1. K-theory - Wikipedia, The Free Encyclopedia In mathematics, ktheory is, firstly, an extraordinary cohomology theory which consists of topological k-theory. It also includes algebraic k-theory. http://en.wikipedia.org/wiki/K-theory

K-theory From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics K-theory is a tool used in several disciplines. In algebraic topology , it is an extraordinary cohomology theory known as topological K-theory . In algebra and algebraic geometry , it is referred to as algebraic K-theory . It also has some applications in operator algebras . It leads to the construction of families of K functors , which contain useful but often hard-to-compute information. In physics , K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes Ramond-Ramond field strengths and also certain spinors on generalized complex manifolds . For details, see also K-theory (physics) edit Early history The subject was originally discovered by Alexander Grothendieck (1957) so that he could formulate his Grothendieck-Riemann-Roch theorem . It takes its name from the German "Klasse", meaning "class" . Grothendieck needed to work with sheaves on an algebraic variety X . Rather than working directly with the sheaves, he gave two constructions. In the first, he used the operation of direct sum to convert the commutative

2. K-theory Preprint Archives Electronic preprint archives for mathematics research papers in ktheory. http://www.math.uiuc.edu/K-theory/

K-theory Preprint Archives Welcome to the preprint archives for papers in K-theory. We accept submissions of preprints in electronic form for storage until publication. Storage after publication may be possible, too. RSS feed for K-theory preprints Search with google: Our mirror site in Germany Our home site in the USA Instructions for authors Instructions for joining the mailing list. Members of the mailing list receive announcements of preprints when they are deposited in the archives. (The list has more than 300 members.) Best Current Practices: Recommendations on Electronic Information Communication A calendar of K-theory events. A bibtex file (2342698 bytes), or the same thing gzipped (557172 bytes) containing 6454 citations to papers related to K-theory and instructions for enlarging bibtex to work with such a large file. MPRESS , a searchable index of preprints, including references to the preprints stored here. Some TeX fonts , stored in a tar image compressed with gzip, including the lams* and xy* fonts, which are needed for some of the preprints. The unprocessed recent submissions bibtex entries Related sites: K-theory Front for the arXiv Preprints: SFB 343 "Diskrete Strukturen in der Mathematik" TMR Network: Algebraic K-Theory, Linear Algebraic Groups and Related Structures

3. SpringerLink Home - Main (Kluwer) Tables of contents from vol.11 (1997). http://www.kluweronline.com/issn/0920-3036

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4. 19: K-theory Encyclopedic reference for ktheory in Dave Rusin s Mathematical Atlas. Includes a brief history along with various links to textbooks, reference works, http://www.math.niu.edu/~rusin/known-math/index/19-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 19: K-theory Introduction K-theory is an interesting blend of algebra and geometry. Originally defined for (vector bundles over) topological spaces it is now also defined for (modules over) rings, giving extra algebraic information about those objects. History Read Atiyah's, "K-Theory Past and Present" at here Applications and related fields Most of the geometric K-theory is treated with Algebraic Topology See also 16E20, 18F25 Subfields Grothendieck groups and K_0, see also 13D15, 18F30 Whitehead groups and K_1 Steinberg groups and K_2 Higher algebraic K-theory K-theory in geometry K-theory in number theory, see also 11R70, 11S70 K-theory of forms, see also 11EXX Obstructions from topology K-theory and operator algebras See mainly 46L80, and also 46M20 Topological K-theory, see also 55N15, 55R50, 55S25 Miscellaneous applications of K-theory K-Theory is the smallest of the 61 active areas of the MSC scheme: only 515 papers with primary classification 19-XX during 1980-1997. But the area 19-XX was only available as a primary classification for Math Reviews papers starting with MR96; hence the count above is an undercount of the true size of the field. (Even granting this, however, K-theory is a fairly small field.) Browse all (old) classifications for this area at the AMS.

5. K-Theory -- From Wolfram MathWorld In general, there are two main types of K theory topological and algebraic. Topological K -theory is the true K -theory in the sense that it came first. http://mathworld.wolfram.com/K-Theory.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Budney K-Theory A branch of mathematics which brings together ideas from algebraic geometry linear algebra , and number theory . In general, there are two main types of -theory: topological and algebraic. Topological -theory is the "true" -theory in the sense that it came first. Topological -theory has to do with vector bundles over topological spaces . Elements of a -theory are stable equivalence classes of vector bundles over a topological space . You can put a ring structure on the collection of stably equivalent bundles by defining addition through the Whitney sum , and multiplication through the tensor product of vector bundles . This defines "the reduced real topological -theory of a space." "The reduced -theory of a space" refers to the same construction, but instead of real vector bundles complex vector bundles are used. Topological -theory is significant because it forms a generalized cohomology theory, and it leads to a solution to the vector fields on spheres problem, as well as to an understanding of the -homeomorphism of homotopy theory Algebraic -theory is somewhat more involved. Swan (1962) noticed that there is a correspondence between the

