For those signed up for the class, Problem Set #9 is to hand in! You can do
problems from the subsequent problem sets instead if you like.
I posted computations from the lecture on Dirac families.
Professor: Dan Freed
Class Meetings: TTh 11:00-12:30,
Discusion/Office Hours: Wednesdays,
2:00-4:00, RLM 9.162
For more details, see the First Day Handout
Problem Set #1
Problem Set #2
Problem Set #3
Problem Set #4
Problem Set #5
Problem Set #6
Problem Set #7
Problem Set #8
Problem Set #9
Problem Set #10
Problem Set #11
Graph paper for the group SU(3)
I will post notes for some of the lectures; others are well-documented in the
Readings below. The course includes guest lectures that are not represented
Lecture 1: Introduction
Lecture 2: Homotopy invariance
Lecture 3: Group completion and the
definition of K-theory
Lectures 4-8: We covered material in the Hatcher reference below
Lectures 9 & 10: Fredholm operators
Lecture 11: Clifford algebras
Lecture 12: Kuiper's theorem, classifying
spaces, Atiyah-Singer loop map, Atiyah-Bott-Shapiro construction
Lecture 13: Topology of skew-adjoint
Lecture 14: Proof of Bott periodicity (con't)
Lecture 15: Groupoids and vector bundles
Computations for the Dirac family in the
Totaro on Algebraic Topology, in
The Princeton Companion to Mathematics. The second half is about
vector bundles and K-theory.
Varadarajan on Historical remarks on
vector bundles and connections.
Hatcher on Vector Bundles and K-theory,
book in progress.
Chapter 1 of Atiyah's K-theory book
on vector bundles.
Warner on partions of unity.
Old notes on fiber bundles (on smooth
manifolds--you can modify for topological spaces).
Hatcher on fiber bundles, including a
proof of the homotopy lifting property.
Atiyah-Bott "elementary" proof
of the periodicity theorem for the unitary group.
Hirzebruch on division algebras and
topology, from the Springer Readings Numbers.
Adams-Atiyah K-theory proof
of the nonexistence of elements of Hopf invariant one.
Earlier proofs of non-parallelizability of
spheres: Kervaire, Letters
between Milnor and Bott, and Milnor's
Palais on Fredholm operators (from
his Seminar on the Atiyah-Singer Index Theorem).
Atiyah on Fredholm operators (from
his K-Theory book).
Atiyah-Bott-Shapiro on Clifford algebras.
Deligne-Morgan on super algebra (and
Deligne on spinors and Clifford algebras,
from Quantum Fields and Strings: A Course for Mathematicians.
Atiyah and Segal on twisted K-theory;
appendices discuss topology of Fredholm operators and also Kuiper's
Atiyah and Singer on spaces of skew-adjoint
Fredholm operators and Bott periodicity.
Deligne and Freed on sign conventions.
Davis and Kirk on spectra and generalized
Freed on groups of Fredholm operators.
Palais on homotopy type of spaces of
Freed-Hopkins-Teleman on Loop groups
and twisted K-theory I.
Hatcher on quasifibrations (based
McDuff proof of Bott periodicity in last
section of article (with many more details
in Aguilar-Prieto and
Raoul Bott, The geometry and
representation theory of compact Lie groups
Adams Lectures on Lie Groups
Sepanski on Borel-Weil
from his book Compact Lie groups.
Freed-Hopkins-Teleman Loop groups
and twisted K-theory II with Dirac families.
Freed-Moore-Segal on Uncertainty of fluxes
including appendix on generalized Heisenberg groups.
Segal on Representations of infinite dimensional
Freed-Hopkins-Lurie-Teleman on Topological field
theories from compact Lie groups, including section 5 on geometry of
cocycles on tori.