arXiv subjects
Publications and source records attributed to Byungdo Park.
We prove that any digraph Brown functor -- i.e. a contravariant functor from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom -- is representable. Furthermore, we show that the first path cohomology functor is a digraph Brown functor.
We prove that any contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying the additivity axiom and the Mayer-Vietoris axiom is representable.
Thurston's equations determine the hyperbolic structure of a 3-manifold with a triangulation. In work by Thistlethwaite and Tsvietkova, an alternative method was developed for link complements in $S^3$ depending on the link diagram, where a set of labels are associated to the vertices and edges of the link diagram, and one attempts to solve a set of equations on the labels. Under certain conditions, there exists a solution to these equations that corresponds to the complete hyperbolic structure, but in general it is difficult to determine which one it is. We generalize this method to 3-manifolds with a polyhedral decomposition, and show that solutions to the equations correspond to $PSL(2,\mathbb{C})$-representations of the fundamental group, and that the solution with the largest volume corresponds to the complete hyperbolic structure. We also consider different classes of complements of links, in particular links in the thickened torus and fully augmented links. For the latter, we establish a correspondence between solutions satisfying some criteria and circle packings realizing the region graph associated to the fully augmented link.
We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth Serre--Swan correspondence. Our construction subsumes the differential K-theory of a smooth manifold when the algebra is complex-valued smooth functions. Furthermore, our construction fits into a noncommutative differential cohomology hexagon diagram.
Let G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.
We introduce a smooth variant of the Hopkins-Singer model of differential K-theory. We prove that our model is naturally isomorphic to the Hopkins-Singer model and also to the Tradler-Wilson-Zeinalian model of differential K-theory.
This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. By differential twists we will mean smooth U(1)-gerbes with connection, and we use twisted vector bundles with connection as cocycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. This paper confirms a long-standing hypothetical idea that twisted vector bundles with connection define twisted differential K-theory.