Non-homogeneous combinatorial manifolds

TL;DR

This paper extends the classical theory of combinatorial manifolds to non-homogeneous cases, showing many properties remain valid and introducing NH-factorization to relate PL-homeomorphic manifolds.

Contribution

It introduces NH-manifolds as a generalization of classical manifolds and establishes NH-factorization as a tool to relate PL-homeomorphic manifolds.

Findings

NH-manifolds are locally like Euclidean spaces of varying dimensions

Many properties of classical manifolds extend to NH-manifolds

PL-homeomorphic manifolds can be connected via NH-factorizations

Abstract

In this paper we extend the classical theory of combinatorial manifolds to the non-homogeneous setting. NH-manifolds are polyhedra which are locally like Euclidean spaces of varying dimensions. We show that many of the properties of classical manifolds remain valid in this wider context. NH-manifolds appear naturally when studying Pachner moves on (classical) manifolds. We introduce the notion of NH-factorization and prove that PL-homeomorphic manifolds are related by a finite sequence of NH-factorizations involving NH-manifolds.