Math 790: Introduction to 4-Manifolds minicourse - Fall 2023

Overview: This course will be an overview of 4-dimensional topology: intersection forms, handle structures, Kirby calculus, minimal genus problems, topological classification theorems, exotic smooth structures, homology cobordism, and so on. In the last few lectures, we will also do a very brief overview of Heegaard Floer homology, with more emphasis on how it is used to study these types of problems than on the details of how it's defined.

Prerequisites: Elementary algebraic topology, at the level of Duke's Math 611.

Time/location: Tuesdays and Thursdays, 1:25-2:40pm, November 2 - December 7, Physics 227. You can also join by Zoom here (please contact me for the password).

References: My primary reference for the 4-manifold topology in the course will be 4-Manifolds and Kirby Calculus by Gompf and Stipsicz. Some other great references for this subject include The Wild World of 4-Manifolds by Scorpan and 4-Manifolds by Akbulut. Some good Heegaard Floer homology introductions include:

• Ozsvath and Szabo's survey articles from 2003, An Introduction to Heegaard Floer Homology and Lectures on Heegaard Floer Homology.
• Jen Hom's 2019 PCMI Lecture notes
• Josh Greene's AMS Notices survey article
• My lecture series on Heegaard Floer homology (Isaac Newton Institute, 2017): Part I, Part II, Part III, Part IV

Lecture videos:

1. November 2: Basic definitions, Poincare conjecture in various dimensions
2. November 7: Algebraic topology of 4-manifolds, intersection form
3. November 9: Intersection forms, Freedman's theorem
4. November 14: Rokhlin's theorem, Donaldson's diagonalization theorem, the E_8 manifold
5. November 16: Homology cobordism, introduction to handles
6. November 21: Handles and framings
7. November 28: Kirby calculus
8. November 30: Kirby calculus, continued
9. December 5: Introduction to Heegaard Floer homology
10. December 7: Heegaard Floer homology, continued