complex K-theory

complex K-theory
Name	Complex K-theory
Field	Algebraic topology
Introduced	1960s
Founders	Michael Atiyah; Friedrich Hirzebruch
Related	Real K-theory; Topological K-theory; Algebraic K-theory

complex K-theory.

Complex K-theory is a generalized cohomology theory arising from vector bundles with complex structure on topological spaces, developed in foundational work by Michael Atiyah and Friedrich Hirzebruch and later extended by Raoul Bott, John Milnor, Daniel Quillen, Jean-Pierre Serre, and others. It plays a central role in index theory related to the Atiyah–Singer index theorem and interfaces with subjects studied by Alexander Grothendieck, Alain Connes, Maxim Kontsevich, Edward Witten, and Vladimir Voevodsky. Complex K-theory connects geometric topology investigated by William Browder and Dennis Sullivan with mathematical physics explored by Edward Witten and Nathan Seiberg.

Introduction

Complex K-theory originates from efforts by Hirzebruch to compute genera of manifolds and by Atiyah to formalize vector bundle invariants, influenced by interactions with Bott's computations on homotopy groups of Lie groups and Adams' work on stable homotopy theory. It is central to developments pursued at institutions such as the Institute for Advanced Study, Princeton University, University of Cambridge, University of Oxford, Massachusetts Institute of Technology, and University of Chicago, and it has informed research at laboratories like Bell Labs and CERN. Figures including Raoul Bott, John Milnor, J. Peter May, Haynes Miller, and Daniel Quillen contributed to its formal structure and computation techniques used in collaborations among groups at Harvard University, Stanford University, Columbia University, and École Normale Supérieure.

Definitions and Basic Properties

Complex K-theory assigns graded abelian groups to spaces via isomorphism classes of complex vector bundles, building on vocabulary developed by Serre, Grothendieck, and Samuel Eilenberg. Foundational expositions by Atiyah, Hirzebruch, and Bott formalized contravariant functors satisfying axioms related to exact sequences studied by Henri Cartan and Jean Leray. The theory uses tools from homotopy theory refined by J. H. C. Whitehead, George Whitehead, and Christopher T. C. Wall, and it interacts with spectral sequence techniques introduced by Jean Leray and John McCleary, as applied in computations by Mark Mahowald and H. F. Blichfeldt. The main groups, denoted K^0 and K^1, are linked via suspension isomorphisms studied by Frederick Adams and Ib Madsen.

Periodicity and Bott Periodicity

Bott periodicity, discovered by Raoul Bott, yields an isomorphism relating K-groups in degrees differing by two, reflecting structures observed in Lie groups like U(n) and SU(n) and in homogeneous spaces studied by Élie Cartan. The periodicity theorem is indispensable in applications examined by Michael Atiyah, Isadore Singer, and Jean-Pierre Serre and used in proofs by Daniel Quillen and Graeme Segal. Bott periodicity connects with work on orthogonal and unitary groups by Hermann Weyl, Élie Cartan, and Wilhelm Killing, and it underpins stable phenomena explored by J. F. Adams, George W. Whitehead, and Haynes Miller.

Cohomology Theory and Generalized Cohomology

As a generalized cohomology theory in the framework articulated by Eilenberg and Steenrod, complex K-theory fits into the landscape alongside cobordism theories developed by René Thom and bordism studied by Thom and Milnor. It interacts with extraordinary cohomology theories examined by Michael Boardman, J. Peter May, and Douglas Ravenel, and it is related to formal groups and complex orientations investigated by Serge Lang, John Tate, and Michel Lazard. Connections to elliptic cohomology pursued by Landweber, Hopkins, and Ando tie K-theory to modularity phenomena studied by Richard Borcherds and Yuri Manin and to modular forms studied by Don Zagier.

Computations and Examples

Classical computations include K-theory of spheres, projective spaces, Grassmannians, and Lie groups computed by Bott, Atiyah, and Hirzebruch, with methods involving Chern classes introduced by Shiing-Shen Chern and characteristic classes studied by René Thom. Calculations for complex projective spaces use results of Grothendieck and Serre and are applied in studies by Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne. Examples in operator algebras use work of Alain Connes and Joachim Cuntz, while applications to string theory and D-branes appeal to insights by Edward Witten, Juan Maldacena, and Joseph Polchinski. Computational frameworks employ Adams spectral sequence developed by J. Frank Adams and Brown–Peterson homology by Edgar Brown and Franklin Peterson.

Operations and Formal Group Laws

Adams operations ψ^k, introduced by J. Frank Adams, act on K-theory and relate to representation rings studied by Issai Schur and Hermann Weyl and to characters investigated by Frobenius and George Mackey. Lambda operations and exterior power operations connect to Schur functors used in work by William Fulton and Joe Harris. Formal group laws in complex-oriented theories relate to Lazard's classification by Michel Lazard, to algebraic geometry frameworks employed by Alexander Grothendieck and Pierre Deligne, and to the chromatic perspective advanced by Douglas Ravenel, Michael Hopkins, and Mark Hovey.

Variants and Related Theories

Variants include real K-theory KO developed by Atiyah and Bott, equivariant K-theory advanced by Segal and Atiyah, and algebraic K-theory initiated by Quillen and furthered by Daniel Quillen, Charles Weibel, and Friedhelm Waldhausen. Twisted K-theory studied by Freed and Hopkins links to work by Graeme Segal and Daniel Freed and has applications to conformal field theory explored by Edward Witten and Alexander Kapustin. Connections extend to cyclic cohomology developed by Alain Connes and Henri Moscovici, to topological modular forms investigated by Hopkins and Miller, and to motivic cohomology advanced by Vladimir Voevodsky and Marc Levine.

Category:Algebraic topology