The K-theory of assemblers

Inna Zakharevich
2017 Advances in Mathematics  
In this paper we introduce the notion of an assembler, which formally encodes "cutting and pasting" data. An assembler has an associated K-theory spectrum, in which π 0 is the free abelian group of objects of the assembler modulo the cutting and pasting relations, and in which the higher homotopy groups encode further geometric invariants. The goal of this paper is to prove structural theorems about this K-theory spectrum, including analogs of Quillen's localization and dévissage theorems. We
more » ... monstrate the uses of these theorems by analyzing the assembler associated to the Grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes. In this paper, however, we concern ourselves with more structural questions. It turns out that assemblers fall into a sweet spot in the definition of algebraic K-theory similar to the sweet spot found by Quillen [16]. When Waldhausen [20] developed a framework for the algebraic K-theory of spaces he had to discard many of the advantages of Quillen's exact categories; in particular, while he 35 had an analog of Quillen's Localization Theorem ([16, Theorem 5], [20, Proposition 1.5.5]) he did not have an analog of Quillen's Dévissage [16, Theorem 4]. This makes computations and analysis using Waldhausen's approach much more difficult than Quillen's. The approach used in this paper, while much more analogous to Waldhausen's combinatorial approach than Quillen's algebraic one, 40 also has both localization and dévissage theorems. Theorem B (Dévissage). Let C be an assembler and D a full subassembler. If for every object A ∈ C there exists a finite disjoint covering family {D i A} i∈I such that D i ∈ D for all i ∈ I then the induced map K(D) K(C) is an equivalence of spectra. 45
doi:10.1016/j.aim.2016.08.045 fatcat:fprl3r6s4bdyljh3qm2bfja3kq