# Differential Topology

## Course information

Code: MAT425F / MAT1340F

Instructor: Marco Gualtieri, office hours by appointment.

Class schedule: W1-3 BA1200 and R11 BA6183

Evaluation:

Exams:

This course is an introduction to the topological aspects of smooth spaces in arbitrary dimension. The main tools will include transversality theory of smooth maps, Morse theory and basic Riemannian geometry, as well as surgery theory. We hope to give a treatment of 4-dimensional manifolds and Kirby calculus. Coursework will involve exercises as well as class presentations.

Field trip to Perimeter Institute to attend Mirror Symmetry conference (Oct 23, 2013)

## Assignments

Treat the assignments as if they were take-home exams. Don’t discuss the problems until after they have been handed in. If you have questions, ask me after any of our classes, or if this is not possible, email me.

Also, don’t hand in any answers which you are not certain are correct. An important part of doing mathematics is to check your own work.

I am providing the TeX source because I would like the assignments to be handed back as a PDF document typeset by latex.

## Resources

### Books

The following is a list of texts which I will be following to various degrees. These are not required texts in the usual sense, but they are very beautiful and important texts which it would not hurt to own a copy of.

- Guillemin-Pollack:
*Differential topology* - Milnor:
*Topology from the differentiable viewpoint* - Milnor:
*Morse theory* - Kirby:
*The topology of 4-manifolds* - Arnol’d:
*Catastrophe theory*(1992 edition) - Notes from my last MAT1300 course

### Background Material

Familiarize yourself with the exterior algebra. There are many good sources for this; you could try, for example, Winitzki’s linear algebra text.

### Related Articles

- Morse theory in the 1990s, M. Guest.
- Morse 2-functions, D. Gay and R. Kirby.
- Catastrophe Theory, E. Zeeman.
- Beyond Rainbows, M. Berry.
- Differential Topology Forty-six Years Later, J. Milnor.