TCC Introduction to threemanifolds
Term II 20172018

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, Ibundles. 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 
JordanSchoenflies 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. KneserMilnor.

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 twosphere in
the threesphere \(S^3\). Then \(S^3  S\) has exactly two
connected components. Furthermore, the closure of each is
diffeomorphic to a threeball.
The important change here is making the separation explicit.

5 
Feb. 7 
Onehalf lives, onehalf dies. Onesided, twosided
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. Twosided surfaces.

Six

Six

See Cameron Gordon's article 3Dimensional Topology up to
1960 for a history of Dehn's Lemma, Kneser's Hilfsatz, and
Papakyriakopoulos' tower argument.
The discussion of twosided surfaces corrects a small mistake
made in Lecture Four.

8 
Feb. 28 




9 
Mar. 7 






