\( \newcommand{\cross}{\times} \newcommand{\connect}{\mathbin{\#}} \newcommand{\isom}{\cong} \newcommand{\RP}{\mathbb{RP}} \newcommand{\CP}{\mathbb{CP}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\Nil}{\operatorname{Nil}} \newcommand{\Sol}{\operatorname{Sol}} \newcommand{\PSL}{\operatorname{PSL}} \)

TCC Introduction to three-manifolds
Term II 2020-2021

Schedule

Week Date of Wednesday Topics Example sheets Lecture notes Comments
2 Jan. 20 Overview and administration details. Manifolds, charts, overlap maps, atlas. \((G, X)\)-structures. Orientations, manifolds with boundary, boundary of manifolds. Closed manifolds. Foliations, oriented foliations. Classification of zero- and one-manifolds. Torus and projective plane. Questions. One One See pages 233-239 of Hatcher's book Algebraic topology for a more detailed discussion of orientability. See the beginning of chapter three of Thurston's notes The geometry and topology of three-manifolds for a discussion of \((G, X)\)-structures.
3 Jan. 27 Connect sums and sphere surgery. Classification of surfaces. Overview of the eight Thurston geometries. \(S^3\) geometry, lens spaces. Surface bundles, Nielsen-Thurston classification. Questions. Two Two Sphere surgery is the "inverse" operation to connect sum. The Nielsen-Thurston classification of surface homeomorphisms states that every homeomorphism is, up to isotopy, either periodic, reducible, or (pseudo-)Anosov. Scott's BLMS article The geometries of three-manifolds is a valuable guide to that topic.
4 Feb. 3 \(S^2 \cross \RR\) geometry. Torus bundles and \(\EE^3\), \(\Nil\), \(\Sol\) geometries. Hyperbolic surfaces and \(\HH^2 \cross \RR\) geometry. Seifert fibered spaces. \(\PSL\) geometry. \(\HH^3\) geometry. Questions. Transversality. Statement of Alexander's theorem. Three Three For an introduction to Seifert fibered spaces from the point of view of tessellations see the book Classical tessellations and three-manifolds by Montesinos. See Thurston's notes, or Purcell's book Hyperbolic knot theory, for an introduction to hyperbolic knot complements. For a discussion of Thurston geometries in dimension four, see Hillman's book Four-manifolds, geometries, and knots.
5 Feb. 10 Schoenflies conjecture. Alexander horned sphere. Collars. Isotopy and ambient isotopy. Proof of Alexander's theorem, surgery/compression of spheres. Irreducible and prime. Independent sphere systems. Questions. Four Four The prime decomposition theorem does not hold in dimension four - see the discussion on page one of the paper Stable prime decompositions of four-manifolds by Kreck, Lück, and Teichner. For information on transversality see the first section of Casson's notes or Chapter Three of Hirsch's book Differential topology. Finally, in reply to the question "do people study analytic manifolds?", all I can say is this is a very large area where I know almost nothing.
6 Feb. 17 Model simplices, triangulations, realisations, skeleta, characteristic maps. Skeletal isotopies. Normal surfaces. \(I\)-bundles, monodromies, orientation \(I\)-bundle. Minimal triangulations, Matveev complexity. Questions. Haken-Kneser finiteness. Five Five See the first four pages of Manolescu's lecture notes on the triangulation conjecture for an recent overview, with references, of what is proved, disproved, and still open regarding triangulations and PL structures on manifolds.
7 Feb. 24 Existence of connect sum decompositions, topological complexity via intersection with skeleta. Uniqueness of connect sum decompositions, invariance under surgery. Begin definition of normal maps and normal families. Six Six Again, connected-sum decompositions are not unique in dimension four. For an explicit example, we have that \(\CP^2 \connect -2\CP^2\) is homeomorphic to \((S^2 \cross S^2) \connect -\CP^2\). For a discussion see the book Four-manifolds and Kirby calculus by Gompf and Stipsicz. Also, the sphere theorem does not hold in dimension four: for example \(S^2 \cross S^2\) is irreducible (again, see Gompf and Stipsicz) but has \(\pi_3 \isom \ZZ^2\).
8 Mar. 3 Matveev complexity. Statement of the sphere theorem. Whitney umbrellas. Least weight maps and double point elimination. Normal maps, normal classes. Hyperbolic faces, PL area. Seven Seven We are following (fairly closely) Casson's notes; the original reference for PL minimal surfaces is PL minimal surfaces in three-manifolds by Jaco and Rubinstein.
9 Mar. 10 PL area. Existence of PL minimal surfaces. Balanced angles at vertices. Diagonals in quads. Transversality for minimal surfaces (corollary). Begin proof of disjointness of embedded minimal spheres. Eight Eight

To avoid "unnecessary" intersections occuring in the interior of tetrahedra Casson chooses diagonals, in a consistent fashion, for all quad normal disks. However, this makes the given statement of the corollary of Lemma 5.4 slightly incorrect. So we alter the statement to take the open neighbourhoods \(U\) and \(V\) to lie in \(\Gamma_f\) and \(\Gamma_g\), respectively.

Instead of choosing diagonals Yi Ni, in his short note Uniqueness of PL minimal surfaces, uses minimal surfaces in hyperbolic tetrahedra. The existence and uniqueness of minimal surfaces there are easier than those in general Riemannian three-manifolds.

10 Mar. 17 Restate corollary of Lemma 5.4. Finish proof of Lemma 5.5, disjointness of embedded minimal spheres. Statement of sphere theorem. Half lives, half dies. \(\pi_2\) for simply connected three-manifolds. Dehn's attempts to prove the disk theorem. The tower construction. Nine Nine

See Hatcher's book Vector bundles and K-theory, and in particular Theorem 1.16, for the beginnings of the connection between vector bundles and classifying spaces. (The classic reference is of course Milnor and Stasheff's book Characteristic classes - see the remark on page 70.)

There are details to be given in Casson's discussion of the Meeks-Yau trick. I will try to turn these into exercises.

11 Mar. 24 Essential and non-peripheral surfaces. Boundary irreducible manifolds, atoroidal manifolds, acylindrical manifolds. Geometrisation of knot complements, after Thurston. Torus knots, satellite knots. JSJ decompositions. Geometrisation after Perelman. Ten Ten

In the course we discussed the sphere, disk, torus, and annulus theorems. There is no corresponding theory for higher genus surfaces. (For example, there are examples of three-manifolds that contain no embedded essential surfaces, but that do admit surface subgroups (in sufficiently high genus).) In particular, the canonical connect-sum and JSJ decompositions are phenomena revolving about embedded surfaces with non-negative Euler characteristic. It is possible to use higher genus surfaces to decompose manifolds (for example Heegaard splitings or fibers of bundle structures) but such decompositions are usually not canonical.