Ktheory

General Introduction
Historically, Ktheory was the first "generalized cohomology theory" to be developed in algebraic topology. Since then, Ktheory has also been developed in algebraic geometry (see Algebraic Ktheory), and there are now many different variations on this theme. We refer to the individual pages for the various versions of Ktheory.
<]]> 
Search results
<]]> 
Online References
Atiyah: Ktheory Past and Present
nLab on Ktheory, and on the Ktheory spectrum
Cortinas (Kth folder): Algebraic vs topological Ktheory, lecture notes from Sedano. Covers in a rather elementary way topological Ktheory of Banach algebras, KaroubiVillamayor Ktheory, Weibel's homotopy Ktheory, and Quillen Ktheory. Focus on rings, not schemes. Stuff on excision and of comparison between top and alg Kth of topological algebras.
<]]> 
Paper References
Many papers by Atiyah, for example: Vector bundles and homogeneous spaces (1961)
Book by Karoubi: Ktheory (1978)
LNM0028 Conner and Floyd. Discusses many different types of cobordism and their relations to Ktheories.
Karoubi lecture notes on Ktheory, in Ktheory folder. Basic intro to top and alg Kth, among other things KV Kth. Final chapters are on hermitian Ktheory 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 Ktheory
Preprint in progress of Rognes: Algebraic Ktheory of the fraction field of topological Ktheory
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 Ktheoretic and related invariants of the algebra C(X) of continuous complexvalued 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, torsionfree, 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 Ktheory 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 Calgebras, and for a homology theory of commutative algebras to vanish on Calgebras. These criteria have numerous applications. For example, the vanishing criterion applied to nilKtheory implies that commutative C*algebras are Kregular. As another application, we show that the familiar formulas of HochschildKostantRosenberg and LodayQuillen 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 BeilinsonSoulé and FarrellJones are also given.
<]]> 
Other Information
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel2007.pdf. See Derived categorical sheaves for more about this article. Contains a remark at the end about algebraic Ktheory determining complex topological Ktheory, with a ref to Walker 2002 in Ktheory.
arXiv:1008.1346 18 Lectures on KTheory from arXiv Front: math.AT by Ioannis P. Zois We present 18 Introductory Lectures on KTheory covering its basic three branches, namely topological, analytic (KHomology) and Higher Algebraic KTheory, 6 lectures on each branch. The skeleton of these notes was provided by the author's personal notes from a graduate summer school on KTheory organised by the London Mathematical Society (LMS) back in 1995 in Lancaster, UK.
<]]> 
Comments Posted
<]]> 
Comments
There are no comments.