Tuesday: Plan for module. Manifolds. Constructions: subspaces, disjoint union, cartesian product, quotients. Examples: sphere, torus, projective space.

Thursday: Manifolds with boundary. The boundary of the boundary is empty. Bundles. Pairs, maps of pairs. Locally flat (versus tame) embedded submanifolds. Constructions: wild knots, tubular neighbourhoods. Examples: ball/disk, Mobius strip, real projective plane, oriented surfaces (closed, bounded, punctured), trefoil.

Friday: Existence and uniqueness of regular neighbourhoods. Knots. Ambient isotopy. Meridian and longitude of knot exteriors. Fundamental group is a knot invariant (up to isotopy and reflection). Gordon-Luecke theorem - knot exterior is a knot invariant. Dehn: left trefoil is not isotopic to right trefoil. Links are trickier. Constructions: Knot exteriors. Examples: unknot, right and left trefoil, figure-eight knot, cinquefoil, third twist knot, fourth twist knot, granny and reef (square) knots.

Tuesday: Removing any one hypothesis from the definition of "manifold" permits strange spaces. For many, many strange examples see the book Counterexamples in topology.

Thursday: The book I brought in was Hyperbolic knot theory by Purcell.

Friday: The three-dimensional print I brought, and another variant, can be found at Shapeways and at SketchFab.

No lecture on Tuesday.

Thursday: Classification of surfaces. The power of the fundamental group in low-dimensions. (Tame) spheres are standard in dimensions one, two, and three. Manifolds (up to dimension three) are triangulable. Constructions: Connect sums of manifolds, of manifold pairs, of knots. Examples: closed surfaces, planar surfaces. Connect sum of projective spaces. Alexander horned sphere.

Friday: Connect sums of non-orientable manifolds. Chiral and achiral manifolds. Non-orientable surfaces, surfaces with boundary. The Jordan-Schoenflies and Alexander theorems (the two- and three-sphere are prime). Unique factorisations of knots and three-manifolds. The Gordon-Luecke theorem. Geometrisation for knot complements. Meridian and longitude in (framing for) the boundary of a knot complement. Construction: Connect sums of knots. Torus knots.

Tuesday: Stereographic projection. The Hopf link and the Clifford torus. Circle actions on the three-sphere with torus knot orbits and cosets of one-parameter subgroups. Companion, pattern, and satellite knots. The cabling conjecture. Constructions: Torus knots. Satellite knots. Cable knots.

Thursday: Double cosets in isometry groups. Actions and their properties. Quotients of manifolds are (often) manifolds. Constructions: Unit tangent bundles of surfaces. Quotient spaces, elliptic manifolds. Examples: Lens spaces.

Friday: The three-sphere is a Lie group. Manifolds viewed from above (as geometric quotients) and from below (as triangulations). Voronoi domains and fundamental domains. Construction: Lie groups modulo lattices. Examples: Poincaré homology sphere, the quarter-turn manifold. Lens spaces.

Thursday: The article I mentioned was The geometries of three-manifolds by Scott.

Friday: The one-half 120-cell print is available at Shapeways and at SketchFab. The books I brought in were A topologist's picturebook by Francis and The shape of space by Weeks.

Tuesday: Handles, handle attachments, and handle decompositions. Construction: Dehn fillings, Dehn surgery.

No lectures Thursday or Friday due to strikes.

No lectures Tuesday or Thursday due to strikes.

Friday: One-half lives, one-half dies. The homological longitude. Spanning surfaces for knots and knot genus. Half-twisted bands. The homological longitude is isotopic to the push-off given by any spanning surface. The meridian. Surgery disks, compressing disks, two-sided surfaces. Algebraic versus geomtric incompressibility.

Tuesday: Framings and slopes. Dehn filling. Homology spheres, and related manifolds. Seifert surfaces and Seifert's algorithm. Spanning surfaces for torus knots. The annular slope of a torus knot. Moser's theorem.

Thursday: Algebraic intersection number. Relation to the determinant when in the torus. Dehn's theorem. The annular slope is reducible. Farey neighbour slopes. Property P and the Poincaré conjecture. Gordon-Luecke theorem, cyclic surgery theorem, Kronheimer-Mrowka. Property R. Cabling conjecture, Berge conjecture.

