To cheer you up in difficult times 13: Triangulating real projective spaces with subexponentially many vertices

Wolfgang Kühnel

Today I want to talk about a new result in a newly arXived paper: A subexponential size $\mathbb RP^n$ by Karim Adiprasito, Sergey Avvakumov, and Roman Karasev. Sergey Avvakumov gave about it a great zoom seminar talk about the result in our combinatorics seminar. Here is the link.

The question is very simple: What is the smallest number of vertices required to triangulate the n-dimensional real projective space?

The short paper exhibits a triangulation with $\exp (\frac {1}{2} \sqrt n \log n)$ vertices.  This is the first construction that requires subexponential number of vertices in the dimension. The best lower bound by Pierre Arnoux and Alexis Marin (from 1999) are quadratic, so there is quite a way to go with the problem. I am thankful to Ryan Alweiss who was the first to tell me about the new result.

Reasons to care

Representing a topological space using simplicial complexes arose in the early days of algebraic topology, but there are certainly more “efficient” representations. What is the reason to to care specifically about representations via simplicial complexes? Here are my answers

1. Constructions of triangulated manifolds with few vertices are occasionally amazing mathematical objects.
2. Simplicial complexes arise in many combinatorial and algebraic contexts,
3. The study of face numbers of simplicial complexes with various topological properties is often related to deep questions in algebra, geometry, and topology,
4. We care also about other types of representation.

Kühnel and Lassmann’s complex projective plane with nine vertices and other miracles

You may all be familiar with the 6-vertex triangulation of the projective space. It is obtained by identifying opposite faces of the icosahedron. It is 2-neighborly, namely every pair of vertices span an edge of the triangulation. (It also played a role in my high dimensional extension of Cayley’s counting trees formula.) The existence of 2-neighborly triangulations of other closed surfaces is essentially the Heawood 1890 conjecture solved by Ringel and Young in the 1960s.

In 1980 Wolfgang Kühnel set out to construct a 3-neighborly triangulation of the complex projective plane with 9 vertices. The construction, achieved in a paper by Kühnel and Lassmann (who also proved uniqueness), and discussed in a paper of the same year by Kühnel and Banchof, is, in my view, among the most beautiful constructions in mathematics with additional hidden structures and many connections. Some motivation to the work came from the study of tight embeddings of smooth manifolds and Morse theory.

Ulrich Brehm and Kühnel proved that if a d-manifold has fewer than 3d/2+3 vertices then it is homeomorphic to a sphere, and if it has exactly 3(d/2)+3 vertices and is not a sphere, then d=2,4,8 or 16 and the manifold has a nondegenerate height function with exactly three critical points. The 6-vertex $\mathbb RP^2$ and the 9-vertex $\mathbbCP^2$ are examples for dimensions 2 and 4. We can expect further such examples for projective planes over the quaternions and octonions. Brehm and Kühnel constructed three 8-dimensional 5-neighborly “manifolds” which are not combinatorially isomorphic. It is conjectured but not known that they all triangulate the quaternionic projective plane.

The result of Adiprasito, Avvakumov, and Karasev as told by Sergey Avvakumov.

Here is a brief description of Sergey Avvakumov’s lecture. (As far as I could see the proof of their result is fully presented along with  3 other proofs for basic related results.)

06:00  The lecture starts. A simple inductive constructions of  a triangulation of $\mathbb R P^n$.

08:00  Equivalent statement: we seek a centrally symmetric triangulation of the n-sphere with diameter 3.

09:00 6-vertex triangulation of  $\mathbb R P^2$

11:00 The construction: triangulate the positive facets of the cross polytope and symmetrically the negative facet. Follow a recipe to triangulate the side facets. Crucial question: what is needed from the triangulation T of the positive facet. Answer: No edge between two disjoint faces.

23:00 Spherical interpretation and Delaunay triangulations of a certain configuration V

31:00   A sufficient combinatorial condition

39:00 Proof of sufficiency

54:00 The construction!

1:09:00 Counting the number of vertices completes the proof of the theorem.