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A 15-vertex triangulation of the quaternionic projective plane. (English) Zbl 1425.57016

U. Brehm and W. Kühnel [Math. Ann. 294, No. 1, 167–193 (1992; Zbl 0734.57017)] found a triangulated 8-manifold \(M^8_{15}\) with 15 vertices which is not a sphere. They gave a heuristic argument supporting that \(M^8_{15}\) is homeomorphic with the quaternionic projective plane, but that was not a proof. From the decomposition into 9:6 vertices (8-simplex vs. boundary of 5-simplex) it was clear that it is a triangulated manifold “like a projective plane” in the sense by J. Eells jun. and N. H. Kuiper [Publ. Math., Inst. Hautes Étud. Sci. 14, 181–222 (1962; Zbl 0109.15701)] and L. Kramer [J. Differ. Geom. 64, No. 1, 1–55 (2003; Zbl 1068.57019)]. The Hirzebruch formula for the signature implies that it is sufficient to compute the first or second Pontryagin class but that was out of range for a long time, even though theoretically there was a formula by Gabrielov-Gelfand-Losik. But that could not practically be evaluated.
In the remarkable paper under review the author manages to compute the first Pontryagin class of \(M^8_{15}\) in a purely combinatorial way. This is based on previous work by A. A. Gaifullin [Izv. Math. 68, No. 5, 861–910 (2004; Zbl 1068.57022); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 5, 13–66 (2004)]. Roughly speaking Gaifullin expressed the first Pontryagin class in terms of bistellar moves between all 3-dimensional links in the triangulated 8-manifold. This defines cycles in the graph \(\Gamma_2\) consisting of all oriented 2-spheres as vertices and bistellar moves as edges. Since \(M^8_{15}\) has \(\binom{15}{5}= 3003\) 4-simplices (with more than 60 combinatorial types even after regarding the automorphism group) one has to examine as many 3-dimensional links. Ultimately the calculation depends on a computer algorithm. This in turn is based on the mathematical software BISTELLAR which is available from the homepage of Frank H. Lutz.

MSC:

57R20 Characteristic classes and numbers in differential topology
57Q15 Triangulating manifolds
52B70 Polyhedral manifolds
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57N65 Algebraic topology of manifolds

Software:

simpcomp; GAP
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References:

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