Bott Periodicity for Fibred Cusp Operators
نویسندگان
چکیده
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 of the important steps is the description of the index of a Fredholm smoothing perturbation of the identity in terms of an associated K-class in K c (T ∗Y ). Thesis Supervisor: Richard B. Melrose Title: Professor of Mathematics
منابع مشابه
Index 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 کاملPseudodifferential Operators on Manifolds with Fibred Boundaries
Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ : ∂X −→ Y, and let x ∈ C∞(X) be a boundary defining function. This data fixes the space of ‘fibred cusp’ vector fields, consisting of those vector fields V on X satisfying V x = O(x2) and which are tangent to the fibres of φ; it is a Lie algebra and C∞(X) module. This Lie algebra is quantized to the ‘small calculu...
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 کاملBott 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...
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...
متن کامل