Year: 2024
Type: preprint
Abstract: In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1096251007, G1399780489, G1652504919, G205497501, SFB 1442: Geometrie: Deformationen und Rigidität, EXC 2044: Mathematik Münster: Dynamik – Geometrie – Struktur, G4130518032, Stability Patterns in the Homology of Moduli Spaces, G6938060324, G6985996716, Algebraic K-Theory in Fixed-Point Theory and Smooth Manifolds, FRG: Collaborative Research: Trace Methods and Applications for Cut-and-Paste K-Theory
OpenAlex ID: W4402698513