The surgery exact sequence, K-theory and the signature operator

Abstract

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.Comment: 29 pages, AMS-LaTeX; v2: small corrections and (hopefully) improved exposition, as suggested by the referee. Final version, to appear in Annals of K-Theor

Rho-classes, index theory and Stolz’ positive scalar curvature sequence

“…We have completed this program in [29], reproving the main results of [11][12][13] with purely operator theoretic methods. The corresponding general index theorems should be useful in other contexts, as well.…”

“…We use the calculus of pseudodifferential operators here just for convenience. In [29], we generalize the assertions to perturbations of Dirac type operators which are not necessarily pseudodifferential, replacing the pseudodifferential arguments by purely functional analytic ones.…”

Adiabatic groupoid and secondary invariants in K-theory

“…where µ f is the Morita equivalence between C * r (N × N ) and C * r (M × M ). The problem here concerns the construction of the dotted arrow in the following diagram Piazza and Schick in [36,37] in the setting of the coarse geometry and in this paper formalized and generalised to the context of Lie groupoids. In order to do it we need to use the b-groupoid…”

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In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of C * r (G • ad ) (where G • ad is the adiabatic deformation G restricted to the interval [0, 1)). Our construction directly generalises the cases treated in [36,37], in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps.

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C∗–algebraic higher signatures and an invariance theorem in codimension two

“…There is a large body of work that studies the homotopy invariance of signatures, and consequences, using signature operator methods rather than triangulations. For example the results of [9,10] are analyzed from this point of view in [19], using methods in part borrowed from [13,26]. It seems to be a challenge to prove Theorem 1.1, and in particular Theorem 2.12, using signature operator methods.…”