Math 8803

Characteristic Classes for
Vector Bundles and
Surface Bundles

Fall 2013
-
----------------------------------------Cassidy Curtis
News: Welcome to Math 8803! We will continue to meet at
the scheduled time, 1:05-1:55 MWF.

General Information

 Click here for the handout from the first day.

Professor

Class Meetings

 Monday, Wednesday, and Friday, 1:05-1:55 pm, Skiles 271.

Books

Office Hours

 In Skiles 244, after class and by appointment.

Homework

 Optional homework will be assigned throughout the semester. Graduate students will also be expected to give one lecture on a related topic in the Geometry-Topology Student Seminar.

Grading

 Grades will be based on attendance, homework, and seminar presentations.

Weekly Schedule
Week Dates Topics Reading Homework Lectures Notes
1 Aug 19 Overview CC in 7 pages Week 1
2 Aug 26 Vector Bundles Hatcher 1.1 Week 2
3 Sep 2 Classifying Vector Bundles Hatcher 1.2 Week 3 Labor day
4 Sep 9 Stiefel-Whitney Classes Hatcher 3.1 Week 4
5 Sep 16 Cohomology of Grassmannian Hatcher 3.1 Week 5
6 Sep 23 Applications of SW Classes Milnor 11 HW1 Week 6
7 Sep 30 Euler class Hatcher AT 4 Week 7
8 Oct 7 Pontryagin classes Hatcher 3.2 Week 8
9 Oct 16 Surfaces and bundles Week 9 Fall break
10 Oct 21 Classifying spaces Hatcher App. A Week 10
11 Oct 28 Contractibility of Diff Hatcher App. B Week 11 No class Fri
12 Nov 4 Degree one Primer on MCG Week 12
13 Nov 11 Surface bundles with nonzero signature Morita 4.3 / Atiyah - Week 13
14 Nov 18 Iterated surface bundles Morita 4.4 Week 14
15 Nov 25 Independence of MMMs Morita 4.4 Week 15 Thanksgiving
16 Dec 2 Madsen-Weiss Hatcher - Week 16
Course notes

References

 ---Characteristic Classes in Seven Pages, Dan Margalit

---Vector bundles and K-theory, Allen Hatcher

---An exposition of the Madsen-Weiss theorem, Allen Hatcher

---Spaces of graphs and surfaces: on the work of Soren Galatius, Ulrike Tillman

---The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces, Nathalie Wahl

---Algebraic Topology, Allen Hatcher

---Cup Product and Intersections, Michael Hutchings

---The signature of fibre-bundles, M.F. Atiyah

---Diffeomorphisms of the 2-sphere, Jacob Lurie

---Some groups of mapping classes not realized by diffeomorphisms, Mladen Bestvina, Thomas Church, Juan Souto

Possible topics for Student Seminars

---Milnor's construction of exotic spheres

---Introduction to K-theory

---Principal bundles and classifying spaces

---Applications to low-dimensional topology

---Homological stability

---Groups acting on the circle

---The J-Homomorphism

---Topological invariance of Pontryagin classes

---Cobordism rings

---Obstruction theory

---A combinatorial formula for normal Stiefel-Whitney classes, Tom Banchoff and Clint McCrory

---Triple Points and Singularities of Projections of Smoothly Immersed Surfaces, Thomas Banchoff (here)

---Some consequences of a theorem of Bott, John Milnor

Resources

---T-Square

---Georgia Tech Honor Code