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
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| 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
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| 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
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| Oct 26 | strong topology on spaces of maps, interesting open sets in Cr with the strong topology: immersions, submersions, embeddings
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| 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
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| 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
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| 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
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| 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
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| Nov 16 | elementary consequences, Brouwer's fixed point theorem, transversality of a map to a submanifold and to another map
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| Nov 21 | transversality for families, examples, mapping degree mod 2, homotopy invariance, first examples and consequences
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| Nov 23 | orientations and integer mapping degree, homotopy invariance, examples and applications, in particular maps to spheres, linking numbers of curves in R3
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| Nov 28 | vector bundles: definitions, first examples, morphisms and trivializations, constructions with vector bundles: direct sum
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| Nov 30 | subbundles, quotient bundles, pullback bundles, definition of the normal bundle of a submanifold, tubular neighborhood theorem
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| Dec 05 | remarks and examples for tubular neighborhood theorem, orientations for vector bundles, intersection numbers of submanifolds of complementary dimension
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| 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
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| 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
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| 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
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| Dec 19 | Thom construction and identification of the cobordism groups with homotopy groups of Thom spaces of universal bundles
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| 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
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| Jan 9 | Mayer-Vietoris sequence and applications, statement of Poincare duality for compact manifolds, introduction of de Rham cohomology with compact supports
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| Jan 11 | Mayer-Vietoris sequence for de Rham cohomology with compact supports, statement of Poincare duality for non-compect manifolds, start of proof
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| Jan 16 | completion of proof of Poincare duality, discussion of examples; definition of integration over the fiber in an oriented vector bundle, basic properties
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| Jan 18 | proof of properties of pushforward, construction of Thom forms, properties, Thom isomorphism theorem
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| 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
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| Jan 25 | definition of Morse functions, Morse lemma and consequences, behavious of topology of sublevel sets of a Morse function
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| Jan 30 | stable and unstable manifolds of critical points, Morse-Smale pairs, spaces of connecting trajectories, compactification by broken flow lines
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| 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
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