## 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

### Citations:

Zbl 0734.57017; Zbl 0109.15701; Zbl 1068.57019; Zbl 1068.57022

simpcomp; GAP
Full Text:

### References:

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