منابع مشابه
Bott 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 کاملIndex in K-theory for Families of Fibred Cusp Operators
A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this...
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 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 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 Geometric Analysis
سال: 2005
ISSN: 1050-6926,1559-002X
DOI: 10.1007/bf02922250