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
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 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, linking numbers of curves in R^{3}

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  MayerVietoris sequence and applications, statement of Poincare duality for compact manifolds, introduction of de Rham cohomology with compact supports

Jan 11  MayerVietoris sequence for de Rham cohomology with compact supports, statement of Poincare duality for noncompect 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, MorseSmale 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

