The Bott Periodicity Theorem
نویسنده
چکیده
The Bott periodicity theorem is of fundamental importance in many areas of mathematics, from algebraic topology to functional analysis. It appears unexpectedly in different guises and I would like to explain some of these as well as the influence it has had on the development of different fields. I will concentrate on two roles that periodicity plays. First, periodicity allows one to deloop classifying spaces and thus define cohomology theories. Second, using periodicity, “wrong way” functoriality maps can be defined and these are of integral importance in the index theorem. Let me begin with the original statement of Bott, [5]. Consider the infinite versions of the matrix groups U = limn U(n), and GL(C) = limn GL(n,C) with the limit topology, as well as the real groups O and GL(R). Then Bott proved
منابع مشابه
A new proof of the Bott periodicity theorem
We give a simplification of the proof of the Bott periodicity theorem presented by Aguilar and Prieto. These methods are extended to provide a new proof of the real Bott periodicity theorem. The loop spaces of the groups O and U are identified by considering the fibers of explicit quasifibrations with contractible total spaces. 2002 Elsevier Science B.V. All rights reserved. AMS classificatio...
متن کاملBott periodicity
1 Description The Periodicity Theorem was proved by Raoul Bott over fifty years ago (cf. survey [3], [4], [9]) and quickly became one of the strongest tools in homotopy theory, topology of manifolds and global analysis. The original theorem asserted that homotopy groups of the linear groups GL(n,F) where F is the field of real, complex or quaternion numbers are periodic i.e. πi(GL(k,F) ' πi+nF(...
متن کاملM392c Notes: K-theory
Part 1. Vector Bundles and Bott Periodicity 2 1. Families of Vector Spaces and Vector Bundles: 8/27/15 2 2. Homotopies of Vector Bundles: 9/1/15 5 3. Abelian Group Completions and K(X): 9/3/15 8 4. Bott’s Theorem: 9/8/15 12 5. The K-theory of X × S2: 9/10/15 15 6. The K-theory of the Spheres: 9/15/15 18 7. Division Algebras Over R: 9/17/15 22 8. The Splitting Principle: 9/22/15 25 9. Flag Manif...
متن کاملErratum Addendum to “ A new proof of the Bott periodicity theorem ” [ Topology Appl . 119 ( 2002 ) 167 – 183 ] ✩
A. Elmendorf has found an error in the approach to Lemmas 2.2 and 2.3 of “A new proof of the Bott periodicity theorem” (Topology and its Applications, 2002, 167–183). There are also errors in the definitions of the maps in Sections 4.2 and 4.5. In this paper we supply corrections to these errors. We also sketch a major simplification of the argument proving real Bott periodicity, unifying the e...
متن کاملA proof of Bott periodicity via Clifford algebras
The purpose of this note is to present a proof of the Bott periodicity theorem that is based on the periodicity of Clifford algebras. Such a proof was first predicted in [2], and then constructed in [8] and in [3]. Here, we give another proof along the same lines as [8], but based on a different model of K-theory. In order to simplify the notation, we only present the periodicity for KO theory....
متن کامل