Reconstructing triangulations of 3-manifolds from their intersection matrix

TL;DR

This paper demonstrates that the intersection matrix uniquely determines the triangulation of a 3-manifold up to isomorphism, extending previous results from surfaces to three-dimensional manifolds.

Contribution

The authors prove that the intersection matrix fully characterizes 3-manifold triangulations, providing a new combinatorial invariant for 3-dimensional topology.

Findings

Intersection matrix determines 3-manifold triangulation up to isomorphism

Extension of surface results to 3-manifolds

Provides a combinatorial tool for 3D topology classification

Abstract

The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this work we show the intersection matrix is enough to determine the triangulation of a 3-manifold up to isomorphism.