumenary · First-year graduate

Vector bundles and characteristic classes

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Date: ______________________

You will learn to

Define vector bundles, sections, pullbacks and Whitney sums, and recognise the tangent bundle, tautological bundles and the Möbius line bundle
Prove that a rank- $k$ bundle is trivial exactly when it has $k$ everywhere independent sections, and decide triviality for the Möbius bundle and the tangent bundles of $S^2$ and $S^3$
Describe bundles by transition functions and clutching, and state the classification of bundles by maps into Grassmannians
Define Stiefel–Whitney and Chern classes axiomatically and use them to obstruct sections, parallelisability and immersions
Use Chern–Weil theory to express characteristic classes through curvature, and state the Gauss–Bonnet–Chern theorem

1.
Which of these vector bundles are trivial? (Select all that apply.)
- A. $TS^1$
- B.the Möbius line bundle over $S^1$
- C. $TS^3$
- D. $TS^2$
- E.the normal bundle of $S^2 \subset \R^3$
2.
For any Riemannian metric on the closed orientable surface of genus $2$ , what is $\frac{1}{2\pi} \int_M K \, dA$ ?
3.
A tangent vector field on $S^2$ has exactly two zeros. One of them is a saddle, of index $-1$ . What is the index of the other zero?
4.
In the proof of the hairy ball theorem, $H(x, t) = \cos(\pi t)\, x + \sin(\pi t)\, v(x)$ . Which property of $v$ guarantees that $H(x, t)$ stays on the sphere?
- A. $v(x) \perp x$ and $|v(x)| = 1$ for every $x$
- B. $v$ is continuous
- C. $v(x) \neq 0$ for every $x$
- D. $n$ is even
5.
$S^5 \subset \C^3$ has the nowhere-zero tangent field $v(z) = iz$ . What can we conclude about $TS^5$ ?
- A. $TS^5 \cong E' \oplus \varepsilon^1$ for a rank- $4$ bundle $E'$ , but $TS^5$ need not be trivial — and in fact it is not.
- B. $TS^5$ is trivial, by Proposition 1.
- C. $w_5(TS^5) \neq 0$ .
- D.Nothing: a nowhere-zero section gives no information about the bundle.
6.
What is the smallest $n \ge 2$ for which $w(T\R P^n) = 1$ ?
7.
The dual Stiefel–Whitney class of $\R P^4$ is $\bar w = 1 + a + a^2 + a^3$ . What does this prove, by Whitney duality?
- A. $\R P^4$ does not immerse in $\R^6$ .
- B. $\R P^4$ does not embed in $\R^8$ .
- C. $\R P^4$ does not immerse in $\R^7$ .
- D. $T\R P^4$ has a nowhere-zero section.
8.
Sections of the Möbius bundle are modelled as continuous s : ℝ → ℝ with s (t + 1) = -s t. Which Lean statement says the Möbius bundle has no nowhere-zero section?
- A.¬ ∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = -s t) ∧ ∀ t, s t ≠ 0
- B.¬ ∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = s t) ∧ ∀ t, s t ≠ 0
- C.∀ s : ℝ → ℝ, Continuous s → (∀ t, s (t + 1) = -s t) → ∀ t, s t = 0
- D.∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = -s t) ∧ ∃ t, s t = 0
9.
Write sw_nonzero(n) returning the sorted list of degrees $i$ with $1 \le i \le n$ for which $w_i(T\R P^n) = \binom{n+1}{i} \bmod 2$ is non-zero.
Computational — Practice

Answer key — Vector bundles and characteristic classes

1.
Trivial: $TS^1$ (section $ix$ ), $TS^3$ (quaternionic frame), and the normal bundle of $S^2$ (section $x$ ). Not trivial: the Möbius bundle (Theorem 2) and $TS^2$ (Theorem 4).
2.
$\frac{1}{2\pi}\int_M K\, dA = \chi(M_2) = 2 - 4 = -2$ , whatever the metric: a genus-two surface must have negative curvature somewhere, and on average.
3.
By Poincaré–Hopf the indices sum to $\chi(S^2) = 2$ , so the other zero has index $2 - (-1) = 3$ . Such a field exists: on the Riemann sphere, $v(z) = \bar z$ has a saddle at $0$ , and in the coordinate $u = 1/z$ near infinity it becomes $\dot u = -\dot z / z^2 = -u^3/|u|^2$ , a zero of index $3$ .
4.
$|H|^2 = \cos^2(\pi t)|x|^2 + 2\cos(\pi t)\sin(\pi t)\langle x, v(x)\rangle + \sin^2(\pi t)|v(x)|^2$ , which is $1$ when $|x| = |v(x)| = 1$ and $\langle x, v(x) \rangle = 0$ .
5.
The section spans a trivial line subbundle, and with a metric $TS^5 \cong E' \oplus \varepsilon^1$ . This does not make $TS^5$ trivial: by Bott–Milnor and Kervaire, $S^n$ is parallelisable only for $n = 0, 1, 3, 7$ . By Adams, $S^5$ has exactly $\rho(6) - 1 = 1$ independent vector field.
6.
$n = 3$ : $(1 + a)^4 = 1 + a^4 = 1$ in $\Z/2[a]/(a^4)$ . Indeed $\R P^3 \cong SO(3)$ is a Lie group and hence parallelisable.
7.
An immersion $\R P^4 \to \R^{4+k}$ has a rank- $k$ normal bundle with $w = \bar w$ , so $\bar w_i = 0$ for $i > k$ . Since $\bar w_3 = a^3 \neq 0$ , $k \ge 3$ : no immersion into $\R^6$ . This is Theorem 8(b) for $r = 2$ , and it is sharp.
8.
(a) is Lumenary.VectorBundlesCharacteristicClasses.mobius_not_trivial. (b) is false (constant $1$ ), (c) is false ( $\sin \pi t$ ), and (d) is true for the trivial reason $s = 0$ .
9.
```
from math import comb

def sw_nonzero(n):
    return [i for i in range(1, n + 1) if comb(n + 1, i) % 2 == 1]
```
The list is empty exactly when $n + 1$ is a power of $2$ ( $n = 1, 3, 7, 15, \dots$ ), the only candidates for parallelisable projective spaces.