Scissors Congruence with Mixed Dimensions

Abstract

We introduce a Grothendieck group $E_n$ for bounded polytopes in ${\mathbb{R}}^n$. It differs from the usual Euclidean scissors congruence group in that lower-dimensional polytopes are not ignored. We also define an analogous group $L_n$ using germs of polytopes at a point, which is related to spherical scissors congruence. This provides a setting for a generalization of the Dehn invariant.

Replaces earlier version called "Total Scissors Congruence". The new version has new material relating this work to previous work of P. McMullen