An Elementary Differential Extension of Odd K-theory

2012/11/19 by Thomas Tradler, Tradler, Thomas, Scott O. Wilson +3
Mathematics · #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AT #math.DG #math.KT

paper · pdf · doi:10.48550/arxiv.1211.4477

23 pages

arxiv created 2012/11/19 · arxiv updated 2012/11/20

Abstract

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Related

• Differential K-theory as equivalence classes of maps to Grassmannians and unitary groups
• A loop group extension of the odd Chern character
• Smooth classifying spaces for differential K-theory
• Chern classes on differential K-theory
• The Caloron Correspondence and Odd Differential K-theory