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	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

Applications and related fields

Most of the geometric K-theory is treated with Algebraic Topology

See also 16E20, 18F25

Subfields

• 19A: Grothendieck groups and K_0, see also 13D15, 18F30
• 19B: Whitehead groups and K_1
• 19C: Steinberg groups and K_2
• 19D: Higher algebraic K-theory
• 19E: K-theory in geometry
• 19F: K-theory in number theory, see also 11R70, 11S70
• 19G: K-theory of forms, see also 11EXX
• 19J: Obstructions from topology
• 19K: K-theory and operator algebras See mainly 46L80, and also 46M20
• 19L: Topological K-theory, see also 55N15, 55R50, 55S25
• 19M05: 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.

Textbooks, reference works, and tutorials

"Reviews in K-theory, 1940--84", edited by Bruce A. Magurn, American Mathematical Society, Providence, 1985, 811 pp., ISBN 0-8218-0088-4

Online chapters of a book-in-progress [Charles Weibel]

Software and tables

Other web sites with this focus

• Preprint server (UIUC); also available there is subscription information for a K-theory mailing list.
• Here are the AMS and Goettingen resource pages for area 19.

Selected topics at this site

• Some simple computations in K-theory.
• How many isomorphism classes of vector bundles are on the spheres?
• Rings with isomorphic modules of different ranks (dimension shifting in K-theory)
• Family of division algebras over a field (plus Milnor conjecture, K-theory)