Janko Latschev
Lecture Course Differential Topology, Summer Semester 2019
Lectures take place on Thursday 8-10 in H1 (with one exception). There will be no exercise classes, but I encourage students to work on exercise sheets which are posted semi-regularly on this page.
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)
- also helpful: some algebraic topology (homology and cohomology)
New:
I have sent out the exam schedule to everyone who expressed interest. If you did not get an exam time but would like to take the exam, please contact me as soon as possible.
To help you in preparing for the exam, I have prepared a list of topics, from which you will be asked to choose one to start the exam with.
In differential topology we study the topology of differentiable manifolds and smooth maps between them. In this course we will start with a discussion of basics such as transversality and degree theory and their applications. We will also discuss vector bundles and tubular neighborhoods. Further topics will depend on audience interest and background knowledge, and may include classification questions for vector bundles, the construction of characteristic classes via Chern-Weil theory, Morse theory and surgery constructions, and the Thom-Pontryagin construction.
The exercise sheets are published here:
Problem Set 1
Problem Set 2
Problem Set 3
Some useful references:
| J. Milnor | Topology from the differentiable viewpoint | University Press of Virginia |
| R. Bott, L. Tu | Differential Forms in Algebraic Topology | Springer Verlag |
| M. Hirsch | Differential Topology | Springer Verlag |
| A. Kosinski | Differential Manifolds | Academic Press |
| I. Madsen, J. Tornehave | From calculus to cohomology | Cambridge University Press |
| J. Robbin, D. Salamon | Introduction to Differential Topology | Book project |
Log of the lectures:
| Apr 04 | introductory remarks; recollections on submanifolds of Rn, definition of topological manifolds, smooth atlases and smooth structures, smooth maps between smooth manifolds, Whitney embedding theorem (first version)
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| Apr 11 | the tangent bundle as a manifold, submersions, immersions and embeddings, regular and critical points and values, preimages of regular values are submanifolds, sets of measure zero, every compact manifold embeds into some euclidean space, statement of Sard's theorem
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| Apr 18 | remarks on Sard's theorem, including remarks on Whitney's example, manifolds with boundary, examples: sublevel sets of regular values, preimages in manifolds with boundary of regular values, smooth Brouwer fixed point theorem
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| Apr 25 | Brouwer's fixed point theorem for continuous maps, transversality of a map to a submanifold, consequences, transversality for families, examples, mapping degree mod 2
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| May 02 | 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
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| May 09 | statement of Hopf's theorem: degree classifies maps of closed connected manifolds into spheres of same dimension up to homotopy, linking number for a pair of knots; vector bundles: definitions, first examples, morphisms and trivializations, constructions with vector bundles: direct sum, subbundles
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| May 16 | quotient bundles, pullback bundles, definition of the normal bundle of a submanifold, tubular neighborhood theorem, remarks on generalizations
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| May 23 | remarks and examples for tubular neighborhood theorem, orientations for vector bundles, intersection numbers of submanifolds of complementary dimension, elementary consequences
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| Jun 06 | Euler characteristic of a closed manifold, computation from vector fields with isolated zeros, examples, extension to compact manifolds with boundary
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| Jun 20 | connected sum of manifolds, examples, prime decomposition of 2- and 3-manifolds, boundary connected sum, gluing two manifolds along a common submanifold
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| Jun 27 | further discussion of gluing along a common submanifold, surgery, examples in dimension 3; Morse theory: Hessian of a function at a critical point, index, definition of Morse functions, statement of Morse Lemma and first corollaries
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| Jul 4 | behaviour of topology of sublevel sets of a Morse function, handle attachment, stable and unstable manifolds of critical points, Morse-Smale pairs, examples
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| Jul 11 | spaces of connecting trajectories, compactification by broken flow lines, description of compactified spaces of connecting trajectories for index differences 1 and 2, definition of the boundary map in the Morse complex, examples, invariance and isomorphism to singular homology
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