\( \newcommand{\cross}{\times} \newcommand{\isom}{\cong} \newcommand{\RP}{\mathbb{RP}} \newcommand{\EE}{\mathbb{E}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \)

TCC Introduction to three-manifolds
Term II 2017-2018

Schedule

Week Date of Wednesday Topics Example sheets Lecture notes Comments
2 Jan. 17 Overview. Homeomorphism problem, review of lower dimensions, connect sum and surgery.

Elliptic examples: lens spaces and prism manifolds.

One One The lecture notes are readable if you zoom in enough!
3 Jan. 24 Orientability, I-bundles. Isometries of spheres. Isometries of \(S^2 \cross \RR\), manifolds with this geometry.

Alexander's horned sphere, locally flat, Alexander's theorem. Irreducible, prime.

Two Two
4 Jan. 31 Jordan-Schoenflies in dimension two. Alexander in dimension three. Morse position, regular levels, classification of components.

Width, thick levels. Surgery does not reduce width, or it does. Space of embeddings. Kneser-Milnor.

Three Three To address Alex's question about Cases 2a and 2b we need a stronger induction hypothesis. Here is a cleaner statement of Alexander's theorem.

Suppose that \(S\) is a smoothly embedded two-sphere in the three-sphere \(S^3\). Then \(S^3 - S\) has exactly two connected components. Furthermore, the closure of each is diffeomorphic to a three-ball.

The important change here is making the separation explicit.

5 Feb. 7 One-half lives, one-half dies. One-sided, two-sided surfaces. Manifolds without double covers.

Knots, wild knots, knot complements. Knot genus, Rolfsen table. Torus knots, satellite knots, hyperbolic knots, Thurston's theorem.

Four Four To clarify the discussion at the very end: a knot \(K \subset S^3\) is hyperbolic if the manifold \(S^3 - K\) admits a complete hyperbolic metric of finite volume. With this in hand we can restate Thurston's theorem.

Any knot in \(S^3\) is either the unknot, a torus knot, a satellite knot, or a hyperbolic knot.

By being a bit more careful with the definitions, we can even make these classes of knots disjoint.

6 Feb. 14 Characterisation of the unknot. General position: simple branch points (Whitney umbrellas), double curves and arcs, triple points. The singular set, sectors, complexity, swaps.

Statement of the Disk Theorem, Dehn's Lemma, characterisation of handlebodies.

Five Five
7 Feb. 21 Proof of the disk theorem. Eliminating simple branch points, double arcs and curves.

Climbing the tower, the planar surface at the top, descending. Consequences. Two-sided surfaces.

Six Six See Cameron Gordon's article 3-Dimensional Topology up to 1960 for a history of Dehn's Lemma, Kneser's Hilfsatz, and Papakyriakopoulos' tower argument.

The discussion of two-sided surfaces corrects a small mistake made in Lecture Four.

8 Feb. 28
9 Mar. 7