MAT 566, Spring 2025

MAT 566: Differential Topology

Stony Brook Spring 2025

General Course Information, A Proof and a Modern ``Proof"

References

All readings listed in the table below are from Milnor&Stasheff's Characteristic Classes, except

Munkres: Munkres, Elements of Algebraic Topology
G&H: Griffiths&Harris, Principles of Algebraic Geometry
Vakil's slides: Vakil, slides for 2005 MAA Inivited Address
Spanier: Spanier, Algebraic Topology
Atiyah-Bott: Atiyah&Bott, The Moment Map and Equivariant Cohomology, Topology 1984
Mirror Symmetry: K. Hori, et. al., Mirror Symmetry
Milnor's paper: Milnor, On manifolds homeomorphic to the 7-sphere, Annals of Math, 1956
Warner: Warner, Foundations of Differentiable Manifolds and Lie Groups
Spin Geometry: Lawson&Michelsohn, Spin Geometry

Tentative Schedule

Date	Topic	Read	Problem Set
01/28, Tu	Review/overview of vector bundles	Sections 1-3; VB Notes	ps1
01/30, Th	Review of (co)homology	Munkres: Sections 31,33 Appendix A: pp257-260, 263bot-270
02/04, Tu	Review of (co)homology	Munkres: Sections 31,33 Appendix A: pp257-260, 263bot-270
02/06, Th	Poincare Duality	Appendix A: pp270-279
02/11, Tu	Stiefel-Whitney classes	Section 4: pp37-50	ps2
02/13, Th	Stiefel-Whitney classes and cobordism Grassmannians	Section 4: pp50-53 Section 5: pp55-70; G&H: pp193-194,207
02/18, Tu	(Co)homology of CW-complexes	Sect. 6: pp73-74; App. A: pp260bot-263
02/20, Th	(Co)homology of Grassmannians	Sections 6,7; G&H: pp194-197
02/25, Tu	Schubert Calculus	Vakil's slides; notes; G&H: pp197-206
02/27, Th	Thom Isomorphism Theorem	Spanier pp255-263; Munkres: pp279-280 Sections 8,10; App. A: pp265bot-268top
03/04, Tu	Applications of TI	Section 9
03/11, Tu	Euler class
03/13, Th	Euler class
03/25, Tu	Topology of smooth manifolds	Section 11
03/27, Th	Topology of smooth manifolds	Section 11
04/01, Tu	Chern classes	Section 12: pp143bot-146 Sections 13,14
04/03, Th
04/08, Tu
04/10, Th	Atiyah-Bott Localization Theorem	Atiyah-Bott's paper: Sections 2,3 Notes on Mirror Symmetry: Chapter 1
04/15, Tu	Pontrjagin classes	Section 15
04/17, Th	Chern and Pontrjagin numbers	Section 16: pp183-193
04/22, Tu	Oriented cobordism ring	Section 16: pp190-195; Section 17
04/24, Th	Some homotopy theory	Spainer: Sections 7.2-7.5
04/29, Tu	Thom-Pontrjagin construction	Section 18
05/01, Th	Multiplicative sequences	Section 19
05/06, Tu	Hirzebruch Signature Theorem	Section 19
05/08, Th	Exotic smooth structures on S⁷	Milnor's paper

This page is maintained by Aleksey Zinger.

Last modified: February 10, 2025.