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Janko Latschev


Lecture Course  Differential Topology, Winter Semester 2017/18

Lectures take place on Tuesday 8-10 in H5 and on Thursday 8-10 in H4. There will be exercise classes Thursdays 10-12 in Geom 142.

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)

There will be oral exams in the week immediately following the semester, i.e. starting February 5.

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:

Set 1   Set 2   Set 3   Set 4   Set 5   Set 6  

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:

Oct 17   introductory remarks; recollections on submanifolds of Rn, definition of topological manifolds, Grassmannians as an example, Cr atlases, Cr structures, differentiable maps between differentiable manifolds
Oct 19   the tangent bundle as a manifold, submersions, immersions and embeddings, every compact manifold embeds into some euclidean space, regular and critical values, preimages of regular values are submanifolds
Oct 24   remarks on sets of (Lebesgue) measure zero, easy Whitney embedding theorem for compact manifolds, remarks about variations; weak topology on spaces of maps
Oct 26   strong topology on spaces of maps, interesting open sets in Cr with the strong topology: immersions, submersions, embeddings
Nov 2   proper maps and diffeomorphisms form open sets in the strong topology; partitions of unity, convolution, smooth maps are dense in Cr(U,V) for any r≥0
Nov 7   relative approximation, denseness of Cs(M,N) in Cr(M,N) for Cs manifolds and r < s, consequences, including uniqueness of smoothing differentiable structures
Nov 9   existence of smoothings of differentiable structures of class Cr, r ≥ 1, discussion of consequences and further remarks; Transversality: initial remarks, statement of Sard's theorem
Nov 14   proof of Sard's theorem in the smooth case; recollection on manifolds with boundary, sublevel sets of regular values are submanifolds with boundary
Nov 16   elementary consequences, Brouwer's fixed point theorem, transversality of a map to a submanifold and to another map
Nov 21   transversality for families, examples, mapping degree mod 2, homotopy invariance, first examples and consequences
Nov 23   orientations and integer mapping degree, homotopy invariance, examples and applications, in particular maps to spheres


 
  Seitenanfang  Impressum 2017-11-22, Janko Latschev