A minimal triangulation of the quaternionic projective plane. (English) Zbl 1369.57027

Gorodkov, D. A.

Russ. Math. Surv. 71, No. 6, 1140-1142 (2016); translation from Usp. Mat. Nauk 71, No. 6, 159-160 (2016).

The problem of finding minimal triangulations of manifolds with respect to the number of vertices is a classical problem in combinatorial topology. The solution of this problem is known for a fairly small set of manifolds. The difficulty of the problem increases especially with the growth of the dimension. The manifolds \(\mathbb{RP}^2\), \(\mathbb{CP}^2\), \(\mathbb{HP}^2\) are some of the fundamental objects in geometry and topology in dimension 2, 4 and 8, respectively. Minimal triangulations of \(\mathbb{RP}^2\) and \(\mathbb{CP}^2\) are known, but the problem remained open for \(\mathbb{HP}^2\). In [Math. Ann. 294, No. 1, 167–193 (1992; Zbl 0734.57017)], U. Brehm and W. Kühnel constructed a minimal triangulation of an \(8\)-dimensional manifold and conjectured that the manifold is PL-homeomorphic to \(\mathbb{HP}^2\). In this article, the author proves this conjecture.

Reviewer: Biplab Basak (Ithaca)

MSC:

57R05	Triangulating
57R20	Characteristic classes and numbers in differential topology
05E45	Combinatorial aspects of simplicial complexes

Keywords:

quaternionic projective plane; triangulation; Morse function

Citations:

Zbl 0734.57017

Software:

GAP