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Problems / geometry-topology

geometry-topology / Combinatorial topology; triangulations of manifolds

A 44-vertex triangulation of RP6\mathbb{RP}^{6}

Construct a simplicial triangulation of RP6\mathbb{RP}^{6} with fewer than 45 vertices, improving the 45-vertex construction of Guyer, Steinerberger and Yang.

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Acceptance criteria: Keep the existence result separate from vertex minimality; preserve the exact finite-certificate scope and the open lower-bound gap.

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geometry-topologySep 6, 2026Significance 58/100

A 44-vertex triangulation of RP6\mathbb{RP}^{6} reported

Prior state unknown→44-vertex RP6\mathbb{RP}^{6} construction source-authenticated; vertex minimum remains open

VibeMathed recorded: A 44-vertex triangulation of RP6\mathbb{RP}^{6} reported. Consult the linked registry entry for the original report and scope.

Source authenticatedA 44-vertex triangulation of RP6\mathbb{RP}^{6} →
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The supplied 44 integer vectors in Z7\mathbb{Z}^{7} and 7404 antipodal facet representatives define a centrally symmetric simplicial 7-polytope with 88 vertices whose antipodal boundary quotient is a simplicial triangulation of RP6\mathbb{RP}^{6} with ff-vector (44,938,7024,22555,34936,25914,7404)(44, 938, 7024, 22555, 34936, 25914, 7404). Thus 29≤ν(RP6)≤4429 \leq \nu(\mathbb{RP}^{6}) \leq 44 and the construction improves the prior 45-vertex bound; no claim that 44 is vertex-minimal is made.

The manuscript's exact supporting-hyperplane, ridge-incidence, antipodal-neighborhood, and quotient-face tests are the stated certificate; this draft does not independently rerun them.

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A 44-vertex triangulation of $\mathbb{RP}^{6}$ — Mathematical Frontier Network