An elementary differential extension of odd K-theory

An Elementary Differential Extension of Odd K-theory

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...

K-THEORY. An elementary introduction

The purpose of these notes is to give a feeling of “K-theory”, a new interdisciplinary subject within Mathematics. This theory was invented by Alexander Grothendieck 1[BS] in the 50’s in order to solve some difficult problems in Algebraic Geometry (the letter “K” comes from the German word “Klassen”, the mother tongue of Grothendieck. This idea of K-theory has invaded other parts of Mathematics...

Loop Differential K-theory

In this paper we introduce an equivariant extension of the ChernSimons form, associated to a path of connections on a bundle over a manifold M , to the free loop space LM , and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M , in much the same way that differential K-theory can be defined using the C...

K - theory of mapping class groups III : Odd torsion Luke

k-odd mean labeling of prism

‎a $(p‎,‎q)$ graph $g$ is said to have a $k$-odd mean‎ ‎labeling $(k ge 1)$ if there exists an injection $f‎ : ‎v‎ ‎to {0‎, ‎1‎, ‎2‎, ‎ldots‎, ‎2k‎ + ‎2q‎ - ‎3}$ such that the‎ ‎induced map $f^*$ defined on $e$ by $f^*(uv) =‎ ‎leftlceil frac{f(u)+f(v)}{2}rightrceil$ is a‎ ‎bijection from $e$ to ${2k - ‎‎‎1‎, ‎2k‎ + ‎1‎, ‎2k‎ + ‎3‎, ‎ldots‎, ‎2‎ ‎k‎ + ‎2q‎ - ‎3}$‎. ‎a graph that admits $k$...