Mildly Interesting Triangulations

This repository contains triangulations in polymake JSON format of various closed, orientable 4-dimensional manifolds. The triangulations were obtained by conversion of Regina isomorphism signatures taken from a census of manifolds into polymake files, and then simplifying them using edge_contraction and bistellar_simplification. For each isomporphism signature, the smallest triangulation by number of vertices was chosen, and then duplicates (combinatorially isomorphic triangulations) were removed.

Contents

The triangulations are sorted by topological/PL-type, which was determined by comparing them to a reference triangulation (or in the case of the topology-ℂP², by computing the intersection form). The triangulations are further sorted by number of vertices.

ℂP²

The triangulation with 9 vertices is combinatorially isomorphic to the minimal triangulation found in Frank H. Lutz's library of triangulations. The triangulations with a larger amount of vertices are triangulations which have been found to be PL-homeomorphic to ℂP² using Regina's retriangulate tool, but elude polymake's simplification capabilities.

ℂP² (top)

This category are four triangulations which have been shown to be homoeomorphic to ℂP² by computing the intersection form, however, it is unknown whether they are also PL-homoeomorphic to ℂP² as they elude all methods tried on them so far.

ℂP²#S³xS¹

These 1464 triangulations have all been shown to be PL-Homeomorphic to each other using retriangulate, however, polymake has big difficulties in finding the PL-homeomorphisms between them.

S⁴

These 1016 triangulations are triangulations of S⁴ shown to be PL-homeomorphic to S⁴ using the retriangulate tool, for which no other methods (bistellar simplification, discrete morse theory) worked.

S⁴-discretemorse

These 48 triangulations have been shown to be PL-homeomorphic to S⁴ by finding a spherical discrete morse vector.