Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory

Abstract

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras Abstract. We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C Ã -algebras.

The Gysin sequence for quantum lens spaces

“…It follows [7,Lem. 6.7] that K 1 (M(F, A)) = 0 and that the index map (5.5) is an isomorphism [7,Prop. 6.8].…”

“…Construction of the sequence. In order to lighten our notation and in a way which is consistent with the notation used in [7,8], we write…”

“…v * v = w * w, without changing their class in V (F, A)/∼. Having done so, an addition is defined [7,Lem. 3.3] (F, A)) are isomorphic as Abelian groups.…”

“…4.13]. It then follows from [7,Prop. 4.14] that the pair ( X, D) determines a class in the bivariant K-theory KK 0 (M(F, A), F ).…”

“…Following [7,Thm. 5.1] and writing E := X m , the internal Kasparov product of the Ktheory K 0 (M (F, A)) with the class of ( X, D) in the bivariant K-theory KK 0 (M(F, A), F ) is represented by the index…”