# TMA4190 Introduction to Topology - Spring 2019

Schedule Room
Lectures: Tuesday 10.15-12.00 R10 Realfagbygget
Thursday 12.15-14.00 MA24 Grønnbygget
Exercises: Friday 10.15-11.00 G012 Gamle Elektro
Exam: see here
Lecturer
Glen Wilson
Office: 1204 Sentralbygg 2
Email: glen [dot] m [dot] wilson [at] ntnu [dot] no

## What this course is about

The aim of the course is to introduce fundamental concepts and examples in differential topology and the idea of using algebraic invariants to study geometric objects. The course discusses the inverse function theorem, 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, the fundamental group and covering spaces. 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.

## 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 Notes 4
5.1 Inverse function theorem GP 1.3 Supplement
5.2 Submersions GP 1.4 Set 4 Notes 5
6.1 Fundamental theorem of algebra GP 1.4 Notes 6
6.2 Transversality GP 1.5 Set 5 Additional FTA notes
7.1 Transversality and homotopy GP 1.6 Notes 7
7.2 Transversality and homotopy GP 1.6 Set 6 Notes 8
8.1 Sard's theorem, Morse functions GP 1.7
8.2 Sard's theorem, Morse functions GP 1.7 Set 7 Notes 9
9.1 Midterm
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
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
11.2 Brouwer Fixed-point theorem, Transversality GP 2.2, 2.3 Set 10 Notes 13
12.1 Transversality GP 2.3 Notes 14
12.2 Mod 2 intersection theory GP 2.4 Set 11 Notes 15
13.1 Winding number, Borsuk-Ulam theorem GP 2.5, 2.6 Notes 16
13.2 Lecture cancelled Set 12
14.1 Orientations GP 3.2 Notes 17
14.2 Oriented intersection number GP 3.3 Set 13
15.1 Lefschetz fixed-point theory GP 3.4 Notes 18
15.2 Poincare-Hopf index theorem GP 3.5 Conway's ZIP proof

Ling Tan

Oskar Goldhahn

Odin Hoff Gardå

## Course material

This course will follow:

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

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