Cubes and Generalized Real Bott Manifolds

TL;DR

This paper introduces facets-pairing structures on cubes and shows that the resulting seal spaces encompass all generalized real Bott manifolds, linking combinatorial cube structures to a class of toric manifolds.

Contribution

It defines facets-pairing structures on cubes and proves that their seal spaces include all generalized real Bott manifolds, establishing a new connection between combinatorics and toric topology.

Findings

Seal spaces of facets-pairing structures on cubes can produce all generalized real Bott manifolds.

A canonical facets-pairing structure $F_A$ is associated with any binary square matrix with zero diagonal.

The class of manifolds obtained from these seal spaces matches exactly the class of generalized real Bott manifolds.

Abstract

We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is a closed manifold. In particular, for any binary square matrix $A$ with zero diagonal in dimension n, there is a canonical facets-pairing structure $F_A$ on the n-dimensional cube. We will show that all the closed manifolds that we can obtain from the seal spaces of such $F_A$'s are neither more nor less than all the generalized real Bott manifolds --- a special class of real toric manifolds introduced by Choi, Masuda and Suh.