Polyhedral Surfaces in Wedge Products

TL;DR

This paper introduces the wedge product of polytopes, explores its combinatorial and geometric properties, and demonstrates how it can generate complex polyhedral surfaces with many moduli, including realizations in three-dimensional space.

Contribution

It defines the wedge product of polytopes, links it to known constructions, and shows how to realize complex surfaces and their duals in R^3 using projections and duality theory.

Findings

Wedge products of polygons with simplices contain interesting regular polyhedral surfaces.

The construction provides polyhedral subdivisions and many local deformations (moduli) for surfaces in R^3.

Realizations of surfaces and their duals in R^3 are achieved via projections and duality of 4-polytopes.

Abstract

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces ``of unusually large genus'' that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via ``projections of deformed wedge products'' we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R^3. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (``moduli'') for the surfaces in R^3. In order to prove that there are many moduli, we introduce the concept of ``affine support sets'' in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R^3 via dual 4-polytopes.