Simplicial Polytope Complexes and Deloopings of $K$-theory

TL;DR

This paper advances the understanding of $K$-theory for polytope complexes by providing formulas for delooping and cofiber constructions, and relates scissors congruence groups to local geometric properties.

Contribution

It introduces formulas for delooping and cofiber of simplicial polytope complexes, extending previous work on their $K$-theory and linking scissors congruence groups to local geometry.

Findings

Formulas for delooping of simplicial polytope complexes

Formulas for cofiber of morphisms between such complexes

Scissors congruence groups determined by local properties

Abstract

This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of simplicial polytope complexes. Along the way we also prove that the (classical and higher) scissors congruence groups of polytopes in a homogeneous $n$-manifold (with sufficient geometric data) are determined by its local properties.