MA4J2, Three-Manifolds
Winter 2011


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.