# Student 3-Manifold Seminar

The aim of this seminar is to learn basic 3-manifold topology, since Berkeley has not recently offered such a course. The topics overlap somewhat with the Student Geometry and Low-Dimensional Topology Seminar, but our focus will be on this particular dimension. This seminar complements Prof. Agol’s 3-manifold seminar, which tends to be more research-focused.

The current plan is to meet weekly in 939 Evans at 4–5:30pm on Fridays for Fall 2019, beginning 9/13/19.

Possible topics:

- 3-manifold decompositions (prime and JSJ)
- Geometric classifications
- The loop and sphere theorems
- Incompressible surfaces and Haken manifolds
- Hyperbolic link complements
- Dehn surgery and the Lickorish–Wallace and Kirby theorems
- Branched covers and the Montesinos trick
- Triangulations and piecewise linear structures
- Normal surface theory and recognition algorithms
- Heegaard splittings
- Sutured manifold hierarchies, Thurston norms
- Thurston’s geometrization conjecture theorem
- Fundamental groups of 3-manifolds
- The surface subgroup conjecture
- Thin position arguments
- Spanning surfaces of knots
- Turaev surfaces
- Khovanov homology

## 1. References

- Hatcher’s notes (overview)
- Calegari’s notes, which are being made into a book
- Lackenby’s notes (at the bottom, “Three-dimensional manifolds”). As a single pdf
- Thurston, “Three-Dimensional Geometry and Topology”
- Thurston’s notes (more thorough than the book)
- Purcell, “Hyperbolic knot theory” (an expansion of some of Thurston’s notes)
- Hempel, “3-Manifolds”
- Jaco, “Lectures on Three-Manifold Topology”
- Rolfsen, “Knots and Links”
- Lickorish, “An Introduction to Knot Theory”
- Rourke and Sanderson, “Introduction to Piecewise-Linear Topology”
- Lurie’s notes about PL topology

## 2. Schedule (Fall 2019)

- 9/13/2019 — Tait’s conjecture — Kyle Miller
This will be an organizational meeting for a student learning seminar about 3-manifolds. We will talk about Tait’s conjecture that reduced alternating link diagrams have minimal crossing number, which was resolved in 1987 by Kauffman, Murasugi, and Thistlethwaite using the Jones polynomial.

## 3. Schedule (Spring 2019)

- 1/25/2019 — Prime decompositions (part 1) — Kyle Miller (Notes)
This is an organizational meeting for this learning seminar. We will additionally discuss a little PL topology and begin with the existence and uniqueness of prime decompositions for connected, oriented, compact 3-manifolds, due to Kneser and Milnor.

- 2/1/2019 — Prime decompositions (part 2) — Kyle Miller (Notes)
We will continue discussing Kneser’s and Milnor’s results on the existence and uniqueness of prime decompositions for connected, oriented, compact 3-manifolds.

- 2/8/2019 — The loop theorem and Dehn’s lemma — Kyle Miller (Notes)
Simplified, the loop theorem states that if the induced map ${\pi}_{1}(\mathrm{\partial}M)\to {\pi}_{1}(M)$ for a $3$-manifold $M$ is not injective, then there is a nullhomotopy of an essential loop in $\mathrm{\partial}M$ that can be represented by an embedded disk. We will go through the proof of Stalling’s formulation of the loop theorem using Papakyriakopoulos’s tower construction and discuss some applications.

- 2/15/2019 — The Sphere Theorem (part 1) — Kyle Miller (Notes)
A version of the Papakyriakopoulos Sphere Theorem states that a compact $3$-manifold with nontrivial ${\pi}_{2}$ has a two-sided embedded sphere or projective plane representing a nontrivial homotopy class. We will discuss ends of groups and how the theorem follows from Stallings’s theorem on finitely generated groups with more than one end.

- 2/22/2019 — The Sphere Theorem (part 2) — Kyle Miller (Notes)
This is a continuation of the previous talk. (References: Stallings,

*Group theory and three-dimensional manifolds*, 1971. Epstein,*Ends*, Topology of 3-manifolds and related topics, 1961. Peschke,*The Theory of Ends*, 1990. Calegari,*Notes on 3-manifold topology*.)- 3/1/2019 — JSJ decompositions — Kyle Miller
The irreducible $3$-manifolds that come from a prime decomposition can be further decomposed along embedded tori. Jaco, Shalen, and Johannson proved there is a minimal collection of such tori, unique up to isotopy, that splits an irreducible compact orientable manifold into pieces that are either Seifert-fibered or atoroidal. We will discuss examples, incompressible surfaces, and Seifert-fibered spaces.

- 3/8/2019 — JSJ decompositions (cont’d) — Kyle Miller
This is a continuation of last week’s talk. We will discuss Seifert-fibered spaces and the existence of JSJ decompositions.

- 3/15/2019 — JSJ decompositions (cont’d) — Kyle Miller
This is a continuation of the discussion about the existence and uniqueness of JSJ decompositions.

- 4/5/2019 — Haken manifolds and their hierarchies — Michael Klug
Compact surfaces with non-positive Euler characteristic can be inductively decomposed by cutting along finitely many properly embedded loops and arcs until one is left with a collection of disks; such a decomposition is called a hierarchy. An analogue up a dimension is called a Haken manifold, which can be inductively decomposed by cutting along two-sided incompressible surfaces until one is left with a collection of balls. Examples of Haken manifolds include link complements and surface bundles over circles. Certain facts about Haken manifolds can be proved by induction on a hierarchy.

- 4/12/2019 — Reidemeister torsion — Calvin McPhail-Snyder
Some background: An early homological invariant of knot complements is the Alexander polynomial (which is the “just” a kind of determinant of ${H}_{1}$ of the knot complement with coefficients twisted by the abelianization), and the polynomial shows up in many guises. One way is through a torsion that Reidemeister introduced to classify lens spaces up to homeomorphism. Using a cellular decomposition of a space, the torsion extracts a little more information from a chain complex, and unlike homology it is not invariant under homotopy equivalence.

- 4/19/2019 — Knot and link complements — Kyle Miller
We will talk about some of the basic theory of knots and links in ${S}^{3}$ with a special focus on the geometry of the complementary space.

- 4/26/2019 — Knot and link invariants — Kyle Miller
We will continue our discussion of spaces that are the complement of a knot or link in ${S}^{3}$, and then we will talk about Conway’s normalization of the Alexander polynomial. If time permits, we will see the definition of the Jones polynomial via the Kauffman bracket.

- 5/3/2019 — All the ways I know how to define the Alexander polynomial – Kyle Miller
In this talk, I will try to go through all the ways I am aware of that one can define the Alexander polynomial. To name a few: Alexander’s original definition, the Fox calculus, the Conway potential function, polynomial extrapolation of ${U}_{q}(\mathfrak{s}\mathfrak{l}(n))$ quantum invariants, Reidemeister torsion, the Burau representation of braid groups, Alexander representations of the knot quandle, and knot Floer homology.

## 4. Other things

There is a list of upcoming conferences in low-dimensional topology at the Low-dimensional topology blog.