K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature. The periodicity lifts to the classifying spaces and makes the representing spectrum KU of complex K-theory be an even periodic ring spectrum. In particular the 2-periodicity in the homotopy groups of the stable unitary group U = lim n U (n) U = \underset{\longrightarrow}{\lim}_n U(n) is thus a shadow of Bott periodicity. 3 Topological K-theory 5 Bott periodicity is a theorem about the matrix groups U(n) and O(n). More speci cally, it is about the limiting behaviour as n!1. For simplicity, we will focus A better way to think about Bott periodicity is to not look at U, but BU. To describe .

Bott periodicity k theory better

3 Topological K-theory 5 Bott periodicity is a theorem about the matrix groups U(n) and O(n). More speci cally, it is about the limiting behaviour as n!1. For simplicity, we will focus A better way to think about Bott periodicity is to not look at U, but BU. To describe . The periodicity lifts to the classifying spaces and makes the representing spectrum KU of complex K-theory be an even periodic ring spectrum. In particular the 2-periodicity in the homotopy groups of the stable unitary group U = lim n U (n) U = \underset{\longrightarrow}{\lim}_n U(n) is thus a shadow of Bott periodicity. Lecture K-theory, KO-theory, and James periodicity 2/20/15 1 Complex (topological) K-theory Let X be a CW-complex, and suppose X is nite. Let Vect(X) denote the set of isomorphism classes of complex vector bundles on X. The Whitney sum gives an operation on Vect(X). De nition K0(X) is the initial group receiving a map from Vect(X) which. K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature. 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 .Summary. This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim . by defining their representing spectra directly using Bott periodicity. . and a more involved computation of the homotopy groups says that πi(Z×BO) is 8- periodic theories are called real and complex topological K-theory. When you pass to the stable vector bundles (i.e. K-theory), you can The Bott periodicity in K-theory then takes the following form: (Hatcher p. The consequences of Bott periodicity for K-theory are explored, corresponding to Chapter 4 are significantly more involved in the real case. Summary. This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim.

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Differential K-theory and its Characters - D. Sullivan,James H. Simons - Лекториум, time: 2:07:05

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