From top left clockwise: Alexander Gaifullin, Denis Gorodkov, Ulrich Brehm, Wolfgang Kühnel
Here is the paper:
Abstract with annotation:
In 1987 Brehm and Kühnel showed that any combinatorial -manifold with less than vertices is PL homeomorphic to the sphere and any combinatorial -manifold with exactly vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for only.
This is a lovely theorem and the proof is based on Morse theoretic ideas, The study was motivated by the theory of tight embeddings of manifolds in differential geometry.
There exist a unique 6-vertex triangulation of ,
This is a wonderful object obtained by identifying opposite faces of an icosahedron.
a unique 9-vertex triangulation of ,
This is an amazing mathematical construction by Kühnel and Lassman from 1983.
and at least three 15-vertex triangulations of . (We mentioned it in this post.)
Those were constructed in 1992 by Brehm and Kühnel. Gorodkov proved in 2019 that the Brehm–Kühnel complexes are indeed PL homeomorphic to .
However, until now, the question of whether there exists a 27-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct 634 vertex-transitive 27-vertex combinatorial 16-manifolds like the octonionic projective plane. Four of them have symmetry group of order 351, and the other 630 have symmetry group of order 27. Further, we construct more than non-vertex-transitive 27-vertex combinatorial 16-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups , , and . We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane . Nevertheless, we have no proof of this fact so far.
(Actually one can even expect that in the Brehm-Kühnel theorems all triangulations of manifolds with vertices are PL projective planes over the Reals, Quaternions or Octanions.)
I heard about the new result from Wolfgang Kühnel who wrote me: “I’m quite surprised that an object with such a huge f-vector can be constructed at all.”
Diversion: Moore Graphs of girth five
The seek for a triangulation of a manifold like a projective plane with vertices (now completed by Gaifullin) reminded me of the situation of Moore graphs of girth five. A Moore graph is a regular -graph with diameter , and girth . (It must have vertices.)
The Hoffman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. There are known examples for degrees 2, 3 and 7, and the existence of a Moore graph of girth 5 and degree 57 is one of the great mysteries of mathematics.
We can ask, more generally, about -neighborly triangulations of manifolds, namely triangulations for which every vertices form a face. (Triangulations of -manifolds with vertices must be -neighborly.) For this is essentially the Heawood conjecture. Ringel and Youngs showed, that 2-neighborly triangulation exists for all surfaces except for the Klein bottle. In dimensions greater than 2 only a small number of examples are known. Beside the triangulations that we talked about above, I am aware of two additional examples: In dimension 4: 16-vertex K3 surface (Casella-Kühnel 2001); In dimension 6: 13-vertex triangulations of (Frank H.Lutz).
Neighborliness is related to upper bound conjectures (that comes in various strengths) proposed by Kühnel in 1993. Here below is a lecture by Kuhnel in a birthday conference honoring the Indian mathematician Basudeb Datta. Kühnel’s conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on $latex n$ vertices
was settled by Isabella Novik and Ed Swartz in 2009. Novik and Swartz (for the orientable case) and Satoshi Murai (for the general case) also proved some strong forms of Kühnel’s conjecture conditioned on the Lefschetz property for the links; the Lefschetz property was proved by Karim Adiprasito in his -conjecture paper. (Things are moving along nicely in the -conjecture front (for older posts see here and here); I hope to write about it soon and in the meanwhile, here are two posts (I,II) by Karim.)
For a wider view of upper bound theorems watch the following ICM22 lecture by Isabella Novik.
Polyhedral and more general complexes: my conjecture.
I also posed a related conjecture to the 1987 Brehm and Kühnel’s theorem. (Related to the 3ᵈ problem)
Conjecture: A polyhedral -manifold (more generally, a strongly regular CW manifold) in dimension d which is not a sphere has at least faces. (Including the empty face.)
We can expect that extremal cases are only in dimensions 2, 4, 8, and 16 and are projective plane like or even genuinly PL projective planes over the reals, complex, quaternions or octonions.