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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   Set 7   Set 8   Set 9   Set 10   Set 11   Set 12   Set 13  

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, linking numbers of curves in R3
Nov 28   vector bundles: definitions, first examples, morphisms and trivializations, constructions with vector bundles: direct sum
Nov 30   subbundles, quotient bundles, pullback bundles, definition of the normal bundle of a submanifold, tubular neighborhood theorem
Dec 05   remarks and examples for tubular neighborhood theorem, orientations for vector bundles, intesection numbers of submanifolds of complementary dimension
Dec 07   examples of intersection numbers, Euler characteristic of a closed manifold, computation from vector fields with isolated zeros, extension to compact manifolds with boundary
Dec 12   classification of vector bundles: universal bundles, every bundle over a compact base is a pullback of a tautological bundle, consequences, classifying maps become isotopic in after stabilisation, homotopic maps induce isomorphic pullbacks
Dec 14   direct limits of compact Hausdorff spaces, examples, compact subsets are contained in a finite part, classifying spaces as limits, classification theorem with examples; cobordism groups
Dec 19   Thom construction and identification of the cobordism groups with homotopy groups of Thom spaces of universal bundles
Dec 21   further remarks on Thom construction, examples, Pontryagin construction: maps to spheres are classified by degree; differential forms: basic definitions, exterior differential, de Rham cohomology, first computations, homotopy invariance and Poincare lemma
Jan 9   Mayer-Vietoris sequence and applications, statement of Poincare duality for compact manifolds, introduction of de Rham cohomology with compact supports
Jan 11   Mayer-Vietoris sequence for de Rham cohomology with compact supports, statement of Poincare duality for non-compect manifolds, start of proof
Jan 16   completion of proof of Poincare duality, discussion of examples; definition of integration over the fiber in an oriented vector bundle, basic properties
Jan 18   proof of properties of pushforward, construction of Thom forms, properties, Thom isomorphism theorem
Jan 23   relationship of Thom forms and intersection of submanifolds, Künneth formula for de Rham cohomology, Euler characteristic as alternating sum of Betti numbers;
Morse theory: the Hessian of a function at a critical point
Jan 25   definition of Morse functions, Morse lemma and consequences, behavious of topology of sublevel sets of a Morse function
Jan 30   stable and unstable manifolds of critical points, Morse-Smale pairs, spaces of connecting trajectories, compactification by broken flow lines
Feb 1   description of compactified spaces of connecting trajectories for index differences 1 and 2, definition of the boundary map in the Morse complex, examples, closing remarks on the Morse complex


 
  Seitenanfang  Impressum 2018-02-09, Janko Latschev