60 lines
2.4 KiB
Markdown
60 lines
2.4 KiB
Markdown
# Mildly Interesting Triangulations
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This repository contains triangulations in [polymake][polymake] JSON
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format of various closed, orientable 4-dimensional manifolds. The
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triangulations were obtained by conversion of [Regina][regina]
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isomorphism signatures taken from a [census of manifolds][lofano] into
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polymake files, and then simplifying them using `edge_contraction` and
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`bistellar_simplification`. For each isomporphism signature, the
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smallest triangulation by number of vertices was chosen, and then
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duplicates (combinatorially isomorphic triangulations) were removed.
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## Contents
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The triangulations are sorted by topological/PL-type, which was
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determined by comparing them to a reference triangulation (or in the
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case of the topology-ℂP², by computing the intersection form). The
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triangulations are further sorted by number of vertices.
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### ℂP²
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The triangulation with 9 vertices is combinatorially isomorphic to the
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[minimal triangulation][lutz-cp2] found in Frank H. Lutz's library of
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triangulations. The triangulations with a larger amount of vertices
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are triangulations which have been found to be PL-homeomorphic to ℂP²
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using Regina's [retriangulate][retriangulate] tool, but elude
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polymake's simplification capabilities.
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#### ℂP² (top)
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This category are four triangulations which have been shown to be
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homoeomorphic to ℂP² by computing the intersection form, however, it
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is unknown whether they are also PL-homoeomorphic to ℂP² as they elude
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all methods tried on them so far.
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### ℂP²#S³xS¹
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These 1464 triangulations have all been shown to be PL-Homeomorphic to
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each other using retriangulate, however, polymake has big difficulties
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in finding the PL-homeomorphisms between them.
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### S⁴
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These 1016 triangulations are triangulations of S⁴ shown to be
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PL-homeomorphic to S⁴ using the retriangulate tool, for which no other
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methods (bistellar simplification, discrete morse theory) worked.
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### S⁴-discretemorse
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These 48 triangulations have been shown to be PL-homeomorphic to S⁴ by
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finding a spherical discrete morse vector.
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[polymake]: https://polymake.org/doku.php/start
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[regina]: https://regina-normal.github.io/
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[lofano]: https://github.com/davelofa/Census6Pentachora
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[lutz-cp2]: https://www3.math.tu-berlin.de/IfM/Nachrufe/Frank_Lutz/stellar/library_of_triangulations/CP2.txt
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[retriangulate]: https://regina-normal.github.io/docs/man-retriangulate.html
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