- CiteKey: “Zak12a” - Type: journalArticle - Title: “Scissors Congruence and K-theory,” - Author: “Zakharevich, Inna;” - Year: 2012 - Collections: “Talbot 2022,”
Scissors Congruence and K-theory
In this thesis we develop a version of classical scissors congruence theory from the perspective of algebraic K-theory. Classically, two polytopes in a manifold X are deﬁned to be scissors congruent if they can be decomposed into ﬁnite sets of pairwise-congruent polytopes. We generalize this notion to an abstract problem: given a set of objects and decomposition and congruence relations between them, when are two objects in the set scissors congruent? By packaging the scissors congruence information in a Waldhausen category we construct a spectrum whose homotopy groups include information about the scissors congruence problem. We then turn our attention to generalizing constructions from the classical case to these Waldhausen categories, and ﬁnd constructions for coﬁbers, suspensions, and products of scissors congruence problems.