Twisted $K$-theory and Poincaré duality

Tu, Jean-Louis

doi:10.1090/s0002-9947-08-04706-5

Cited by 21 publications

(26 citation statements)

References 15 publications

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“…Given a twisted-by-A K-cycle .M;'; / on X, then by applying Poincaré duality in twisted K-theory (Cf. [12] and [23]), It is routine to check that Á X is well-defined on K top j .X;A/ and is functorial in the following sense. Let f W Y !…”

Section: K-cycles For Twisted K-homologymentioning

confidence: 99%

K-cycles for twisted K-homology

2013

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We summarise the construction of geometric cycles and their use in describing the Kasparov K-homology of a CW-complex X . When Kasparov Khomology is twisted by a degree three element of theČech cohomology of X then there is a corresponding construction of twisted geometric cycles for the case where X is a smooth manifold however the method that was employed does not apply in the case of CW-complexes. In this article we propose a new approach to the construction of twisted geometric cycles for CW-complexes motivated by the study of D-branes in string theory.

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Section: K-cycles For Twisted K-homologymentioning

confidence: 99%

K-cycles for twisted K-homology

2013

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“…The algebra A H is not generally a PD algebra, but it is a member of the class D and a dual algebraÃ H may be constructed as follows [12,52]. Let Cliff(X) be the Clifford algebra bundle of the cotangent bundle of X with respect to a chosen riemannian metric.…”

Section: Twisted D-branesmentioning

confidence: 99%

Non-commutative correspondences, duality and D-branes in bivariant K-theory

Brodzki¹,

Mathai²,

Rosenberg³

et al. 2009

Advances in Theoretical and Mathematical Physics

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Abstract. We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C * -algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

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“…In [22] (see also [42]), a natural isomorphism, called the Poincaré duality in analytical twisted K-theory, KK.C; C c .X; P˛.K// Ő C c .X/ C c .X; Cliff.TX/// Š KK.C c .X; P ˛. K//; C/ is constructed using the Kasparov product with the weak dual-Dirac element associated to P˛.K/; see Definition 1.11 and Theorem 1.13 in [22] for details.…”

Section: Topological Index D Analytical Indexmentioning

confidence: 99%

Geometric cycles, index theory and twisted K-homology

Wang¹

2008

J. Noncommut. Geom.

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Abstract. We study twisted Spinc -manifolds over a paracompact Hausdorff space X with a twisting˛W X ! K.Z; 3/. We introduce the topological index and the analytical index on the bordism group of˛-twisted Spin c -manifolds over .X;˛/, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this article is to establish the equality between the topological index and the analytical index for closed smooth manifolds. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of .X;˛/ analogous to Baum-Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold .X; F / with a twisting˛W X ! K.Z; 3/, which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer families index theorem to twisted cases. (2000). 19K56, 55N22, 58J22. Mathematics Subject Classification

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Twisted $K$-theory and Poincaré duality

Abstract: Abstract. Using methods of KK-theory, we generalize Poincaré K-duality to the framework of twisted K-theory.

Cited by 21 publications

References 15 publications

K-cycles for twisted K-homology

K-cycles for twisted K-homology

Non-commutative correspondences, duality and D-branes in bivariant K-theory

Geometric cycles, index theory and twisted K-homology

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