Topological and Bivariant K-Theory pp 1-18 | Cite as

# The elementary algebra of K-theory

Chapter

## Abstract

Originally, K-theory was the study of vector bundles on topological spaces. But it was soon realised that the notion of vector bundle can be formulated more algebraically: Swan’s Theorem identifies the monoid of vector bundles over a compact space *X* with the monoid of finitely generated projective modules over the algebra *C*(*X*) of continuous functions on *X*; we take real- or complex-valued functions here to get real or complex vector bundles, respectively.

## Keywords

Abelian Group Vector Bundle Euler Characteristic Projective Module Invertible Element
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## Copyright information

© Birkhäuser Verlag AG 2007