Friday: Triangulations, model tetrahedra, model cells, face pairings, realisations. The homeomorphism problem and Kuperberg's theorem. NP and FNP for recognition problems, Lackenby-Schleimer theorem, Jackson's theorem. Knot genus problem. Polytopes in dimension four - 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell. Conjecture on minimal triangulations of lens spaces.

Tuesday: See Chapter Five of Rolfsen's book Knots and links for a discussion of Seifert surfaces. The 3D prints of Seifert surfaces (for torus knots) can also be found at SketchFab and at Shapeways.

Thursday: Dehn, in his 1910 paper Uber die Topologie des dreidimensionalen Raumes (see also Stillwell's translation) shows that the $1/1$ filling of the right trefoil is the Poincaré homology sphere. (Amusingly, Dehn claims that the fundamental group of the PHS is the icosahedral group extended by a reflection. This is incorrect; the special order two element of the binary icosahedral group is central so cannot be a reflection.) Dehn's result is greatly extended in Moser's 1971 paper Elementary surgery along a torus knot. In lecture we reproduced Cameron Cordon's exposition of (a part of) Moser's work. Further results mentioned in the lecture include: Gordon and Luecke, Culler-Gordon-Luecke-Shalen (the cyclic surgery theorem), Kronheimer-Mrowka, Perelman, and Gabai. See also the expository paper Eight faces of the Poincaré homology three-sphere by Kirby and Scharlemann. (Curiously, they omit Poincaré's original presentation via Heegaard diagram.)

Friday:

Tuesday: Doubles. Edges and model edges. Ideal triangulations, vertex links and ends of non-compact manifolds. Angle structures.

Thursday: Giant diagram of relations among properties of triangulations. (Transverse) taut ideal triangulations, relation to Thurston norm minimizing surfaces. Veering triangulations and mapping tori of pseudo-Anosov mapping classes (Agol's theorem). Veering triangulations and pseudo-Anosov flows.

Friday: Models of hyperbolic geometry: upper half space and ball. Isometries. Conformality. Types of isometries (assumed). Geodesics: classification, asymptotic. Topology at infinity. PSL(2, C) action on boundary at infinity is simply three-transitive.

Thursday: Here is a photo of the giant diagram. The census of veering tringulations contains all (transverse) examples with at most 16 tetrahedra. The census includes (geometric and combinatorial) pictures of the induced triangulations of the boundary tori.

Friday: Scott's BLMS article The geometries of three-manifolds gives an excellent overview.

Tuesday: Ideal hyperbolic tetrahedra. The 24 cross-ratios and Mobius transformations. Edge parameters and labelling of the cusp triangles. Complex angles and their relations.

Thursday: (G, X)-structures, the developing map, and holonomy in the special case of Sim(C). Dil(C), Euc(C), and Trans(C). Punctured (G, X)-structures. The holonomy about a vertex and "unpuncturing". Example: Similarity surfaces coming from quadrilaterals in the plane.

Friday: Example: Presentations of the figure-eight knot complement, its vertex link. Thurston's hyperbolic structure on the figure-eight knot complment.

Tuesday: Chimneys and cusps of hyperbolic manifolds. Completeness criterion. The incompleteness locus is (sometimes) a geodesic. "Completion is a Dehn filling" criterion. Geometric triangulations.

No lectures Thursday or Friday due to strikes.

Tuesday: See Figure 4 of Thurston's article Three-dimensional manifolds, Kleinian groups, and hyperbolic geometry for an image of the tiling of the plane corresponding to the point $(12, 13/2)$ in the Dehn surgery space of the figure-eight knot complement.

Friday: Shape variety, tetrahedron and edge equations, dimension of the shape variety. Map of shapes to holonomies. Representation variety. Mostow-Prasad rigidity and the discrete and faithful representation. Dehn surgery space. Hyperbolic Dehn surgery theorem and its proof. Examples: Shape variety of figure-eight knot complement.

Friday: In the first break I showed the program snappy (and in particular the "inside view" available from the "browse" command). In the second break I showed pictures of Dehn surgery space made (using snappy) by Tracy Hall, Henry Segerman, and myself.