| 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, Jordan-Schoenflies 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
I-bundles. |
Two |
Two |
|
| 3 |
Jan. 25 |
Triangulations, normal surfaces, Haken-Kneser finiteness.
Fundamental group, Seifert-van Kampen theorem. |
Three |
Three |
Free product with amalgamation (sometimes called the
amalgamated product) is a consquence of the Seifert--van 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. Non-uniqueness of torus
decompositions. Fibered solid tori, Seifert fibered spaces, base
orbifolds. |
Six |
Six |
SO(2) is a circle. SO(3) = Isom+(S2) is
real projective space. (The group of unit quaternions is the
three-sphere.) PSL(2,R) = Isom+(H2) is an
open solid torus, as is Isom+(R2).
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 I-bundles. |
| 8 |
Mar. 1 |
Orbifolds, Euler characteristic, and covering maps. Cutting
along horizontal surfaces yields I-bundles. Structure of torus
knots. SFS's are irreducible or have S2 × 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 three-sphere, 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. |
