# Math392C: K-theory

## Announcements

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.

## Basic Information

Professor: Dan Freed

Class Meetings: TTh 11:00-12:30, RLM 9.166

Discusion/Office Hours: Wednesdays, 2:00-4:00, RLM 9.162

For more details, see the First Day Handout

## Problem Sets

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)

## Notes

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 here.

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 Fredholm operators

Lecture 14: Proof of Bott periodicity (con't)

Lecture 15: Groupoids and vector bundles

Computations for the Dirac family in the 11/24 lecture

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 paper .

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 supermanifolds).

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 theorem.

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 cohomology theories.

Freed on groups of Fredholm operators.

Palais on homotopy type of spaces of operators.

Freed-Hopkins-Teleman on Loop groups and twisted K-theory I.

Hatcher on quasifibrations (based on May).

McDuff proof of Bott periodicity in last section of article (with many more details in Aguilar-Prieto and Behrens and Behrens).

Raoul Bott, The geometry and representation theory of compact Lie groups