Higher Spherical Scissors Congruence I: Hopf Algebra

Abstract

In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_{\infty },E_1)$-Hopf algebra spectrum whose $\ddot{I}{€}_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups ${\tilde{K}}_{2n}\left({\mathcal{P}}_{O(2k+2)}^{S^{2k+1}}\right)$ are nonzero for all nonnegative integers $n$ and $k$.

v1: Comments welcome!