Triangulations of Real Projective Space Combinatorial Approaches - - PowerPoint PPT Presentation

Triangulations of Real Projective Space

Combinatorial Approaches Sonia Balagopalan

Institute of Mathematics Hebrew University s.balagopalan@gmail.com

Plzeň, October 6, 2016

Equiangular Lines in Elliptic Space

A question of Seidel

Let Er−1 denote elliptic space of dimension r − 1. Think of this as the sphere Sr−1 with antipodal points identified. We say that a set of points in Er−1 is equilateral if the (absolute) cosines

• f the pairwise angles between them (as vectors in Rr) are all equal.

Problem [van Lint, Seidel (’65)]

Examples

Example 0: E1

1 ω2 ω 1 ω2 ω

Figure: The long diagonals of a regular hexagon.

Examples

Example 1: E2

1 2 2 1 3 4 4 3 5 6 6 5

Figure: The long diagonals of a regular icosahedron.

Motivation for our interest in this problem

Projective space

Real projective space of dimension d, RPd is defined as the sphere Sd with antipodal points identified. (x ∼ −x) We are interested in triangulations of projective space.

◮ An abstract simplicial complex K is a collection of finite sets closed

under taking subsets.

◮ Given a simplicial complex with vertex set V = v1, v2, v3, . . . , vn, we

can define its geometric realization via the map vi → ei ∈ Rd+1.

◮ A triangulation of RPd is a simplicial complex K whose geometric

realization is homeomorphic to RPd. A natural question: What are the “smallest” triangulations of RPd? Not much is known about this, or minimal triangulations in general.

Triangulations

A fun example

Consider the simplicial complex with facets consisting of two vertices each above and below the line of 5 disjoint copies of K2, such the no edge contributes more than one vertex. S3 on 10 vertices with automorphism group Sym(5) × C2. Vertex links are symmetric triangulations of the cube.

A general framework

for constructing triangulations of projective space

◮ Recall that RPd is the antipodal quotient of Sd. ◮ We try to construct triangulated spheres which have antipodal

quotients.

Antipodal sphere

Let T be a triangulated sphere. We say that T is antipodal if

◮ there is an involution σ on T with a fixed-point free action on the

vertices V (T). (We usually denote σ(x) = −x).

◮ the graph-distance between v and σ(v) is at least 3 for all

v ∈ V (T). The last condition allows us to take the quotient T/σ without introducing “double edges”. It is easy to see that the problem of finding triangulations of projective space is the same as that of finding antipodal spheres.

Examples of antipodal spheres

Example 0: E1

1 −ω2 ω −1 ω2 −ω

Figure: The hexagon

Examples

Example 1: E2

1 2 −2 −1 3 4 −4 −3 5 6 −6 −5

Figure: Faces of the icosahedron

Quotient is RP2

6.

Its facets are the blocks of the unique 2 − (6, 3, 2) design.

The general example

Barycentric subdivision of the simplex

Let ∆d denote the d-dimensional simplex with vertices v1, v2, . . . , vd+1. The (first) barycentric subdivision of ∆d is the complex obtained by subdividing ∂∆d at each of its proper faces.

v1 v2 v4 v3 v12 v14 v13 v24 v23 v34 v124 v123 v134 v234

◮ Antipodal (d − 1)-sphere ◮ Each vertex a nonempty, proper

subset of [d + 1]

◮ 2d+1 − 2 vertices ◮ Each facet ∼ an ordering of [d + 1] ◮ (d + 1)! facets

State of the art on triangulations of RPd

◮ The above construction uses n = 2d+1 − 1 points to triangulate

RPd.

◮ All known infinte families are exponential. ◮ Best known lower bound for RPd is n =

d+2

2

• + 1, d ≥ 3, due to

Arnoux, Marin (’91).

◮ We know better constructions for d = 4, 5

Better Constructions

Maximal equilateral point set in E5.

van Lint, Seidel: Consider a two-coloured 6-dimensional hypercube Q6, with vertices {±1}6. Let S be the colour-class {(x1, x2, . . . x6) ∈ V (Q6)|

6

• i=1

xi = −1}. S is an equilateral point set in E5 of size 16. We use this configuration to construct a 4-dimensional antipodal sphere.

Antipodal S4 on 32 vertices

◮ Let ˜

S be the lifting of S into R6, a set of 32 points on the standard S5.

◮ Write ˜

S as S1 ⊔ S3 ⊔ S5, where exactly i coordinates of Si are equal to −1.

◮ We have |S1| = |S5| = 6, |S3| = 20, these are orbits under the

Sym(6)-action permuting the coordinates.

◮ We need to project the points of ˜

S into R5, i.e.the hyperplane x = 0.

◮ Transform (−1, 1, 1, 1, 1, 1) ∈ S1 to (−5, 1, 1, 1, 1, 1), and

(1, −1, −1, −1, −1, −1) ∈ S5 to (5, −1, −1, −1, −1, −1).

◮ S3 is already in R5, but we could scale it by

√ 5 to lie in the sphere

• f radius

√ 30.

◮ Call the new vertex set V32. The boundary of the convex hull of V32

is an antipodal sphere of dimension 4, with automorphism group C2 × Sym(6).

◮ This can be quotiented to obtain a triangulated RP4 on 16 vertices.

Antipodal S4 on 32 vertices

Another approach

◮ Observe that V32 is a two-angle set. The graph G32 obtained by

joining points in V32 with positive dot product is the graph of our antipodal sphere.

◮ In fact, G32 is just the halved 6-cube graph, a double cover of K16. ◮ In this case, as well as for d = 1, 2, the sphere is a flag sphere and

can be obtained by taking the clique complex of the graph.

◮ This can be quotiented to obtain a triangulated RP4 on 16 vertices.

The object we just constructed satisfies the theoretical lower bound on the number of vertices. It was first discovered via computer search by F.H. Lutz (’99).

RP4

16

Remarks

◮ Sym(6) acts on RP4 16 intransitively on its 6 + 10 vertices. ◮ The vertex-orbit of size 10 corresponds to the set of bisections of 6

points into 3 + 3 points.

◮ The action of Aut(RP4 16) is a “total” Sym(6) action. ◮ RP4 16 “sits naturally in a neighbourhood of” the nicest 16-point

biplane and the 22-point Witt design.

One more example

Maximal equilateral point set in E6.

van Lint, Seidel: Let M the incidence matrix of the Fano plane.           1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1           Consider the set S with elements the rows of M, but with each entry 1 replaced with ±1. |S| = 56. Identifying x with −x gives an equilateral point set in E6

RP5

28

◮ The “clique trick” does not work here. We just get the graph of an

E7 polytope.

◮ We split S into four subsets by the number of coordinates equal to

−1.

◮ Do a similar transformation as above:

(1, 1, 1, 0, 0, 0, 0) → (20, 20, 20, −15, −15, −15, −15) and (1, −1, −1, 0, 0, 0, 0) → (24, −18, −18, 3, 3, 3, 3)

◮ Calculate the convex hull. We get an antipodal S7. ◮ RP5 28 is not optimal. Lutz has discovered a 24-vertex example. ◮ Nevertheless our object has better symmetry. Sym(7) acting on the

vertices and edges of K7, and just three orbits on the facets.

Open questions

◮ Does RP21 276 exist? ◮ Known quadratic equiangular line families (with quadratic number of

points) are too small. What is the right relaxation?

◮ Is it two-angle sets? k-angle sets for small k? ◮ Spherical designs??

Thank you!