Practice: Low-dimensional topology: 3- and 4-manifolds

Low-dimensional topology: 3- and 4-manifolds

Warm-upHuman

Let $K$ be the figure-eight knot. What is the order of $H_1\big(S^3_{7/3}(K)\big)$ , the first homology of $7/3$ -surgery on $K$ ?

Low-dimensional topology: 3- and 4-manifolds

Warm-upHuman

Integral surgery on the Hopf link (two unknots with linking number $1$ ) with framings $3$ and $4$ gives a lens space. What is the order of its first homology group?

Low-dimensional topology: 3- and 4-manifolds

CoreHuman

For which $q$ is $L(11, q)$ homeomorphic to $L(11, 2)$ ? (Choose all that apply.)

Choose all that apply

Low-dimensional topology: 3- and 4-manifolds

CoreHuman

Rokhlin's theorem: if a closed smooth simply connected 4-manifold has an even intersection form, its signature is divisible by 16. Which example shows that the smoothness hypothesis cannot be dropped?

Choose one

Freedman's

E_8

-manifoldThe K3 surface

\mathbb{C}P^2

S^2 \times S^2

Low-dimensional topology: 3- and 4-manifolds

CoreHuman

Which of these statements follow from the geometrisation theorem? (Choose all that apply.)

Choose all that apply

A closed simply connected 3-manifold is homeomorphic to

S^3

.A closed 3-manifold with finite fundamental group is a quotient

S^3/\Gamma

.Every closed 3-manifold is hyperbolic.Every closed prime 3-manifold is Seifert fibred.

Low-dimensional topology: 3- and 4-manifolds

CoreFormal

The Lean theorem lens_seven proves LensHomotopyRel (p := 7) 1 2 ∧ ¬ LensHomeoRel (p := 7) 1 2 (with 2 as a unit of ZMod 7). What exactly has been machine-checked?

Choose one

Two congruence statements modulo 7:

1 \cdot 2 \equiv \pm n^2

for some

n

, and

2 \not\equiv \pm 1^{\pm1}

.That the lens spaces

L(7,1)

and

L(7,2)

are homotopy equivalent but not homeomorphic.Whitehead's homotopy classification of lens spaces.That Reidemeister torsion distinguishes

L(7, 1)

from

L(7, 2)

.

Low-dimensional topology: 3- and 4-manifolds

StretchHuman

A closed smooth simply connected 4-manifold with even, indefinite intersection form has form $\pm\big(a E_8 \oplus b H\big)$ with $a \ge 0$ , $b \ge 1$ , and signature $\pm 8a$ . By Rokhlin's theorem, what is the smallest positive value of $a$ that can occur?

Low-dimensional topology: 3- and 4-manifolds

StretchHuman

Which statement is a consequence of Mostow rigidity?

Choose one

The volume of a closed hyperbolic 3-manifold is a topological invariant: homeomorphic hyperbolic 3-manifolds have equal volumes.A closed surface of genus 2 has a unique hyperbolic metric.Every closed 3-manifold with infinite fundamental group is hyperbolic.Homotopy equivalent closed 3-manifolds are homeomorphic.

Low-dimensional topology: 3- and 4-manifolds

CoreComputational

Write lens_classes(p) returning the number of homeomorphism classes of lens spaces $L(p, q)$ with $0 < q < p$ and $\gcd(p, q) = 1$ , using the criterion $L(p, q) \cong L(p, q') \iff q' \equiv \pm q^{\pm 1} \pmod p$ . (In Python, pow(q, -1, p) is the inverse of $q$ modulo $p$ .)

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