6. An Introduction To Algebraic K-theory An incomplete graduate textbook in Algebraic ktheory, by Charles Weibel. Several chapters are available online. http://www.math.rutgers.edu/~weibel/Kbook.html

``The K-book: An introduction to algebraic K-theory'' by Charles Weibel (a graduate textbook in progress) Introduction here Chapter I : Projective Modules and Vector Bundles Here is the searchable PDF file Last major update March 1997 (section 4). Minor updates July 2000, Sept. 2004, June 2005, May 2007. 1. Free and stably free modules; p.1 2. Projective modules; p.6 3. The Picard group of a ring; p.15 4. Topological vector bundles and Chern classes; p.26 5. Algebraic vector bundles. p.38 Chapter II : The Grothendieck group K_0 (101 pp.) Here is the searchable PDF file Last major updates Dec. 2003 (sec.6, 7, 9), July 2004 (sec.2, 7, App.). Minor updates Sept. 2004, June 2005 (sec.3), August 2006 (Burnside ring), Jan 2007 (compatibility with Chapter V). 1. group completion of a monoid; p.1 2. K_0 of a ring; p.5 3. K(X) of a topological space; p.17 Lambda and Adams operations; p.24 5. K_0 of a symmetric monoidal category; p.37 6. K_0 of an abelian category; p.45 7. K_0 of an exact category; p.59 8. K_0 of schemes and varieties; p.73

7. Vector Bundles & K-Theory Book The plan is for this to be a fairly short book focusing on topological ktheory and containing also the necessary background material on vector bundles and http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

The plan is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. Here is a provisional Table of Contents At present about half of the book is in good enough shape to be posted online, approximately 110 pages. What is included in this installment is: Chapter 1, containing basics about vector bundles. Part of Chapter 2, introducing K-theory, then proving Bott periodicity in the complex case and Adams' theorem on the Hopf invariant, with its famous applications to division algebras and parallelizability of spheres. Not yet written is the proof of Bott Periodicity in the real case, with its application to vector fields on spheres. Most of Chapter 3, constructing Stiefel-Whitney, Chern, Euler, and Pontryagin classes and establishing their basic properties. Part of Chapter 4 on the stable J-homomorphism. What is written so far is just the application of complex K-theory, using the Chern character, to give a lower bound on the order of the image of the stable J-homomorphism. Much of this material is already well covered in other sources, notably the classic books of Atiyah (

8. SpringerLink - Journal springerlink.metapress.com/content/15730514/ - Similar pages Lectures on Topological k-theoryThis will be an expository seminar on the elements of topological k-theory at a level suitable for graduate students in mathematics and physics. http://springerlink.metapress.com/content/1573-0514/

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9. K-Theory - Algebra Journals, Books & Online Media | Springer ktheory - Algebra. k-theory is devoted to developments in the mathematical sciences which are related to one of the various aspects of k-theory. http://www.springer.com/west/home/math/algebra?SGWID=4-10043-70-35677085-0

10. K-Theory And Homology Authors/titles Recent Submissions Subjects Rings and Algebras (math.RA); ktheory and Homology (math.KT) Title Homological algebra in bivariant k-theory and other triangulated http://arxiv.org/list/math.KT/recent

arXiv.org math math.KT Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages K-Theory and Homology Authors and titles for recent submissions Thu, 13 Mar 2008 Wed, 12 Mar 2008 Tue, 11 Mar 2008 Thu, 6 Mar 2008 ... Wed, 5 Mar 2008 [ total of 9 entries: [ showing up to 25 entries per page: fewer more Thu, 13 Mar 2008 arXiv:0803.1681 ps pdf other Title: Authors: Joseph Gubeladze Comments: 5 pages Subjects: K-Theory and Homology (math.KT) ; Algebraic Geometry (math.AG) Wed, 12 Mar 2008 arXiv:0803.1641 ps pdf other Title: Some remarks on Nil groups in algebraic K-theory Authors: James F. Davis Subjects: K-Theory and Homology (math.KT) ; Algebraic Topology (math.AT) arXiv:0803.1639 ps pdf other Title: Algebraic K-theory over the infinite dihedral group Authors: James F. Davis Qayum Khan Andrew Ranicki Subjects: K-Theory and Homology (math.KT) ; Algebraic Topology (math.AT) arXiv:0803.1635 ps pdf other Title: Authors: Serge Romeo Tagne Pelap Subjects: K-Theory and Homology (math.KT) arXiv:0803.1614 ps pdf other Title: On 0-homology of categorical at zero semigroups Authors: B. V. Novikov

