The Lumenary · Research pathways

Low-dimensional topology: 3- and 4-manifolds

Name: ______________________

Date: ______________________

You will learn to

Describe closed 3-manifolds by Heegaard splittings and by Dehn surgery on links, computing π₁ and H₁ of lens spaces and of surgeries from the gluing data
Use the arithmetic classification of lens spaces to exhibit 3-manifolds that are homotopy equivalent but not homeomorphic
State the geometrisation theorem and the Poincaré conjecture, describe the eight Thurston geometries, and explain Mostow rigidity for hyperbolic 3-manifolds
Contrast topological and smooth 4-manifolds through intersection forms and the theorems of Freedman, Rokhlin and Donaldson, using the E₈ form
Separate established results from open problems such as the smooth 4-dimensional Poincaré conjecture and the 11/8 conjecture

1.
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 $K$ ?
2.
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?
3.
For which $q$ is $L(11, q)$ homeomorphic to $L(11, 2)$ ? (Choose all that apply.)
- A. $q = 6$
- B. $q = 9$
- C. $q = 5$
- D. $q = 3$
- E. $q = 4$
4.
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?
- A.Freedman's $E_8$ -manifold
- B.The K3 surface
- C. $\mathbb{C}P^2$
- D. $S^2 \times S^2$
5.
Which of these statements follow from the geometrisation theorem? (Choose all that apply.)
- A.A closed simply connected 3-manifold is homeomorphic to $S^3$ .
- B.A closed 3-manifold with finite fundamental group is a quotient $S^3/\Gamma$ .
- C.Every closed 3-manifold is hyperbolic.
- D.Every closed prime 3-manifold is Seifert fibred.
6.
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?
- A.Two congruence statements modulo 7: $1 \cdot 2 \equiv \pm n^2$ for some $n$ , and $2 \not\equiv \pm 1^{\pm1}$ .
- B.That the lens spaces $L(7,1)$ and $L(7,2)$ are homotopy equivalent but not homeomorphic.
- C.Whitehead's homotopy classification of lens spaces.
- D.That Reidemeister torsion distinguishes $L(7, 1)$ from $L(7, 2)$ .
7.
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?
8.
Which statement is a consequence of Mostow rigidity?
- A.The volume of a closed hyperbolic 3-manifold is a topological invariant: homeomorphic hyperbolic 3-manifolds have equal volumes.
- B.A closed surface of genus 2 has a unique hyperbolic metric.
- C.Every closed 3-manifold with infinite fundamental group is hyperbolic.
- D.Homotopy equivalent closed 3-manifolds are homeomorphic.
9.
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 $p$ .)
Computational — Practice

Answer key — Low-dimensional topology: 3- and 4-manifolds

1.
$H_1(S^3 \setminus N(K)) \cong \mathbb{Z}\langle\mu\rangle$ and $[\lambda] = 0$ . The new solid torus kills $7\mu + 3\lambda = 7\mu$ , so $H_1 \cong \mathbb{Z}/7$ , of order $7$ . This holds for every knot: the homology of a surgery does not see the knot type.
2.
$\Lambda = \begin{pmatrix} 3 & 1 \\ 1 & 4 \end{pmatrix}$ , $\det \Lambda = 12 - 1 = 11$ . So $H_1 \cong \mathbb{Z}/11$ (it is cyclic, since $\gcd$ of the entries is $1$ ), and the result is a lens space $L(11, q)$ for a suitable $q$ .
3.
The class of $2$ is $\{2, -2, 2^{-1}, -2^{-1}\} = \{2, 9, 6, 5\}$ modulo 11. So $q \in \{5, 6, 9\}$ (besides $2$ itself). The remaining units form the classes $\{1, 10\}$ and $\{3, 4, 7, 8\}$ : three homeomorphism classes, but only one homotopy class.
4.
By Freedman there is a topological 4-manifold with form $E_8$ , which is even, unimodular and has signature 8. If smoothness were not needed in Rokhlin's theorem, we would get $16 \mid 8$ . So the theorem is genuinely about smooth manifolds — and the $E_8$ -manifold has no smooth structure.
5.
Pieces with finite $\pi_1$ are spherical, which gives elliptisation and, as the special case $\pi_1 = 1$ , the Poincaré conjecture. Geometrisation does not say every manifold is hyperbolic or Seifert fibred: those are two of the possible kinds of pieces.
6.
The definitions are relations on units of $\mathbb{Z}/7$ , and decide checks them by exhausting the six units. So Lean certifies the arithmetic of Corollary 5. The step from arithmetic to topology is Theorem 4, cited from the literature. Saying precisely what is and is not formalised is part of honest formal mathematics.
7.
$\sigma = \pm 8a$ , and Rokhlin requires $16 \mid 8a$ , so $a$ is even; the smallest positive value is $a = 2$ . It occurs: the K3 surface has form $2(-E_8) \oplus 3H$ . Whether $2E_8 \oplus bH$ is realised smoothly for $b = 1, 2$ is decided by Furuta's 10/8 theorem (no: it needs $b_2 = 16 + 2b \ge 22$ , so $b \ge 3$ ).
8.
If $M \cong N$ are closed hyperbolic 3-manifolds, the induced isomorphism $\pi_1(M) \cong \pi_1(N)$ is realised by an isometry, so $\mathrm{vol}(M) = \mathrm{vol}(N)$ . Hence volume is a topological invariant — the basis of computer tables of hyperbolic 3-manifolds ordered by volume. In dimension 2 rigidity fails.
9.
```
from math import gcd

def lens_classes(p):
    reps = set()
    for q in range(1, p):
        if gcd(q, p) == 1:
            inv = pow(q, -1, p)
            reps.add(min(q % p, -q % p, inv, -inv % p))
    return len(reps)
```
The set $\{q, -q, q^{-1}, -q^{-1}\}$ is exactly the class of $q$ (the relation is an equivalence relation, as the Lean file proves), so its minimum is a canonical representative. For prime $p$ the classes have size 4, except the class of $\pm 1$ and, when $-1$ is a square, the class of a square root of $-1$ , which have size 2.