Janko Latschev
Lecture "Differential Topology", Winter semester 2013/14
In this 2h-per-week lecture course we will cover the foundations of differential topology, which are often assumed to be known in more advanced classes in geometry, topology and related fields. Possible topics are: transversality, Sard's theorem, de Rham cohomology, vector bundles and their classification, characteristic classes, Morse theory. The precise choice will depend on background and needs of the attending students.
Basic prerequisites are the concepts of multivariable calculus, including differential forms, vector fields and the implicit function theorem, as well as the definition of differentiable manifolds.
In particular, the course is also suitable for advanced bachelor students.
Problems for exercises will be mentioned during class. Here is the first set of problems, which were mostly selected from the textbooks below. And here is the second set of problems.
Books for background and/or additional reading:
The link only works from inside the campus network.
| J. Milnor | Topology from the differentiable viewpoint |
| V. Guillemin, A. Pollack | Differential Topology |
| M. Hirsch | Differential Topology | |
| T. Bröcker, K. Jänich | Einführung in die Differentialtopologie | |
| A. Kosinski | Differential Manifolds | |
| J. Lee | Introduction to smooth manifolds |
| J. Milnor | Characteristic classes |
Diary:
| 16.10. | topological manifolds, differentiable structures, differentiable maps, Whitney embedding theorem, proof of the easiest version (compact manifolds, large ambient dimension)
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| 23.10. | tangent spaces and the tangent bundle, differential of a map, regular values, fire alarm, statement of Sard's theorem
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| 30.10. | first application of Sard's theorem: the easy Whitney embedding theorem (compact manifolds, dimension 2d+1); tranversality: motivation, definition, examples, transversality for families ("transversality is generic")
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| 6.11. | further remarks on transversality, short discussion on manifolds with boundary, smooth homotopy and isotopy, moving points by diffeomorphisms isotopic to the identity, local degree mod 2, invariance under homotopy, independence of regular value for connected target manifolds, homotopy invariance of mapping degree mod 2
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| 13.11. | orientations of manifolds, integer valued mapping degree, applications and examples
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| 20.11. | vector bundles: definition, basic notions, constructions with vector bundles: subbundles, pullbacks, direct sums, quotients
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| 27.11. | normal bundle of a submanifold, tubular neighborhood theorem, oriented bundles, cooriented submanifolds, intersection numbers for submanifolds of complementary dimension
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| 4.12. | example for intersection number, Euler characteristic as a special case, computation via indices of isolated zeros of a vector field, examples on surfaces; classification of vector bundles: universal bundle, existence of classifying map
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| | Here are some problems related to the material discussed so far.
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| 11.12. | homotopy covering theorem, homotopic classifying maps give isomorphic bundles, direct limits of families of compact Hausdorff spaces and their compact subsets, the classifying space, short outlook on characteristic classes, every vector bundle is a direct summand (and hence also a quotient) of a trivial bundle
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| 18.12. | differential forms, exterior differentiation, de Rham cohomology, understanding H0, statements of Poincaré lemma and homotopy invariance, computation for S1, Mayer-Vietoris sequence, cohomology of compact manifolds is finite dimensional, computation for Sn
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| 8.1. | statement of Poincaré duality, discussion of mapping degree in terms of differential forms, intersection form, signature of a manifold; geometric distributions, integrability for geometric distributions, examples, statement of the theorem of Frobenius
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| 15.1. | brief discussion of foliations, proof of the theorem of Frobenius, brief outlook on contact structures
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| | Here is a second set of problems related to the material of the recent lectures.
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| 22.1. | Hessian form of a function at a critical point, nondegenerate critical points, Morse functions, Morse Lemma and its consequences, for a connected set of regular values the sublevel sets are diffeomorphic
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| 29.1. | remark on denseness of Morse functions, topology change when passing a critical level; stable and unstable manifolds, Morse-Smale condition, definition of the Morse complex, outline of proof for ∂2=0
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