11. Front: Math.KT K-Theory And Homology arXiv0801.1207 Adjamagbo Determinant and Serre conjecture for linear groups over Weyl algebras. Kossivi Adjamagbo. math.KT (math.RA). http://front.math.ucdavis.edu/math.KT

Front for the arXiv Fri, 14 Mar 2008 Front math KT search register submit journals ... iFAQ math.KT K-Theory and Homology Calendar Search Atom feed Search Author Title/ID Abstract+ Category articles per page Show Search help Recent New articles (last 12) 13 Mar arXiv:0803.1681 On Bass' question for finitely generated algebras over large fields. Joseph Gubeladze math.KT math.AG 12 Mar arXiv:0803.1635 Poisson (co)homology of polynomial Poisson algebras in dimension four : Sklyanin's case. Serge Romeo Tagne Pelap math.KT 12 Mar arXiv:0803.1641 Some remarks on Nil groups in algebraic K-theory. James F. Davis math.KT math.AT 12 Mar arXiv:0803.1639 Algebraic K-theory over the infinite dihedral group. James F. Davis , Qayum Khan , Andrew Ranicki math.KT math.AT 12 Mar arXiv:0803.1614 On 0-homology of categorical at zero semigroups. B. V. Novikov , L. Yu. Polyakova math.KT math.RA Cross-listings 12 Mar arXiv:0803.1320 Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology. Henri Moscovici (OSU), Bahram Rangipour (UNB). math.QA math.DG math.KT math.OA 11 Mar arXiv:0803.1200

12. Not Even Wrong » Blog Archive » Another Journal Board Resigns Ars Mathematica » Blog Archive » ktheory editorial board resigns Says Woit reports that the entire editorial board of the journal k-theory (published http://www.math.columbia.edu/~woit/wordpress/?p=581

13. K-theory -- Britannica Online Encyclopedia He was one of the pioneers, along with the Frenchman Alexandre Grothendieck and the German Friedrich Hirzebruch, in the development of ktheory—culminating http://www.britannica.com/eb/topic-309101/K-theory

document.writeln(''); document.writeln('Initializing application...'); Username Password Remember me Forgot your password? Browse Shop Log In Search Site: K-theory mathematics Main Aspects of this topic are discussed in the following places at Britannica. Assorted References algebraic topology in mathematics: Mathematical physics and the theory of groups ...in the 1960s, work chiefly by Grothendieck and the English mathematician Michael Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory (called K theory). More significantly still, in the 1960s Atiyah, the American Isadore Singer, and others found ways of connecting this work to the study of a wide variety of questions involving partial... work of Atiyah in Atiyah, Sir Michael Francis ...in Moscow in 1966 for his work on topology and analysis. He was one of the pioneers, along with the Frenchman Alexandre Grothendieck and the German Friedrich Hirzebruch, in the development of K Citations MLA Style: K-theory http://www.britannica.com/bps/topic/309101/K-theory

14. The Math Forum - Math Library - K-Theory The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/k_theory/

Browse and Search the Library Home Math Topics Algebra Modern Algebra : K-Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category K-Theory - Dave Rusin; The Mathematical Atlas A short article designed to provide an introduction to K-theory, a blend of algebra and geometry. Originally defined for (vector bundles over) topological spaces it is now also defined for (modules over) rings, giving extra algebraic information about those objects. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus. more>> All Sites - 8 items found, showing 1 to 8 arXiv.org e-Print archive - Los Alamos National Laboratory (LANL) A major site for mathematics preprints that has incorporated many formerly independent specialist archives including alg-geom, funct-an, dg-ga, q-alg, auto-fms, cd-hg, MAGNUS, Several Complex Variables, Logic E-prints, Commutative Algebra, Dynamical Systems, ...more>> Front for the Mathematics ArXiv - Univ. of California, Davis

15. PlanetMath: Algebraic K-theory Algebraic ktheory is a series of functors on the category of rings. Broadly speaking, it classifies ring invariants, i.e. ring properties that are Morita http://planetmath.org/encyclopedia/AlgebraicKTheory.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Algebraic K-theory (Topic) Algebraic K-theory is a series of functors on the category of rings . Broadly speaking, it classifies ring invariants , i.e. ring properties that are Morita invariant The functor Let be a ring and denote by the algebraic direct limit of matrix algebras under the embeddings . The zeroth K-group of , is the Grothendieck group abelian group of formal differences ) of idempotents in up to similarity transformations . Let and be two idempotents. The sum of their equivalence classes and is the equivalence class of their direct sum where . Equivalently, one can work with finitely generated projective modules over The functor Denote by the direct limit of general linear groups under the embeddings . Give the direct limit topology , i.e. a

