MT 831: Algebraic topology and low-dimensional topology
Time/Location: MW 10-11:15, Fulton 210
Instructor: Eli Grigsby
Office hours: By appointment (or poke your head in), Carney 357
Course description: This will be something of a second-semester algebraic topology course, but focused on those topics that I have personally found indispensable. Since I am a low-dimensional topologist, I will try my best to show how the topics relate to questions about low-dimensional (3- and 4-) manifolds by inserting examples/applications that have been relevant to my own research.
Here is what I hope to cover.
What we actually did:
- Glen Bredon, Topology and Geometry
- Robert Gompf and András Stipsicz, 4-manifolds and Kirby calculus
- Allen Hatcher, Algebraic topology
- W, 9/3: Motivation: 3D Poincare conjecture, Floer homology; Definition of n-th based homotopy groups of a topological space; Group structure (abelian, when n > 1), map on homotopy groups induced by a (homotopy class of) path
- F, 9/5: Maps on homotopy groups induced by maps between topological spaces; higher homotopy groups as modules over group ring of fundamental group; isomorphism on higher homotopy groups induced by covering space projection; statement of Whitehead's theorem
- M, 9/8: Toward the proof of Whitehead's theorem: Definition of relative homotopy groups and the LES on a pair; compression lemma; homotopy extension property for cofibrations
- W, 9/10: Finished proof of compression lemma, Whitehead theorem, and cellular approximation theorem (modulo this week's HW problems and the technical Lemma 4.10 in Hatcher)
- M, 9/15: Pontryagin-Thom construction (one-to-one correspondence between framed bordism classes of imbedded k-manifolds in R^n+k and based homotopy classes of maps S^n+k to S^n) (Another great reference: Milnor's Topology from the differentiable viewpoint)
- W, 9/17: Finished sketch of proof that Pontryagin-Thom construction gives a 1:1 correspondence, used construction to compute pi_n(S^n), and described nice model for degree k map S^n to S^n as an (n-1)-fold suspension of the standard degree k map S^1 to S^1
- M, 9/22: Homotopy classes of based maps of S^n to wedges of S^n; description of Hurewicz map homomorphism relating homotopy groups and homology groups of the same dimension; statement of Hurewicz theorem and most of the proof when the space is a CW complex
- W, 9/24: Finished proof of Hurewicz theorem (for CW cpxs) by proving that an (n-1)-connected CW complex is homotopy-equivalent to a CW complex with (n-1)-skeleton a point; Defined fibration, Serre fibration, via the homotopy lifting property, gave a simple non-example, showed that a fibre bundle is a Serre fibration
- M, 9/29: Proved that the homotopy groups of the fiber, total space, and base space of a Serre fibration fit into a LES, discussed covering spaces, some other examples.
- W, 10/1: Detailed example: Hopf fibration; Introduction to mapping class groups (Reference: Farb and Margalit's A Primer on mapping class groups
- M, 10/6: Applications of LES of fibration: Birman exact sequence, boundary capping sequence
- W, 10/8: Open book decompositions of 3-manifolds, a relationship between a (nonstandard) notion of reducibility for mapping classes and for the corresponding open book: following this paper of Ghiggini-Lisca (see Prop. 2.2).
- M, 10/13: No class: Columbus Day
- W, 10/15: Towards studying imbedded surfaces representing homology classes in 3- and 4-manifolds. Definition Eilenberg-Maclane space. Intro to Obstruction theory: Defined obstruction cocycle associated to extending a map from n-skeleton to (n+1)-skeleton, difference co-chain associated to a pair of maps from n-skeleton agreeing on (n-1)-skeleton.
- M, 10/20: Finished proof that a map from the n-skeleton of a CW complex can be changed in the complement of the (n-1)-skeleton iff the cohomology class represented by the obstruction cocycle is zero. Proved that the first cohomology of any CW complex is isomorphic to the group of homotopy classes of maps to the circle, indicating how these arguments extend to give a proof of Hopf theorem on maps to spheres and cohomology classes as homotopy classes of maps to general E-M spaces (I heart M. Hutching's notes on this. Also Section 1.2 of Chapter 3 of V.V. Prasolov's Elements of homology theory.)
- W, 10/22: Defined knot genus (knots in the three-sphere), described Seifert's algorithm for producing a smoothly-imbedded rep. of the (dual of) a primitive first cohomology class of S^3 - N(K) (aka a Seifert surface!). Generalization: Thurston norm.
- M, 10/27: First properties of the Thurston norm: It can be extended to a pseudo-norm on 1st cohomology (with real coefficients) of a compact, connected, oriented, irreducible 3-manifold with boundary a disjoint union of tori. If manifold is also atoroidal, then it is a norm. (Reference: Calegari's Foliations and the Geometry of 3-manifolds).
- W, 10/29: Introduction to handlebodies and surgery (Gompf-Stipsicz, Chp. 4)
- M, 11/3: More elementary handlebody theory: basics of Morse theory; handles can be added in order of increasing index, handleslides, handle cancellation (geometric analogues of change of basis, Gaussian elimination)
- W, 11/5: Dualizing a handlebody decomposition; Handlebody decompositions of 3-manifolds: Heegaard decompositions, Dehn surgery
- M, 11/10: 4-dimensional handlebodies: Kirby diagrams and integral Dehn surgery
- W, 11/12: More on 4-D handlebodies: dotted circle notation; some fundamental examples: disk bundles over surfaces
- M, 11/17: General intersection theory for smoothly-imbedded submanifolds of a smooth manifold (Thom class, Thom isomorphism theorem) + Defn. of Intersection form for 4-manifolds
- W, 11/19: If homology classes are represented by smoothly-imbedded submanifolds, their geometric intersection pairing (homology class of transverse intersection) agrees with their algebraic intersection pairing (dualize, cup, cap with fundamental class); For a 4D hbody without 1- and 3-handles, intersection form agrees with linking form; Example: boundary of E8 plumbing + how to construct a 3D handlebody decomposition of a link complement
- M, 11/24: Background on symmetric bilinear integral forms, 2-handle slides and change of basis for intersection form of 4-manifold, statement of Milnor-Whitehead theorem (Intersection form determines homotopy-equivalence class of simply-connected, closed, topological 4-manifolds) and set-up of proof.
- M, 12/1: Finish proof of Milnor-Whitehead theorem, The word problem for groups an obstruction to classifying 4-manifolds, statement of Smale's h-cobordism theorem (Another reference: Scorpan's The wild world of 4-manifolds)
- W, 12/3: Relationship between Smale's h-cobordism theorem and topological version of Poincare conjecture in higher dimensions, Andrews-Curtis conjecture; Back to 3D: Alexander module of a classical knot
- M, 12/8: Seifert form + how to obtain a presentation for the Alexander module (Reference: Rolfsen's Knots and links)
- W, 12/10: Defns of knot condordance group and knot signature (plus why signature is a homomorphism from concordance group to the even integers)