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Cycles and relative cycles in analytic K-homology. (English) Zbl 0697.58050
This paper is a continuation of the authors’ work on elliptic operators and K-homology. Their intention is to produce K-cycles from elliptic operators on Riemannian manifolds with boundary and to calculate the boundary map of the exact sequence explicitly. After recalling the definitions of KK and \(K^*\)-groups of \(C^*\)-algebras or pairs of \(C^*\)-algebras and ideals a technical result is proved which helps verifying the axioms for cycles arising from elliptic differential operators with boundary conditions. The content of the second section is that an elliptic differential operator on a (not necessarily complete) Riemannian manifold M produces a cycle in \(KK(C_ 0(M),{\mathbb{C}})\) independent of the closure used. In the third section the authors construct cycles of \(K_ 0(M,\partial M)\) in the case of Riemannian manifolds with boundary and show that these are equivalent for any boundary condition using the result of section 2 and the isomorphism \(K_ 0(M,\partial M)\cong KK(C_ 0(M),{\mathbb{C}})\). In the fourth section they calculate the boundary \(\partial: K_ 0(M,\partial M)\to K_ 1(\partial M)\) and identify the boundary of a cycle arising from an elliptic operator with boundary conditions with the cycle defined by the Calderon projector of the boundary value problem and also the cycle produced by a certain differential operator on the boundary. Together with the independence from the boundary conditions result of section 3 this leads to relations in \(K_ 1(\partial M)\). In an example the index theorem of Boutet de Monvel is extended to a broader class of pseudoconvex domains. In the fifth section the boundary \(KK^ 1(C_ 0(M),{\mathbb{C}})\to KK^ 0(C(\partial M),{\mathbb{C}})\cong K_ 0(\partial M)\) is derived from the map already calculated via Bott periodicity. The boundary of cycles defined by Dirac operators on odd dimensional manifolds is identified by cycles produced by Dirac operators on the boundary. From this the cobordism invariance of the index is deduced. The authors conclude their paper with some remarks on how index theory should be understood and a discussion of nonlocal boundary conditions.
Reviewer: J.Eichhorn

58J20 Index theory and related fixed-point theorems on manifolds
58J32 Boundary value problems on manifolds
55N15 Topological \(K\)-theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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