K-Theory of Vector Bundles, of Modules, and of Idempotents

2007 ◽  
pp. 45-54
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
Vector Bundles ◽  
K Theory

scholarly journals The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra

2019 ◽  
Vol 30 (11) ◽  
pp. 1950057 ◽  
Author(s):  
M. Izumi ◽  
T. Sogabe
Keyword(s):  
Automorphism Group ◽  
Vector Bundles ◽  
Group Structure ◽  
Cuntz Algebra ◽  
Classifying Space ◽  
Cuntz Algebras ◽  
K Theory ◽  
Continuous Fields

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra [Formula: see text] for finite [Formula: see text] in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite [Formula: see text] is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

scholarly journals Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

2013 ◽  
Vol 13 (1) ◽  
pp. 83-113 ◽  
Author(s):  
El-Kaïoum M. Moutuou
Keyword(s):  
Vector Bundles ◽  
Isomorphism Theorem ◽  
Locally Compact ◽  
Bott Periodicity ◽  
C Algebras ◽  
Thom Isomorphism ◽  
K Theory ◽  
Thom Isomorphism Theorem

AbstractWe develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

scholarly journals Topological phases: Isomorphism, homotopy and K-theory

2015 ◽  
Vol 12 (09) ◽  
pp. 1550098 ◽  
Author(s):  
Guo Chuan Thiang
Keyword(s):  
Topological Insulators ◽  
Vector Bundles ◽  
Equivalence Classes ◽  
Topological Phases ◽  
Winding Number ◽  
Symmetry Constraints ◽  
K Theory

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modeled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and K-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of K-theory.

scholarly journals Equivariant K-theory, groupoids and proper actions

2011 ◽  
Vol 9 (3) ◽  
pp. 475-501
Author(s):  
Jose Cantarero
Keyword(s):  
Vector Bundles ◽  
Cohomology Theory ◽  
Dimensional Vector ◽  
Lie Groupoids ◽  
Proper Actions ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Cw Complexes ◽  
K Theory ◽  
Completion Theorem

AbstractIn this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

scholarly journals A Thom-Porteous formula for connective K-theory using algebraic cobordism

2014 ◽  
Vol 14 (2) ◽  
pp. 343-369 ◽  
Author(s):  
Thomas Hudson
Keyword(s):  
Vector Bundles ◽  
Geometric Interpretation ◽  
Schubert Varieties ◽  
Chow Ring ◽  
Grothendieck Ring ◽  
Push Forward ◽  
Cobordism Ring ◽  
K Theory ◽  
Flag Bundle ◽  
Algebraic Cobordism

AbstractUnder the assumption that the base field k has characteristic 0, we prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double β-polynomials of Fomin and Kirillov by computing the fundamental classes of schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.

Massey products in K-theory

1970 ◽  
Vol 68 (2) ◽  
pp. 303-320 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Vector Bundles ◽  
Higher Order ◽  
Massey Products ◽  
Cohomology Operations ◽  
K Theory

The aim of this paper is to reformulate the work of Massey(6), and Spanier(9, 10), on higher order cohomology operations, in terms of vector bundles in such a way as to produce geometrically some higher order operations in K-theory, which we will call Massey products.

scholarly journals Orientations on 2-vector Bundles and Determinant Gerbes

2013 ◽  
Vol 113 (1) ◽  
pp. 63 ◽  
Author(s):  
Thomas Kragh
Keyword(s):  
Vector Bundles ◽  
Magnetic Monopole ◽  
Monoidal Categories ◽  
Continuous Map ◽  
K Theory ◽  
Definition Of

In a paper from 2009, a half magnetic monopole was discovered by Ausoni, Dundas, and Rognes. This describes an obstruction to the existence of a continuous map $K(ku) \to B(ku^*)$ with determinant like properties. This magnetic monopole is in fact an obstruction to the existence of a map from $K(ku)$ to $K(\mathsf{Z},3)$, which is a retract of the natural map $K(\mathsf{Z},3) \to K(ku)$; and any sensible definition of determinant like should produce such a retract. In this paper we describe this obstruction precisely using monoidal categories. By a result from 2011 by Baas, Dundas, Richter and Rognes $K(ku)$ classifies 2-vector bundles. We thus define the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. We use this to define an oriented K-theory of 2-vector bundles with a lift of the natural map from $K(\mathsf{Z},3)$. It is then possible to define a retraction of this map and since $K(\mathsf{Z},3)$ classifies complex gerbes we call this a determinant gerbe map.

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