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Differential Topology [MA5122]

Wintersemester 2018/19

Prof. Dr. Michael M. Wolf

Lecturer: Prof. Dr. Michael M. Wolf
Assistant: Margret Heinze
Lecture: Mo 14:15 - 15:45 (room 00.09.022) Anmeldung
Exercises: Mi 10:15-11:45 und Do 08:15-09:45 (room 03.12.020A) Anmeldung

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Content

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.

Date Content Black board notes Further reading
Oct 15 Intro, Topological spaces, subspace-, product- and quotient-topologies lec 1 A. Hatcher's notes Pfeil with more details and proofs
Oct 22 Compactness, homeomorphisms, topological manifolds and their embeddings into Rn, differential calculus lec 2  
Oct 29 Classification of low-dim. manifolds, manifolds with boundary, constant rank thm., smooth structures lec 3  
Nov 5 Smooth maps and diffeomorphisms, smooth invariance of domain and its consequences lec 4  
Nov 12 Embeddings, immerions, submersions, submanifolds, preimage theorems lec 5  
Nov 19 Tangent-vectors, -spaces and -bundles lec 6  
Nov 26 Differentials, measure zero sets, Sard's theorem with and without boundary lec 7  
Dec 3 Density of Morse functions, Morse's Lemma, no-retraction theorem, Brouwer's fixed point theorem lec 8  
Dec 10 Invariance of domain and dimension, Whitney's embedding and immersion theorem lec 9  

Exercises

File Content Week Solution
Ex 1 Basic notions of topologies Oct. 22nd - Oct. 26th Sol 1
Ex 2 Topological and smooth manifolds Nov. 5th - Nov. 9th Sol 2
Ex 3 Lie groups, embeddings, immersions, submanifolds Nov. 19th - Nov. 23th Sol 3
Ex 4 tangent space, more Lie groups Dec. 3th - Dec. 7th Sol 4
Ex 5 Brouwer's fixed point theorem, classifiaction of 1-dim compact manifolds Dec. 17th - Dec. 21th  

Additional literature