scholarly journals Equivariant $K$-theory and the Chern character for discrete groups

2012 ◽  
Vol 140 (3) ◽  
pp. 745-747
Author(s):  
Efton Park
Keyword(s):  
Chern Character ◽  
Discrete Groups ◽  
K Theory

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals Chern character for twisted K-theory of orbifolds

2006 ◽  
Vol 207 (2) ◽  
pp. 455-483 ◽  
Author(s):  
Jean-Louis Tu ◽  
Ping Xu
Keyword(s):  
Chern Character ◽  
K Theory

scholarly journals Chern Character in Twisted K -Theory: Equivariant and Holomorphic Cases

2003 ◽  
Vol 236 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Varghese Mathai ◽  
Danny Stevenson
Keyword(s):  
Chern Character ◽  
K Theory

CHERN CHARACTER FOR DISCRETE GROUPS

1988 ◽  
pp. 163-232 ◽  
Author(s):  
Paul Baum ◽  
Alain Connes
Keyword(s):  
Chern Character ◽  
Discrete Groups

On the β-Construction in K-Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 819-824
Author(s):  
C. M. Naylor
Keyword(s):  
Lie Group ◽  
Chern Character ◽  
Complex Representation ◽  
Group Homomorphism ◽  
Unique Element ◽  
Compact Lie Group ◽  
Fundamental Representation ◽  
K Theory ◽  
Simply Connected

The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ.

Cohomology for Generalized Bredon Coefficient Systems and Higher K-Theory

2012 ◽  
Vol 12 (1) ◽  
pp. 99-113
Author(s):  
Aderemi Kuku
Keyword(s):  
Commutative Ring ◽  
Cohomology Theory ◽  
Discrete Groups ◽  
Finitely Generated ◽  
Bredon Cohomology ◽  
K Theory

AbstractLet be a generalized based category (see definition 1.2). In this paper, we construct a cohomology theory in the category of contravariant functors: where R is a commutative ring with identity, which generalizes Bredon cohomology involving finite, profinite or discrete groups.We also study higher K-theory of the category of finitely generated projective objects in and the category of finitely generated objects in and obtain some finiteness and other results.

Intersection K-theory for isolated conical singularities

2011 ◽  
Vol 9 (1) ◽  
pp. 177-200
Author(s):  
André Legrand ◽  
David Poutriquet
Keyword(s):  
Chern Character ◽  
Intersection Cohomology ◽  
Conical Singularities ◽  
K Theory

AbstractStarting from the Karoubi multiplicative K-theory, we construct a Chern-Weil theory adapted to isolated conical singularities. The Chern character takes its values in the intersection cohomology of Goresky-MacPherson. We also propose an integer intersection K-theory for such singularities.

RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULAS

2009 ◽  
Vol 01 (03) ◽  
pp. 207-250 ◽  
Author(s):  
PIERRE ALBIN ◽  
RICHARD MELROSE
Keyword(s):  
Compact Manifold ◽  
Pseudodifferential Operators ◽  
Chern Character ◽  
Explicit Formulas ◽  
Manifold With Boundary ◽  
Index Bundle ◽  
Normal Operators ◽  
K Theory ◽  
Index Formulas ◽  
Character Mapping

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have "geometric K-theory", namely the "transmission algebra" introduced by Boutet de Monvel [5], the "zero algebra" introduced by Mazzeo in [9, 10] and the "scattering algebra" from [16], we give explicit formulas for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fiber operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

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