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2022 On Fredholm boundary conditions on manifolds with corners I: Global corner cycles obstructions
Paulo Carrillo Rouse, Jean-Marie Lescure, Mario Velásquez
Ann. K-Theory 6(4): 607-628 (2022). DOI: 10.2140/akt.2021.6.607

Abstract

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner cycles.

Our main theorem is that, for any manifold with corners X of any codimension, there is a natural and explicit morphism K(𝒦(X))THpcn(X,) between the K-theory group of the algebra 𝒦(X) of b-compact operators for X and the periodic conormal homology group with rational coefficients, and that T is a rational isomorphism.

As shown by the first two authors in a previous paper, this computation implies that the rational groups Hevpcn(X,) provide an obstruction to the Fredholm perturbation property for compact connected manifolds with corners.

This paper differs from that previous paper, in which the problem is solved in low codimensions, in that here we overcome the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the K-theory groups of the algebra 𝒦(X).

Citation

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Paulo Carrillo Rouse. Jean-Marie Lescure. Mario Velásquez. "On Fredholm boundary conditions on manifolds with corners I: Global corner cycles obstructions." Ann. K-Theory 6 (4) 607 - 628, 2022. https://doi.org/10.2140/akt.2021.6.607

Information

Received: 24 October 2019; Revised: 22 February 2021; Accepted: 22 March 2021; Published: 2022
First available in Project Euclid: 8 May 2022

Digital Object Identifier: 10.2140/akt.2021.6.607

Subjects:
Primary: 19K56 , 55N15 , 58H05 , 58J22 , 58J40

Keywords: Index theory , ‎K-theory , Lie groupoids , Manifolds with corners

Rights: Copyright © 2022 Mathematical Sciences Publishers

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