Week 
Date of Tuesday 
Topics 
Example sheet 
Lecture notes 
Comments 
1 
Jan. 11 
S, P, T, connect sum, classification of surfaces. Ambient
isotopy, prime, irreducibity, JordanSchoenflies theorem. Begin
proof of Alexander's theorem. 
One 
One 
Conway's ZIP
proof, by Francis and Weeks, discusses the classification
theorem for surfaces. A.A. Markov's 1958 paper "Insolubility of
the problem of homeomorphy" proves that the homeomorphism problem
for manifolds is undecidable. See also Chapter 9 of Stillwell's
book "Classical topology and combinatorial group theory". 
2 
Jan. 18 
Finish proof of Alexander's theorem. Incompressible surfaces,
handlebodies. Bundles, regular neighborhoods, classification of
Ibundles. 
Two 
Two 

3 
Jan. 25 
Triangulations, normal surfaces, HakenKneser finiteness.
Fundamental group, Seifertvan Kampen theorem. 
Three 
Three 
Free product with amalgamation (sometimes called the
amalgamated product) is a consquence of the Seifertvan Kampen
theorem. A closely
related topic is HNN extensions. 
4 
Feb. 1 
Computing fundamental groups, rank, begin existence and
uniqueness of sphere decompositions (prime factorization),
surgery. 
Four 
Four 

5 
Feb. 8 
The baseball move, finish sphere decomposition. Normalization
of incompressible surfaces in irreducible manifolds. Boundary
parallel, atoroidal, torus bundles. Existence of torus
decomposition (JSJ). Lens spaces. 
Five 
Five 
Lens spaces were introduced by Tietze in 1908. The JSJ
decomposition is due to Jaco, Shalen and independently Johannson,
around 1979. 
6 
Feb. 15 
Torus knots and essential annuli. Nonuniqueness of torus
decompositions. Fibered solid tori, Seifert fibered spaces, base
orbifolds. 
Six 
Six 
SO(2) is a circle. SO(3) = Isom^{+}(S^{2}) is
real projective space. (The group of unit quaternions is the
threesphere.) PSL(2,R) = Isom^{+}(H^{2}) is an
open solid torus, as is Isom^{+}(R^{2}).
PSL(2,R)/PSL(2,Z) is the trefoil knot exterior. 
7 
Feb. 22 
Essential surfaces. Vertical and horizontal surfaces in
SFSs. 
Seven 
Seven 
Exercise: classify essential surfaces in Ibundles. 
8 
Mar. 1 
Orbifolds, Euler characteristic, and covering maps. Cutting
along horizontal surfaces yields Ibundles. Structure of torus
knots. SFS's are irreducible or have S^{2} × R
geometry.
 Eight 
Eight 

9 
Mar. 8 
Finish discussion of uniqueness of torus decomposition.
Poincare conjecture, Poincare homology sphere. Characterisations
of the unknot. Dehn's lemma, the loop, disk, and sphere theorems.
Hierarchies. 
Nine 
Nine 
Dante,
Fra Angelico, the threesphere, and the Hopf fibration,
according to Ralph Abraham. 
10 
Mar. 15 
Compression bodies. Short hierarchies. Boundary patterns.
Hierarchy for the figure eight knot. Proof sketch of the disk
theorem. 
Ten
Eleven 
Ten 
Assorted other topics: Heegaard splitings, surface
bundles. Thurston's geometrization program. 