16. K-Theory And Homology Authors/titles Recent Submissions Subjects ktheory and Homology (math.KT); Number Theory (math.NT) Title Homological algebra in bivariant k-theory and other triangulated categories. http://aps.arxiv.org/list/math.KT/recent

aps.arXiv.org math math.KT Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text K-Theory and Homology Authors and titles for recent submissions Thu, 13 Mar 2008 Wed, 12 Mar 2008 Tue, 11 Mar 2008 Thu, 6 Mar 2008 ... Wed, 5 Mar 2008 [ total of 9 entries: [ showing up to 25 entries per page: fewer more Thu, 13 Mar 2008 arXiv:0803.1681 ps pdf other Title: Authors: Joseph Gubeladze Comments: 5 pages License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Subjects: K-Theory and Homology (math.KT) ; Algebraic Geometry (math.AG) Wed, 12 Mar 2008 arXiv:0803.1641 ps pdf other Title: Some remarks on Nil groups in algebraic K-theory Authors: James F. Davis License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Subjects: K-Theory and Homology (math.KT) ; Algebraic Topology (math.AT) arXiv:0803.1639 ps pdf other Title: Algebraic K-theory over the infinite dihedral group Authors: James F. Davis Qayum Khan Andrew Ranicki License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Subjects: K-Theory and Homology (math.KT) ; Algebraic Topology (math.AT)

17. Ovid: Welcome To OvidSP of the book Introduction to Algebraic k-theory. (AM-72) by Milnor, J., published by Princeton University Press.......gateway.ovid.com/ovidweb.cgi?T=JS MODE=ovid NEWS=n PAGE=toc D=ovft AN=00133300000000 - Similar pages Milnor, J. Introduction to Algebraic k-theory. (AM-72). http://gateway.ovid.com/ovidweb.cgi?T=JS&MODE=ovid&NEWS=n&PAGE=toc&D=ovft&AN=001

18. K-theory For Dummies, I | The String Coffee Table I feel like talking about ktheory, K-homology, and the like. Here I start leisurely with a bird’s eye overview of some central ideas. http://golem.ph.utexas.edu/string/archives/000627.html

@import url("/string/styles-site.css"); The String Coffee Table A Group Blog on Physics Skip to the Main Content Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main August 12, 2005 K-theory for dummies, I Posted by urs So this are the main concepts, informally: Pick some topological space X . Consider all complex vector bundles over X. Including their homomorphisms these form a category Vect X There is sort of a semi-ring structure on that category in that vector bundles can be added, using the direct sum, and multiplied, using the direct product. When we take equivalence classes in Vect X the result is in fact an ordinary semi-ring. There is a standard way, called the Grothendieck group completion , to throw in additive inverses such as to obtain an honest ring. This ring is denoted

19. JSTOR Algebraic And Etale $K$-Theory We define etale ktheory, interpret various conjectures of Quillen and Lichtenbaum in terms of a map from algebraic k-theory to etale k-theory, http://links.jstor.org/sici?sici=0002-9947(198511)292:1<247:AAE>2.0.CO;2-T

20. Ars Mathematica » Blog Archive » K-Theory Editorial Board Resigns Peter Woit reports that the entire editorial board of the journal ktheory (published by Springer) has resigned. The current editors have already formed a http://www.arsmathematica.net/archives/2007/08/06/k-theory-editorial-board-resig

Ars Mathematica Dedicated to the mathematical arts. Education without Permission End of News Era K-Theory editorial board resigns August 6th, 2007 by Walt Peter Woit reports that the entire editorial board of the journal K-Theory (published by Springer) has resigned. The current editors have already formed a new journal, Journal of K-Theory , which will be distributed by Cambridge University Press. The subscription price for the Springer journal was $1590. The subscription price for the new journal is 380 British pounds, which is roughly half the price. This entry was posted on Monday, August 6th, 2007 at 2:15 pm and is filed under Mathematics . You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response , or trackback from your own site. Name (required) Says: August 7th, 2007 at 12:01 pm Is it open access? Price is not the real problem, intellectual freedom is. Walt Says: August 7th, 2007 at 12:45 pm Says: August 7th, 2007 at 3:06 pm Andrei Says: August 7th, 2007 at 5:41 pm

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