Practice: Vector bundles and characteristic classes

Vector bundles and characteristic classes

Warm-upHuman

Which of these vector bundles are trivial? (Select all that apply.)

Choose all that apply

Vector bundles and characteristic classes

Warm-upHuman

For any Riemannian metric on the closed orientable surface of genus $2$ , what is $\frac{1}{2\pi} \int_M K \, dA$ ?

Vector bundles and characteristic classes

CoreHuman

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?

Vector bundles and characteristic classes

CoreHuman

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?

Choose one

v(x) \perp x

and

|v(x)| = 1

for every

x

v

is continuous

v(x) \neq 0

for every

x

n

is even

Vector bundles and characteristic classes

CoreHuman

$S^5 \subset \C^3$ has the nowhere-zero tangent field $v(z) = iz$ . What can we conclude about $TS^5$ ?

Choose one

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.

TS^5

is trivial, by Proposition 1.

w_5(TS^5) \neq 0

.Nothing: a nowhere-zero section gives no information about the bundle.

Vector bundles and characteristic classes

CoreHuman

What is the smallest $n \ge 2$ for which $w(T\R P^n) = 1$ ?

Vector bundles and characteristic classes

StretchHuman

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?

Choose one

\R P^4

does not immerse in

\R^6

.

\R P^4

does not embed in

\R^8

.

\R P^4

does not immerse in

\R^7

.

T\R P^4

has a nowhere-zero section.

Vector bundles and characteristic classes

CoreFormal

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?

Choose one

¬ ∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = -s t) ∧ ∀ t, s t ≠ 0¬ ∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = s t) ∧ ∀ t, s t ≠ 0∀ s : ℝ → ℝ, Continuous s → (∀ t, s (t + 1) = -s t) → ∀ t, s t = 0∃ s : ℝ → ℝ, Continuous s ∧ (∀ t, s (t + 1) = -s t) ∧ ∃ t, s t = 0

Vector bundles and characteristic classes

CoreComputational

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.

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