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Introduction to Differential Topology

Instructors: Peter Franek and Mirko Klukas, Teaching Assistant: Stephan Zhechev

Description

 

The course (roughly) follows Milnor's book Topology from the differential viewpoint. Our focus is to develop an intuitive (yet solid) understanding of basic concepts in differential topology such as tangent bundles, differentials, degree of a map, and framed cobordisms.

The first goal is the Pontryagin-Thom construction which enables us to understand the space of homotopy classes of maps into spheres in some simple cases. Depending on the time-frame, other topics might be chosen in January.


Requirements: Regular attendence + written final exam.

 

Schedule (subject to change)

Date Topic
Nov. 29 Def. of manifold, tangent space, reg. values. Proof of FTA based on counting #f^{-1}(y).
Dec. 1 Preimage of regular values is a submanifold. Sard's theorem and consequences: disc has FPP.
Dec. 6 Proof of Sard's theorem, steps 1 and 3 from Milnor (p. 16--19). Degree mod 2.
Recitation Classification of 1-manifolds.
Dec. 13 Brouwer degree: definition and basic properties.
Recitation sketch of classification of surfaces
Dec. 15 no class (we can organize a make-up class later after agreement)
Dec. 20 Framed cobordisms and Pontryagin-Thom construction: statement and consequences
Recitation Grassmanians, linking number and its properties
Dec. 22 Pontryagin-Thom construction: proof

Homework

File Comment
Exercises1 Some if this will be discussed next Tuesday