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Sep, 2019
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K-theory

K -theory appears in various fields in many different forms. Connections with physics, e.g. D -brane in string theory and Verlinde algebra , have been discovered recently.

See, for example, Atiyah’s “ K -theory Past and Present” [ Ati ] . Another very concise introductory text is Karoubi’s [ Kar ] , which seems to be a lecture note of talks at the Clay Mathematics Institute. There is a lecture note by Corti驕s [ Cor ] on relations among algebraic K -theory, operator algebraic K -theory, and topological K -theory.

One of the reasons of this ubiquitiy of K -theory might be that taking the Grothendieck group is a decategorification, an operation inverse to categorifications ,

One of the most important properties of topological K -theory is the Bott periodicity. Higher order periodicities are one of the central themes in stable homotopy theory .

References