Abstract

Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. This can be done in Euclidean, spherical, or hyperbolic geometry. In the 2010s Zakharevich defined a “higher” version of scissors congruence, where we don’t just ask whether two polytopes are scissors congruent, but also about the space of scissors congruences between polytopes. Zakharevich’s definition is a form of algebraic K-theory.   Classically, Sah defined Hopf algebra and comodule structures on various scissors congruence groups. In this talk, I will focus on scissors congruence K-theory of spherical geometry. Specifically, I will discuss joint work with Josefien Kuijper, Cary Malkiewich, David Mehrle, and Thor Wittich, in which we upgrade Sah’s structure to a commutative Hopf algebra structure on the (reduced) spherical scissors congruence K-theory spectrum.