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

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.

### MSC:

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

Zbl 0734.57017

GAP
Full Text: