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15-vertex triangulations of an 8-manifold. (English) Zbl 0734.57017

A combinatorial 8-manifold which is not PL homeomorphic to the sphere must have at least 15 vertices. Such an 8-manifold with 15 vertices must be a “manifold like the quaternionic projective plane” in the sense of J. Eells and N. H. Kuiper. In this paper we construct three such triangulations, the most symmetric one being invariant under a vertex transitive action of \(A_ 5\). Each of the three triangulations is 5- neighborly, i.e. it contains all \(15\choose 5\) 4-dimensional simplices. This induces a tight polyhedral embedding into \(E^{14}\) as a subcomplex of the 14-dimensional simplex. The three triangulations are PL homeomorphic to each other, we conjecture they are triangulations of the quaternionic projective plane.
Reviewer: W.Kühnel

MSC:

57Q99 PL-topology
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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