## Structured Vector Bundles Define Differential K-Theory (810)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Simons810structuredvector,

author = {James Simons and Dennis Sullivan},

title = {Structured Vector Bundles Define Differential K-Theory},

year = {810}

}

### OpenURL

### Abstract

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring ˆ K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f. Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.

### Citations

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(Show Context)
Citation Context ...ll be called flat if some ∇ ∈ {∇} is flat. Since any two such of dim n are isomorphic, we shall denote this isomorphism class by [n] ∈ Struct(X). The following theorem is based on a related result in =-=[11]-=-, stated without giving the proof. We employ that proof here in Lemma 1.16 below. Theorem 1.15: Given any V ∈ Struct(X) there is a W ∈ Struct(X) such that V ⊕ W = [n] for some n. Any such W will be ca... |

85 |
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(Show Context)
Citation Context ...re ˆ K or ˆ KR uniquely determined by the diagram, as shown in [7] in the case of ordinary differential cohomology? 2. Can one enrich the families index theorem by passing from K to ˆ K or ˆ KR? c.f. =-=[3]-=-, [4], [6]. Finally, this model of ˆ K or ˆ KR might be helpful for certain quantum theories and M-theory, in which it has already been observed that actions can be written more appropriately in the l... |

79 |
Homologie cyclique et K-theorie. Asterisque 149
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(Show Context)
Citation Context ...K is measured by a complex torus, the dimension of which is the sum of the odd Betti numbers of the base manifold. In showing that the kernel of ch is K(C/Z) we were influenced by the work of Karoubi =-=[2]-=- and Lott [1], which gave a related description of K(C/Z) involving a bundle with connection and an extra total odd form whose d is the Chern character form. Our proof is based on the characterization... |

32 | R/Z index theory
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(Show Context)
Citation Context ... by a complex torus, the dimension of which is the sum of the odd Betti numbers of the base manifold. In showing that the kernel of ch is K(C/Z) we were influenced by the work of Karoubi [2] and Lott =-=[1]-=-, which gave a related description of K(C/Z) involving a bundle with connection and an extra total odd form whose d is the Chern character form. Our proof is based on the characterization given in App... |

14 |
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(Show Context)
Citation Context ...f: ker(ch) of a point is zero. ker(ch) is a homotopy functor satisfying Mayer-Vietoris by Corollary 3.10. ker(ch) also sends finite disjoint unions to finite products. It follows from Brown’s theorem =-=[10]-=- that ker(ch) on compact manifolds with corners has a classifying space which we denote GL(C/Z). � Proposition 4.3: map GL(C/Z) is homotopy equivalent to the homotopy fibre of the Chern character BGL ... |

7 |
Quadratic functions in geometry
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(Show Context)
Citation Context ...ith connection. Introduction This paper grew out of the effort to construct a simple geometric model for differential K-theory, the fibre product of usual K-theory with closed differential forms, [4],=-=[5]-=-,[6]. The model which finally emerged also fulfilled our long standing wish for a simple and straightforward codification of complex vector bundles with connection. Considering pairs of connections wh... |

6 | Axiomatic Characterization of Ordinary Differential Cohomology
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(Show Context)
Citation Context ...ue up to adding factors with trivial holonomy. Our model of ˆ K or ˆ KR may relate to two questions: 1. Up to a natural transformation, are ˆ K or ˆ KR uniquely determined by the diagram, as shown in =-=[7]-=- in the case of ordinary differential cohomology? 2. Can one enrich the families index theorem by passing from K to ˆ K or ˆ KR? c.f. [3], [4], [6]. Finally, this model of ˆ K or ˆ KR might be helpful... |

2 |
Pions and generalized cohomology,” arXiv:hep-th/0607134
- Freed
(Show Context)
Citation Context ...connection. Introduction This paper grew out of the effort to construct a simple geometric model for differential K-theory, the fibre product of usual K-theory with closed differential forms, [4],[5],=-=[6]-=-. The model which finally emerged also fulfilled our long standing wish for a simple and straightforward codification of complex vector bundles with connection. Considering pairs of connections whose ... |

1 |
Smooth K-Theory”. arXiv : 0707.0046
- Bunke, Schick
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(Show Context)
Citation Context ...es with connection. Introduction This paper grew out of the effort to construct a simple geometric model for differential K-theory, the fibre product of usual K-theory with closed differential forms, =-=[4]-=-,[5],[6]. The model which finally emerged also fulfilled our long standing wish for a simple and straightforward codification of complex vector bundles with connection. Considering pairs of connection... |