\( \newcommand{\EE}{\mathbb{E}} \newcommand{\RR}{\mathbb{R}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \)

MA4J2, Three-Manifolds
Winter 2012


Week Date of Tuesday Topics Example sheet Lecture notes Comments
1 Jan. 10 Manifolds, S, P, T, classification of surfaces. Euler characteristic. Homogeneous and locally homogeneous metrics. 2-dimensional geometries. One Lecture 1 Lecture 2 Lecture 3
Conway's ZIP proof [Francis and Weeks] discusses the classification theorem for surfaces. Markov's 1958 paper "Insolubility of the problem of homeomorphy" proves that the homeomorphism problem for manifolds is undecidable. See also Chapter 9 of the book "Classical topology and combinatorial group theory" [Stillwell].
2 Jan. 17 Length elements, geodesics, isometries. Group actions, reflector points, the four quotients of \(\EE^1\). Tilings, fundamental domains, classification of isometries of \(\EE^2\), deck groups. Two Lecture 4 Lecture 5 Lecture 6
3 Jan. 24 Orbifolds, discrete groups, and tilings. Classification in dimension one. Classification of frieze groups. Wallpaper groups, orbifolds in dimension two. Three Lecture 7 Lecture 8 Lecture 9 In a 2-orbifold we may find: regular points, mirror boundary, cone points of angle \(2\pi/n\), corner reflectors of angle \(\pi/n\), regular boundary, and half-mirrored corners. The last necessarily have angle \(\pi/2\).

A video [Hart] that mentions frieze patterns and fruit by the foot.

4 Jan. 31 Interlude on spherical and hyperbolic geometry, the projection argument. Local groups, orbifold Euler characteristic, orbifold coverings. Good versus bad. Four Lecture 10 Lecture 11 Lecture 12 Dror Bar-Natan has many lovely pictures of wallpaper groups. Doug Ravenel's meta-page is also useful.
5 Feb. 7 Geometric orbifolds, universal covers, orbifold fundamental group. Seifert-van Kampen theorem, gluing along 1-orbifolds, computation of \(\pi_1^\orb\). Five Lecture 13 Lecture 14 Lecture 15 Question from class: Suppose that \(X\) is a complete, connected, simply-connected, homogeneous metric space. Must \(X\) be a manifold? Answer: Not necessarily. One example is an \(\RR\)-tree [Young] arising as an asymptotic cone of the hyperbolic plane. I don't know a simpler construction.
6 Feb. 14 Review of 2-orbifolds and orbifolds as quotients of surfaces. Fibered solid tori and Klein bottles, orientations and invariants, Seifert fibered spaces, isomorphism, regular and critical fibers, trivial bundles, a non-trivial example. Six Lecture 16 Lecture 17 Note that class and office hours on Tuesday are cancelled.
7 Feb. 21 Review of orbifold fundamental group. Quotient orbifolds of Seifert fibered spaces. Covers of SFS's and their quotient orbifolds. Circle actions and orientability of the fibers. Seven Lecture 18 Lecture 19 Lecture 20 Calegari makes a connection between his son's train track set and indiscrete subgroups of \(\Isom(\EE^2)\). In his set all the segments of track are exactly one-eighth of a circle. Suppose we had a set with segments that were, instead, one-quarter of a circle. Show that the group \(G\) for these pieces is discrete. There are four more that give discrete groups; find them and their groups.
8 Feb. 28 Fiber bundles, isomorphism. Sterographic projection, Hopf fibration, lens spaces. Embeddings and isotopies and their proper variants. Sections of bundles. Eight Lecture 21 Lecture 22 Lecture 23 Watch Dimensions for a visual introduction to sterographic projection and the Hopf fibration.
9 Mar. 6 Disks with handles, \(S^1\)-bundles over surfaces with boundary, their fundamental groups. Nine Lecture 24 Lecture 25 Friday's class will include a showing of the videos The shape of space and Not knot.
10 Mar. 13 Slopes in the torus. Meridional, longitudinal, fiber slopes for a fibered solid torus. Decomposing Seifert fibered spaces, sectional slopes, Seifert invariants. Euler number and covers. Ten Eleven Lecture 26 Lecture 27 Lecture 28 Scott refers to Neumann and Raymond's lecture notes for many details concerning the Euler number. The notes by Jankins and Neumann are also useful.

Jeff Weeks has written several programs for visualising tilings, groups, and manifolds. We played with his program Curved spaces in the last class.