Triangulations of Conal Manifolds

TL;DR

This paper explores how triangulations of conal manifolds encode causal and topological structures, characterizing when such manifolds can be triangulated based on properties of their cone bundles.

Contribution

It introduces a framework for triangulating conal manifolds and characterizes their triangulability through cone bundle properties.

Findings

Triangulations encode causal and topological structures.

Triangulability depends on fibrewise freeness of cone bundles.

Characterization of triangulability via cone bundle properties.

Abstract

Casual structure can take the form of cone bundles on a manifold, more general local preorders on a topological space, or simplicial orientations implicit in a simplicial set. This note takes a triangulation of a conal manifold M to mean an isomorphism between M and the locally preordered geometric realization of a simplicial set. Such triangulations completely encode causal and topological structure combinatorially. This note characterizes the triangulability of closed conal manifolds as the fibrewise freeness and generativity of the cone bundle.