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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  
Jan 7 Smooth homotopy and isotopy, mod2-degree lec 10  
Jan 14 Mod-2 winding numbers, Jordan-Brouwer separation, Borsuk-Ulam 1st version lec 11  
Jan 24 Borsuk-Ulam variants, optimal non-linear approximations, parametrized Sard for probability-one homotopy methods lec 12  
Jan 28 Ham-sandwich theorem, Orientability, Brouwer degree theorem lec 13  
Feb 4 Properties of Brouwer degree, hedgehog theorem, Euclidean degree, boundary theorem, Rothe's theorem lec 14 Chap.1 in Vandervorst's online book Pfeil

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 Sol 5
Ex 6 embeddings, Nash equilibrium Jan. 16th - Jan. 17th Sol 6
Ex 7 stack of records, smooth homotopy, mod-2 degree Jan. 28th - Feb. 1st Sol 7
Ex 8 degree, Borsuk-Ulam Feb. 4th - Feb. 8th Sol 8

Additional literature