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Differential Topology [MA5122]

Sommersemester 2013

Prof. Dr. Michael M. Wolf

Dozent: Prof. Dr. Michael M. Wolf
Vorlesung: Montag, 10:15 - 11:45, Raum 00.07.014 Anmeldung
Zentralübung: Montag, 14:15 - 15:45, Raum 03.10.011 , zweiwöchig Anmeldung
Tutorübungen:   Anmeldung


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 15.04.2013 Introductory remarks, reminder of topological spaces
lecture 2 22.04.2013 Differential calculus, constant rank thm., topological manifolds, 1st embedding thm.
lecture 3 04.05.2013 Connected sums, differentiable structures, smooth manifolds, manifolds with boundary
lecture 4 06.05.2013 Smooth maps and smooth invariance of domain
lecture 5 13.05.2013 Smooth submanifolds and their construction via maps of constant rank or embeddings
lecture 6 27.05.2013 Tangent spaces, the tangent bundle as smooth manifold, differentials
lecture 7 03.06.2013 Transversality
lecture 8 10.06.2013 Sard's theorem, no-retraction-theorem, Brouwer's fixed point theorem
lecture 9 17.06.2013 Embeddings and immersions into Euclidean space
lecture 10 01.07.2013 Homotopy and stability, degree mod 2
lecture 11 08.07.2013 Degree and winding number mod 2, Jordan-Brouwer separation theorem
lecture 12 15.07.2013 Borsuk-Ulam type theorems, Ham-Sandwich theorem

Exercises Due date Topics
Exercise 1 22.04.2013 Homeomorphisms, Invariance Groups, Matrix Analysis, Bump Functions, A riddle
Exercise 2 06.05.2013 Connectedness, Basic Topology, Projective Spaces, Manifolds with Boundary
Exercise 3 27.05.2013 Smooth Maps, Complex Projective Space, Lie Groups, A Swindle, Immersions and Embeddings
Exercise 4 03.06.2013 Lie Group Actions, Orbits
Exercise 5 17.06.2013 Classification of smooth and compact 1-manifolds, Transversality, Perron-Frobenius
Exercise 6 01.07.2013 Stack of Records Theorem, Morse Functions, An embedding
Exercise 7 15.07.2013 Smooth Homotopy, Transversality, Smooth-no-retraction theorem


Additional literature:


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