de Rham theory and cocycles of cubical sets from smooth quandles

TL;DR

This paper develops a de Rham theory for cubical manifolds, explores the rational homotopy type of classifying spaces of smooth quandles, and links secondary characteristic classes to quandle cocycles.

Contribution

It introduces a de Rham framework for cubical manifolds and connects secondary characteristic classes to quandle cocycles, advancing the understanding of smooth quandles.

Findings

Established a de Rham theory for cubical manifolds

Analyzed the rational homotopy type of quandle classifying spaces

Linked secondary characteristic classes to quandle cocycles

Abstract

We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in \cite{Dup2,DK} produce cocycles of quandles.