K-theory and elliptic operators

Details K-theory and elliptic operators K-theory and elliptic operators

2005-04-27

arxiv.org/abs/math/0504555v1

Mathematics

Algebraic Topology

50 pages

Scientific paper

This expository paper is an introductory text on topological K-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah's classic "K-Theory" text, as well as his series of seminal papers "The Index of Elliptic Operators" with Singer. Additional topics include equivariant K-theory, the G-index theorem, and Bott's paper "The Index Theorem for Homogeneous Differential Operators". It also includes an appendix with a proof of Bott periodicity, as well as sketches of proofs for both the standard and equivariant versions of the K-theory Thom isomorphism theorem, in terms of the index for families of elliptic operators. A second appendix derives the Atiyah-Hirzebruch spectral sequence. This text originated as notes from a series of lectures given by the author as an undergraduate at Princeton. In its current form, the author has used it for graduate courses at the University of Oregon.

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