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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

Course information

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
9.1	Midterm
9.2	Embeddings in Rn	GP 1.8
10.1	Manifolds, partitions of unity	GP 1.8
10.2	Manifolds with boundary	GP 2.1
11.1	Brouwer Fixed-point theorem	GP 2.2
11.2	Transversality	GP 2.3
12.1	Transversality	GP 2.3
12.2	Mod 2 intersection theory	GP 2.4
13.1	Winding number, Jordan-Brouwer	GP 2.5
13.2	Borsuk-Ulam theorem	GP 2.6
14.1	Orientations	GP 3.2
14.2	Oriented intersection number	GP 3.3
15.1	Oriented intersection number	GP 3.3
15.2	Lefschetz fixed-point theory	GP 3.4
	Poincare-Hopf index theorem	GP 3.5

Reference Group

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.