Janko Latschev
Lecture Course Differential Topology, Winter Semester 2017/18
Lectures take place on Tuesday 810 in H5 and on Thursday 810 in H4. There will be exercise classes Thursdays 1012 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 ChernWeil theory, Morse theory and surgery constructions, and the ThomPontryagin 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 R^{n}, definition of topological manifolds, Grassmannians as an example, C^{r} atlases, C^{r} 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 C^{r} 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 C^{r}(U,V) for any r≥0

Nov 7  relative approximation, denseness of C^{s}(M,N) in C^{r}(M,N) for C^{s} manifolds and r < s, consequences, including uniqueness of smoothing differentiable structures

Nov 9  existence of smoothings of differentiable structures of class C^{r}, 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

