K-theory
-
General Introduction
Historically, K-theory was the first "generalized cohomology theory" to be developed in algebraic topology. Since then, K-theory has also been developed in algebraic geometry (see Algebraic K-theory), and there are now many different variations on this theme. We refer to the individual pages for the various versions of K-theory.
<]]> -
Search results
<]]> -
Online References
Atiyah: K-theory Past and Present
nLab on K-theory, and on the K-theory spectrum
Cortinas (K-th folder): Algebraic vs topological K-theory, lecture notes from Sedano. Covers in a rather elementary way topological K-theory of Banach algebras, Karoubi-Villamayor K-theory, Weibel's homotopy K-theory, and Quillen K-theory. Focus on rings, not schemes. Stuff on excision and of comparison between top and alg K-th of topological algebras.
<]]> -
Paper References
Many papers by Atiyah, for example: Vector bundles and homogeneous spaces (1961)
Book by Karoubi: K-theory (1978)
LNM0028 Conner and Floyd. Discusses many different types of cobordism and their relations to K-theories.
Karoubi lecture notes on K-theory, in K-theory folder. Basic intro to top and alg K-th, among other things K-V K-th. Final chapters are on hermitian K-theory and cyclic homology.
<]]> -
Definition
<]]> -
Properties
<]]> -
Standard theorems
<]]> -
Open Problems
<]]> -
Connections to Number Theory
<]]> -
Computations and Examples
<]]> -
History and Applications
<]]> -
Some Research Articles
Preprint in progress of Rognes: A note on monochromatic K-theory
Preprint in progress of Rognes: Algebraic K-theory of the fraction field of topological K-theory
arXiv:0912.3635 Algebraic Geometry of Topological Spaces I from arXiv Front: math.KT by Guillermo Cortiñas, Andreas Thom We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case when M=Z^n. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C-algebras, and for a homology theory of commutative algebras to vanish on C-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil-K-theory implies that commutative C*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild-Kostant-Rosenberg and Loday-Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beilinson-Soulé and Farrell-Jones are also given.
<]]> -
Other Information
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel-2007.pdf. See Derived categorical sheaves for more about this article. Contains a remark at the end about algebraic K-theory determining complex topological K-theory, with a ref to Walker 2002 in K-theory.
arXiv:1008.1346 18 Lectures on K-Theory from arXiv Front: math.AT by Ioannis P. Zois We present 18 Introductory Lectures on K-Theory covering its basic three branches, namely topological, analytic (K-Homology) and Higher Algebraic K-Theory, 6 lectures on each branch. The skeleton of these notes was provided by the author's personal notes from a graduate summer school on K-Theory organised by the London Mathematical Society (LMS) back in 1995 in Lancaster, UK.
<]]> -
Comments Posted
<]]> -
Comments
There are no comments.