Differential Topology is the study of differentiable manifolds, spaces that locally look like an Euclidean space, but as a whole might look much different, such as for instance a sphere. Central questions we will address in this course include:

When can a manifold be embedded into another manifold, for instance, Euclidean space?
What can we say, globally, about the space of smooth maps from one manifold to another?
How do submanifolds in a larger manifold intersect one another?
How can we decompose a smooth manifold into smaller parts and obtain a more combinatorial description?

Prerequisites

No background in algebraic topology is required. Some background in point-set topology is expected. Knowledge about the basic definitions of manifolds and their tangent spaces is useful, but can be acquired during the course.

Literature

We will largely be following the following textbook:

Guillemin, Pollack: Differential topology

Victor Guillemin, Alan Pollack, Differential Topology, AMS Chelsea publishing, 2010 (reprint)

I will provide or point to additional material.

Examination

The examination is based on weekly homework and presentations of problems in the exercise classes. There will be no final exam.

There will be 12 homework sets, each worth 10 points. At the end of the course, the two worst-scoring homework sets will be disregarded (this includes missed homework), for a maximum of 100 points. The grade boundaries are:

F	Fx	E	D	C	B	A
0	45	50	60	70	80	90

In addition, you are required to present solutions to homework problems in the exercise sessions, three times in total during the courses duration. This is a pass-fail criterion. You may not choose which exercise you want to present, but you have some control over it, as follows:

On every exercise sheet you turn in, you clearly mark the problems you would be willing to present in the following exercise session with an asterisk (*). The assistant may or may not call on you to present this problem. To pass the course, you have to mark at least 50% of the problems with an asterisk. If you mark any problems on an exercise sheet with an asterisk, you have to go to the corresponding exercise session, otherwise the asterisks are not counted.

Times and dates

All lectures will be online only. There will be lectures and exercise sessions, always on Tuesdays and Fridays. First lecture: January 19, 1pm. The exercise sessions are led by Eric Ahlqvist.

The zoom link for all lectures and exercises is https://kth-se.zoom.us/j/69184586345

T, Jan 19	lecture	Introduction; definition of smooth submanifolds of Rⁿ and abstract smooth manifolds; smooth maps between manifolds	Guillemin-Pollack ([GP]): I.1 For the definition of abstract smooth manifolds: Madsen-Tornehave: From Calculus to Cohomology, Cambridge 1997 ([MT]) Chapter 8 Lecture Notes
F, Jan 22	lecture	Tangent vectors and derivatives of maps; immersions, submersions, and embeddings	[GP]: I.2-3 For the definition of tangent vectors of abstract manifolds: [MT]: Chapter 9 Lecture Notes
T, Jan 26	exercises	Examples to understand the definitions of manifolds, immersions etc. Jonathan, Adam, and Ludvig presented.	Exercises 1
F, Jan 29	lecture	Standard form theorems for immersions and submersions; implicit function theorem; proper injective immersions; transversality; statement of Sard's Theorem; proof of Fubini's measure-zero Theorem	[GP]: I.3-5, Appendix I Lecture Notes (including half of next lecture)
T, Feb 2	lecture	Proof of Sard's Theorem; stability; the "easy" Whitney embedding theorem.	[GP]: I.6-7, Appendix I [MT]: Chapter 8, in particular pp. 60–61 Lecture Notes (including parts of next lecture)
F, Feb 5	exercises	Examples of regular/critical values and a proof of the fundamental theorem of algebra. Daniel, Jonathan, and Lina presented.	Exercises 2
T, Feb 9	lecture 10am instead of 1pm!	Existence of smooth partitions of unity; the Whitney embedding theorem (into R²ⁿ⁺¹) for compact and noncompact manifolds	[MT]: Appendix A, Chapter 8 Bredon: Topology and Geometry, Springer GTM 139, 1993 [B], Chapters I.12, II.10 Lecture Notes (including parts of next lecture)
F, Feb 12	exercises	Discussed previous homework, the structure of the tangent bundle, and Lie groups. Daniel, Jonathan and Lina presented.	Exercises 3
T, Feb 16	lecture	Homotopy and Stability; Manifolds with boundary	[GP] I.6, II.1 Lecture Notes (including next lecture)
F, Feb 19	exercises	Discussed the previous homework in detail. Daniel and Ludvig presented.	Exercises 4
T, Feb 23	lecture	Classification of 1-dimensional manifolds, Brouwer's fixed point theorem	[GP] II.2, Appendix 2
F, Feb 26	exercises	Discussed solutions to previous homework and looked at oriantations. Lina and Ludvig presented.	Exercises 5
T, Mar 2	lecture	Transversality and transversality homotopy theorems; smooth approximation	[GP] II.3 [B] Chapter II.11 Lecture notes
F, Mar 5	exercises	Solutions to Homework 6. Compared definitions of orientations and solved exercise about deformations.	Exercises 6
Exam weeks
T, Mar 23	lecture cancelled
F, Mar 26	exercises cancelled
T, Mar 30	lecture	Unoriented intersection theory, mapping degrees, winding numbers, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem	[GP] II.4-6 Lecture notes
W, Mar 31	extra exercise session at 1pm-3pm	Solved exercises 1-4 of the previous homework.	Exercises 7
Easter
T, Apr 13	lecture	proof of the Borsuk-Ulam theorem; orientations; oriented intersection theory	[GP] III.1-3 Lecture notes
F, Apr 16	exercises	Solved previous homework and compared the new definition of orientation with the old one.	Exercises 8
T, Apr 20	lecture	Lefschetz fixed point theory	[GP] III.4 Lecture notes
F, Apr 23	exercises	Discussed orientations and solved exercises from homework 9.	Exercises 9
T, Apr 27	lecture	Vector fields and indices; triangulations and Euler characteristics of simplicial complexes; Poincaré-Hopf theorem; Statement and preparations for the Hopf degree theorem.	[GP] III.5-7 Lecture notes, containing parts of next lecture
Walpurgis night
T, May 4	lecture	proof of the Hopf degree theorem, introduction to Lie groups	[GP] III.7 Adams, J.F.: Lectures on Lie groups [A], Chapter 1 Lecture notes
F, May 7	exercises	Solved homework 10 and discussed triangulations. Seuri presented problem 2.	Exercises 10
T, May 11	lecture	The exponential map for Lie groups, classification of abelian Lie groups, Cartans theorem on closed subgroups of Lie groups.	[A] Chapter 2 Lecture notes
Kristi klämdag
T, May 18	lecture	The quotient manifold theorem and applications	Lee, J. M: Introduction to smooth manifolds, Ch. 7 Lecture notes
F, May 21	exercises	Solved homework 11 and 12. Seuri presented problem 1 and 2 on HW12.	Exercises 11-12