Bott periodicity
نویسنده
چکیده
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(GL(k,F) where nC = 2, nR = nH = 8, and i k. Almost decade later Atiyah and Bott found a surprising generalization of the periodicity theorem in context of newly developed topological K–theory. Moreover, they gave conceptually a very different proof of it [1]. Since that time the theorem attracts interest of many mathematicians and several different proofs were given (e.g. [2], [5],[6]) shading new light on the classical theorem. The student is expected to analyze proofs of the Periodicity Theorem of her/his choice and describe them in modern terms.
منابع مشابه
Bott Periodicity in Topological, Algebraic and Hermitian K-theory
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 at completeness. For instance, the variants of Bott periodicity related to bivariant K-theory are described by Cuntz in this handbook. As another example, we don’t emphasize here the relation between motivic homotopy theory and Bott periodicit...
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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...
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