Janko Latschev
Lecture Course Geometric Topology, Winter Semester 2023/24
Lectures take place on Tuesdays, 2-4pm in room 205 in Sedanstrasse 19, and Thursdays 12-2pm in lecture hall H4.
Exercise classes take place Wednesdays 4-6pm in room 428 of the Geomatikum
Here is a summary of the topics we covered, which might be helpful in preparation of the exam. It also contains a few problems. You may choose to start your exam by presenting a solution to one of these.
Prerequisites: - necessary: topology (including fundamental group and covering spaces), basics about manifolds (tangent bundle, flows of vector fields, differential forms)
- recommended: some differential geometry (Riemannian geometry, exponential map)
- recommended: some algebraic topology (homology and cohomology)
The aim of this course is to discuss some topics that have no room in the standard differential geometry or algebraic topology classes, but are useful knowledge for anyone interested in either topic. The term geometric topology is used as a general reference to any kind of geometric arguments in the study of the topology of manifolds. In low-dimensional cases, these often involve 'picture proofs'. Our goal is to introduce some of the common techniques, along with the background that explains their validity.
We will start with some basic results in differential topology, including transversality and its uses, tubular neighborhoods and the like. Further topics will depend on audience interest and background knowledge, and may include Morse theory, Heegaard decompositions of 3-manifolds, handle attachments and surgery constructions, classification questions for vector bundles, the construction of characteristic classes via Chern-Weil theory, or an introduction to h-principles.
The selection and presentation of the material will not follow any specific source, and hopefully, there will be many pictures. Therefore, I strongly encourage you to take handwritten notes in class.
The exercise sheets are published here:
Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6
Problem Set 7
Problem Set 8
Problem Set 9
Problem Set 10
Problem Set 11
Some useful references: (this list will be updated as needed)
| J. Milnor | Topology from the differentiable viewpoint | University Press of Virginia |
| M. Hirsch | Differential Topology | Springer Verlag |
| A. Kosinski | Differential Manifolds | Academic Press |
| J. Robbin, D. Salamon | Introduction to Differential Topology | Book project |
| V.V. Prasolov, A.B. Sossinsky | Knots, Links, Braids and 3-manifolds | AMS |
| M. Audin, M. Damian | Morse Theory and Floer homology | Springer |
| W.P. Thurston | Three-dimensional Geometry and Topology | Springer |
| D. Rolfsen | Knots and Links | AMS |
| R. Gompf, A. Stipsicz | 4-manifolds and Kirby calculus | AMS |
Log of lecture content:
| Oct 17 |
introductory remarks; reminder on definition of submanifolds of RN, definition of topological manifolds, smooth atlases and smooth structures, submersions, immersions and embeddings, Whitney embedding theorem (simplest version)
|
| Oct 19 | sets of measure zero, behavior under smooth maps, easy Whitney embedding theorem, regular and critical points and values, preimages of regular values are submanifolds
|
| Oct 24 | statement of Sard's theorem and remarks, manifolds with boundary, examples: sublevel sets of regular values, preimages of regular values in manifolds with boundary, application: Brouwer fixed point theorem
|
| Oct 26 | transversality of a map to a submanifold, consequences and examples, transversality for families, statement of perturbation theorem
|
| Nov 02 | proof of the perturbation theorem, remarks on applications; local mapping degree mod 2
|
| Nov 07 | mapping degree mod 2: homotopy invariance, first examples and consequences, orientations and integer mapping degree, homotopy invariance, examples and applications, in particular maps to spheres
|
| Nov 09 | linking number for a pair of submanifolds in Rn; vector bundles: definitions, first examples, morphisms and trivializations, non-uniqueness of trivializations
|
| Nov 14 | S3 has trivial tangent bundle, more examples and remarks; subbundles and quotient bundles, pullback bundles, definition of the normal bundle of a submanifold, statement of tubular neighborhood theorem
|
| Nov 16 | proof of the tubular neighborhood theorem, remarks and examples, coorientations for submanifolds, intersection numbers of submanifolds of complementary dimension
|
| Nov 21 | remarks on intersection numbers, elementary consequences, Euler characteristic of a closed manifold Q as self-intersection of zero section in TQ, 2-dimensional examples
|
| Nov 23 | computation of the Euler characteristic from vector fields with isolated zeros (Poincaré-Hopf), Euler characteristic determines existence of non-vanishing sections of TQ, maps into spheres are classified by degree (Hopf)
|
| Nov 28 | Euler characteristic for compact manifolds with boundary, examples; strong Whitney embedding theorem, start of proof up to matching double points with opposite signs
|
| Nov 30 | continuation of the proof: the Whitney trick
|
| Dec 05 | conclusion of the proof; discussion of other uses of the Whitney trick: h-cobordism theorem and the generalized Poincaré conjecture; Morse theory: basic definitions, Morse functions are dense, topology of sublevel sets changes only at critical values
|
| Dec 07 | Morse Lemma, example: height function on T2; interlude: gluing manifolds along submanifolds, connected sum, statement of well-definedness up to diffeomorphism
|
| Dec 12 | proof of the isotopy lemma for disk embeddings, prime decomposition of 2- and 3-manifolds
|
| Dec 14 | gluing two manifolds along a common submanifold, discussion on S3 as a union of two solid tori, relation to the Hopf fibration
|
| Dec 19 | gluing along submanifolds continued, gluing along submanifolds of the boundary, examples: double, boundary connected sum
|
| Dec 21 | handle attachment, handle attachment associated to a critical point in Morse theory, stable and unstable manifolds, first application of Morse theory ideas: closed oriented 4-manifolds with arbitrary finitely presented fundamental group
|
| Jan 09 | Morse-Smale condition on gradient flow
|
| Jan 11 | Rearrangement Lemma, self-indexing Morse functions, compactification of space of flow lines by broken flow lines, statement of consequences for index difference 1 and 2
|
| Jan 16 | proof of compactification of spaces of flow lines for index difference 1 and 2, construction of Morse complex over Z2, remarks on independence of Morse homology from the pair (f,g), comparison to other theories, remarks on definition of Z
|
| Jan 18 | closed 3-manifolds: list of example classes, Heegaard splittings and Heegaard diagrams
|
| Jan 23 | remarks on Heegaard diagrams, equivalent diagrams give homeomorphic 3-manifolds, Dehn twists, Dehn-Lickorish theorem, start of the proof
|
| Jan 25 | completion of the proof of the Dehn-Lickorish theorem, statement of main surgery theorem for 3-manifolds
|
| Jan 30 | description of data necessary for Dehn surgery, reformulation for surgery on S3 (rational surgery coefficients), examples, relation to handle attachment, more precise restatement of surgery theorem, start of proof
|
| Feb 01 | proof of the surgery theorem, example: +1 surgery on the right-handed trefoil
|
|
