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+# Mildly Interesting Triangulations
+
+This repository contains triangulations in [polymake][polymake] JSON
+format of various closed, orientable 4-dimensional manifolds. The
+triangulations were obtained by conversion of [Regina][regina]
+isomorphism signatures taken from a [census of manifolds][lofano] 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][lutz-cp2] 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][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. 
+
+
+
+
+[polymake]: https://polymake.org/doku.php/start
+[regina]: https://regina-normal.github.io/
+[lofano]: https://github.com/davelofa/Census6Pentachora
+[lutz-cp2]: https://www3.math.tu-berlin.de/IfM/Nachrufe/Frank_Lutz/stellar/library_of_triangulations/CP2.txt
+[retriangulate]: https://regina-normal.github.io/docs/man-retriangulate.html