# TMA4192 Differential Topology - Spring 2020

Schedule Room
Lectures: Tuesday 12:15–14:00 S23 Sentralbygg 2
Thursday 14:15–16:00 R21 Realfagbygget
Office Hours: Wednesday 13:00–15:00
Exam: see here
Lecturer
Glen Wilson
Office: 1204 Sentralbygg 2
Email: glen [dot] m [dot] wilson [at] ntnu [dot] no

The final lectures for this course will take place on April 21 and April 23 at the usual time on Blackboard Collaborate.

Bonus online lecture on Friday, March 27 at 12:15 on blackboard collaborate. This will discuss the technical transversality results needed to make our intersection theory work.

There will be a rescheduled lecture on February 14 at 14:15–16:00 in Gamle Fysikk F4. Classes on the 18th and 20th of February will be cancelled. There will be a take home midterm exam for that week.

First lecture backup loaction: Sentralbygg 2, 7. etasje, 734. 10:15–12:00 on January 7.

Please enter your preferences for our course meeting time here: when 2 meet website.

## What this course is about

The aim of the course is to introduce fundamental concepts and examples in differential topology. Key concepts that will be discussed include differentiable structures and smooth manifolds, tangent bundles, embeddings, submersions and regular/critical points. Important examples of spaces are surfaces, spheres, and projective spaces. Other key concepts are homotopy, transversality, intersection numbers and cobordism. Applications presented in the course may range from Brouwer's fixed point theorem to vector fields on spheres. These methods and ideas have been influential to and are used in many other parts of mathematics, but also in physics and other areas of application. See the study handbook for more information.

## Prerequisites

You should have taken Calculus 1, 2 and 3, or something similar. For example, you should have seen what it means for a function from $f \colon \mathbb{R}^n \to \mathbb{R}^m$ to be differentiable and what it means for a subset $U\subseteq \mathbb{R}^n$ to be open, closed, or compact. We will review the basics we need at the start of the course.

## Course structure

Students will receive weekly problem sets to help them practice the material and gain experience with the most important concepts. Students are encouraged to write up their solutions to the exercises and submit them to receive feedback. This will be good practice for the final exam. I will also draft a midterm that can either be taken in class or at home.

Your grade will be solely determined by the written final exam. The final exam will be a home exam through Inspera and will be graded on a pass/fail basis. See the message below from NTNU.

NTNU has decided that no exams with physical attendance will be held in April/May/June of 2020. Because if this, several subjects have changed their form of assessment. Some of these have also changed their form of grading from letter grades to pass/fail.

TMA4192 - Differential Topology has a changed form of assessment. The assessment form Written examination has been changed to Home exam

The form of grading is changed to pass/fail

The course description and information regarding assessment will be updated in StudentWeb by April 8th 2020

You will find more information concerning: • Tool for exam implementation (Inspera) • Improvement of grade and change in form of grading • Deadline for cancellation of exam registration • Consequences of change in form of assessment from letter grading to pass/fail • Exam dates and duration • Special needs accomodation • Examination aids for home examination • Re-sit exam • Self-certification for absence from assessment • Explanation of grades and appeals • FAQ - Frequently asked questions regarding exams and Corona • Academic information about exams

on this website: https://innsida.ntnu.no/wiki/-/wiki/Norsk/Digital+vurdering+-+IE

## Lecture Plan

This is a rough outline of the course structure that will be updated throughout the semester.

Week Topic Reference Exercises Notes
2.1 Introduction Notes 1
2.2 Topology of Rn Set 1 Notes 2
3.1 Subspaces, connectedness
3.2 Cut points, compactness Set 2
4.1 Smooth manifolds, tangent spaces GP 1.1 Notes 3
4.2 Tangent spaces GP 1.2 Set 3 Lecture 6-7, Old Notes 4
5.1 Inverse function theorem GP 1.3 Supplement
5.2 Immersions GP 1.3 Set 4 Notes 5
6.1 Submersions GP 1.4 Lecture 9 , Old Notes 6
6.2 Fundamental Theorem of Algebra GP 1.4 Set 5 New FTA notes
7.1 Transversality and homotopy GP 1.5 Notes 7
7.2 Transversality and homotopy GP 1.6 Set 6 Notes 8
7.3 Sard's theorem GP 1.7 Notes 9
8.1 Midterm Midterm
9.1 Sard's theorem, Morse functions GP 1.7 Set 7 Morse Theory
9.2 Embeddings in Rn GP 1.8 Set 8,Prob 4 Notes 9
10.1 Abstract Manifolds, Projective Space GP 1.8 Notes 10,Old Notes 10
10.2 Partitions of Unity GP 1.8 Set 9 Notes 11,P.O.U. supplement
11.1 Manifolds with boundary GP 2.1 Notes 12, Old Notes 12
11.2 Brouwer Fixed-point theorem, Transversality GP 2.2, 2.3 Set 10 Notes 13
12.1 Transversality GP 2.3 Notes 14, Old Notes 14
12.2 Mod 2 intersection theory GP 2.4 Set 11 Notes 15
13.1 Winding number GP 2.5 Notes 16
13.2 Borsuk-Ulam theorem GP 2.6 Set 12
14.1 Orientations GP 3.2 Notes 17
14.2 Oriented intersection number GP 3.3 Set 13
15.1 Summary GP 3.4 Summary 2020
Conway's ZIP proof, Notes 18

## Reference Group

Tallak Manum

Andreas Palm Sivertsen

Elias Klakken Angelsen

## Course material

This course will follow:

• [GP] V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.

A highly recommended companion to this book is:

• [T] L. W. Tu, An Introduction to Manifolds, Springer Verlag.

Other interesting and related books (not required for the course):

• [M69] J. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1969.
• [M63] J. Milnor, Morse Theory, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963.
• [D] B. Dundas, Differential Topology. Version January 2013
• [L] J.M. Lee, Introduction to Smooth manifolds, Springer-Verlag.
• [S] D. Spivak, Calculus on Manifolds, Addison-Wesley, 1965.
• [Mu91] J.R. Munkres, Analysis on Manifolds, Addison-Wesley, 1991.
• [Mu75] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.

Jeff Weeks's geometry games.