منابع مشابه
Bott 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 کاملThe Bott Periodicity Theorem
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 cla...
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 کاملBott Periodicity for Fibred Cusp Operators
In the framework of fibred cusp operators on a manifold X associated to a boundary fibration Φ : ∂X → Y , the homotopy groups of the space G−∞ Φ (X;E) of invertible smoothing perturbations of the identity are computed in terms of the K-theory of T ∗Y . It is shown that there is a periodicity, namely the odd and the even homotopy groups are isomorphic among themselves. To obtain this result, one...
متن کاملA Noncommutative Proof of Bott Periodicity
Bott periodicity in K-theory is a rather mysterious object. The classical proofs typically consist of showing that the unitary groups form an Ω-spectrum from which to get a cohomology theory; then showing that that theory is K-theory; and most formidably showing that U(n) is homotopic to U(n+ 2) for all n. However, Cuntz showed that Bott periodicity can be derived in a much simpler way if one r...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1980
ISSN: 0021-8693
DOI: 10.1016/0021-8693(80)90194-5