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Differential Topology [MA5122]

Sommersemester 2014

Prof. Dr. Michael M. Wolf

Dozent: Prof. Dr. Michael M. Wolf
Übungsleitung: Alexander Müller-Hermes
Vorlesung: Mittwoch, 10:15 - 11:45, Raum 00.07.011 Anmeldung
Zentralübung: Montag, 14:15 - 15:45, Raum 03.08.011 Anmeldung
Zweite Zentralübung: Freitag, 08:30-10:00, Raum 03.10.011



We will follow a direct approach to differential topology and to many of its applications without requiring and exploiting the abstract machinery of algebraic topology. The course will cover immersion, submersions and embeddings of manifolds in Euclidean space (including the basic results by Sard and Whitney), a discussion of the Euler number and winding numbers, fixed point theorems, the Borsuk-Ulam theorem and respective applications. At the end of the course, students should be able to analyse topological problems from a differentiable viewpoint and to see differential problems from a topological perspective.

File Version Topics
lecture 1 09.04.2014 Introductory remarks, reminder of topological spaces, topological manifolds
lecture 2 16.04.2014 quotient topology, embedding of topological manifolds, differential calculus
lecture 3 30.04.2014 manifolds with boundary, differentiable structures, smooths manifolds and smooth maps
lecture 4 07.05.2014 smooth invariance of domain and its consequences, embeddings, immersions, submersions
lecture 5 14.05.2014 smooth submanifolds
lecture 6 21.05.2014 tangent space, tangent bundle, differential
lecture 7 28.05.2014 Sard's theorem, preimage theorem with boundary, no-retraction theorem
lecture 8 04.06.2014 Brouwer's fixed point theorem, Whitney's embedding theorem I
lecture 9 11.06.2014 Whitney's embedding theorem II, homotopy, isotopy, mod-2 degree
lecture 10 25.06.2014 mod-2 winding number, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem
lecture 11 02.07.2014 Corollaries from Borsuk-Ulam, Ham-Sandwich theorem, orientability of manifolds
lecture 12 09.07.2014 Brouwer degree, fixed points on S^n, hairy ball theorem, Hopf's characterization of homotopy classes into S^n

Exercises Solutions Due date Topics
Exercise 1 23.04.2014 Topology, real projective Space, matrix groups, a riddle
Exercise 2 07.05.2014 Partition of unity, bump functions, differential calculus, incompatible differential structures
Exercise 3 21.05.2014 Classification of smooth and compact 1-manifolds, projective spaces, manifolds with boundary, immersions and embeddings
Exercise 4 04.06.2014 Lie groups, Lie group actions, embedding of RP4, maxima and minima, Grassmann manifold
Exercise 5 18.06.2014 Transversality, Stack of records theorem, Morse functions, Brouwer
Exercise 6 02.07.2014 Embedding problems, rectangle on curve, Homotopy, smooth-no-retraction, mod 2 fundamental theorem of algebra
Exercise 7 09.07.2014 Ham-Sandwich, Orientations, Degree of a map


Additional literature:


MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra 1, MA1102 Linear Algebra 2. Helpful but not essential: MA2004 Vector Analysis.