Simplicial complexes from finite projective planes and colored configurations☆
Introduction
The torus has a 7-vertex triangulation, arising from Fig. 1.1. To see this, identify any two vertices with the same label, and identify any two edges whose ends have the same label. Combinatorially, this identification produces a simplicial complex on 7 vertices. Topologically, this identification is equivalent to first identifying the leftmost and rightmost edges to obtain a cylinder, and then identifying the top and bottom circles to obtain .
This triangulation X of has several notable properties:
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X has exactly 7 vertices. (In fact, X is vertex-minimal; any triangulation of has at least 7 vertices; see [11], [16].)
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X contains the Fano plane; the triangles pointing up, can be viewed as the lines of the Fano plane on points (see Fig. 1.2). (Similarly for the triangles pointing down.)
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X is cyclic; the cyclic group acts on X by cyclically permuting labels.
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X is 2-neighborly; that is, each pair of vertices in X form an edge.
The Fano plane is just one of a family of finite projective planes for prime powers q. Hence we ask:
Question 1.1 Does the 7-vertex triangulation of , along with its notable properties, generalize to all prime power dimensions q?
Our main result is a construction (Theorem 5.3), which takes as its input a colored -configuration (see Definition 2.8), and produces a simplicial complex with . The complex is not in general, but can be made homeomorphic to by adding vertices and faces.
For example, if is , then we obtain the following:
Corollary 6.6 Let q be a prime power. Then there exists a connected q-dimensional simplicial complex with , such that: has exactly vertices. contains two copies of , each consisting of facets of . These two copies fully describe , in that these facets are all of the facets of , and every face of is contained in a facet. is cyclic; the cyclic group acts freely on . is 2-neighborly.
In the general construction, the complex contains one copy of , and one copy of its dual , which is obtained from by switching points and lines. Since is isomorphic to its dual, in this case we obtain two copies of . See Section 6 for details.
We make no claim of vertex-minimality. However, we note that the smallest known triangulations of use vertices ([15]; see [16]). Our complex from Corollary 6.6 uses fewer vertices but lacks the full structure of . We conjecture that is vertex-minimal in the following sense:
Conjecture 9.4 Suppose X is a simplicial complex on n vertices, such that , and such that X admits a free -action. Then , with equality attainable only for prime powers k.
This conjecture (along with Theorem 6.4) implies that every cyclic planar difference set has prime power order, an open problem in design theory (see [1], Chapter VII). In this way, our work suggests possible topological obstructions to the existence of planar difference sets and finite projective planes.
Our construction is closely related to a construction of linear codes from Sidon sets (see [14]). In our construction, we assign labels to a k-dimensional lattice and then take a quotient according to that labeling. The labeling of the lattice can be viewed as a linear code in the lattice . This observation leads to the following correspondences: Theorem 8.5 We have one-to-one correspondences (up to isomorphism) between any two of the following three structures: Pairs , where G is an abelian group with , and B is a Sidon set of order 2 in G with . Linear codes with radius 1 in , with . Colored -configurations with n points and n lines.
We also obtain restatements of open problems on planar difference sets. For example, the conjecture that all planar difference sets are cyclic becomes:
Conjecture 9.1 Let be a colored k-configuration which is also a projective plane. Then admits a free cyclic group action.
Section snippets
Colored k-configurations
Following Grünbaum [7], we define a k-configuration as a certain kind of k-regular incidence structure (where ):
Definition 2.1 A k-configuration consists of finite sets (whose elements are called “points” and “lines,” respectively) and an incidence relation , satisfying the following conditions: There do not exist distinct and distinct with for all . (That is, no two points are on more than one common line; equivalently, no two lines contain more than one common
Finite projective planes
In our definition of k-configuration, condition (1) says that any two points lie on at most one common line, and vice versa. By replacing “at most one” with “exactly one,” we obtain a definition of a finite projective plane. We recall the usual definition of a projective plane:
Definition 3.1 A projective plane consists of sets (whose elements are called “points” and “lines,” respectively), and an incidence relation , satisfying the following conditions: For distinct , there exists a unique [8]
The lattice and the simplicial complex
We first introduce the lattice , which is well known from the sphere packing literature (see [4]): Definition 4.1 For , the lattice is defined by
For example, is the hexagonal lattice, and is the face-centered cubic lattice. Locally, has the structure of the expanded simplex (see [5]). We may consider a metric space by using a scaled -metric, . (This metric d can be viewed as a graph metric, where we consider to be adjacent if
Construction of simplicial complexes from colored k-configurations
We now give a construction taking a colored -configuration and giving a labeling of the complex defined in Section 4. After taking a quotient of according to this labeling, we obtain a simplicial complex with properties analogous to the 7-vertex triangulation of (Theorem 5.3).
Lemma 5.1 Let be a colored -configuration. Then there exists a surjective labeling of the vertices of , such that: If are adjacent, then . Given and a point ,
Construction of colored k-configurations from planar difference sets
In Theorem 5.3, we constructed simplicial complexes with specific properties from colored k-configurations. In this section, we give a construction of colored k-configurations from planar difference sets. As a special case, we will obtain our generalization of the 7-vertex triangulation of (Corollary 6.6).
We begin by defining planar difference sets (see [1]):
Definition 6.1 Let G be an abelian group. A planar difference set in G is a subset , such that for each other than the identity, there exist
Construction of colored k-configurations from commutative semifields
In this section, we give a construction of colored k-configurations from commutative semifields, giving another possible input to our simplicial complex construction (Theorem 5.3). A semifield has the properties of a field, except that multiplication is not required to be associative or commutative:
Definition 7.1 A semifield consists of a set S and operations , such that: S forms a group under addition; we call the additive identity 0. If , , then there exists unique with . If , , [12]; see [22]
Relationships with Sidon sets & linear codes
We now explain the connection between Sidon sets, linear codes, and colored k-configurations. To begin, we define a Sidon set:
Definition 8.1 Let G be an abelian group, written additively. A set is a Sidon set of order h if the sums , , are all different.[18]; see [14]
For , this condition is equivalent to the condition that the differences for are all distinct, so . Moreover:
Remark 8.2 Let G be an abelian group with . Then a planar difference set in G is exactly a
Directions for further research
By Corollary 8.6, we can restate open problems on planar difference sets. For example, the conjecture that all planar difference sets are cyclic becomes:
Conjecture 9.1 Let be a colored k-configuration which is also a projective plane. Then admits a free cyclic group action.
The conjecture that all planar difference sets are Desarguesian becomes:
Conjecture 9.2 Let be a colored k-configuration which is also a projective plane. Then is Desarguesian (that is, isomorphic to ).
More generally, we can ask the
Declaration of Competing Interest
No conflicts of interest.
Acknowledgements
We thank Florian Frick for showing the author what became the motivating observation of this work, that the 7-vertex triangulation of consists of two copies of , and for many helpful conversations throughout.
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Supported by NSF grant DMS 1855591.