Simplicial complexes from finite projective planes and colored configurations

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Abstract

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as T1T2, such that each Ti represents the lines of a copy of the Fano plane PG(2,F2). We generalize this observation by constructing, for each prime power q, a simplicial complex Xq with q2+q+1 vertices and 2(q2+q+1) facets consisting of two copies of PG(2,Fq). Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring χ:E(G)[k], such that for all vV(G), a,b,c[k], following edges of colors a,b,c,a,b,c from v brings us back to v. We give one-to-one correspondences between (1) Sidon sets of order 2 and size k+1 in groups with order n, (2) linear codes with radius 1 and index n in the lattice Ak, and (3) colored (k+1)-configurations with n points and n lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.

Introduction

The torus T2 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 T2.

This triangulation X of T2 has several notable properties:

  • X has exactly 7 vertices. (In fact, X is vertex-minimal; any triangulation of T2 has at least 7 vertices; see [11], [16].)

  • X contains the Fano plane; the triangles pointing up,{0,1,3},{1,2,4},{2,3,5},{3,4,6},{4,5,0},{5,6,1},{6,0,2}, can be viewed as the lines of the Fano plane on points {0,,6} (see Fig. 1.2). (Similarly for the triangles pointing down.)

  • X is cyclic; the cyclic group Z7 acts on X by cyclically permuting labels.

  • X is 2-neighborly; that is, each pair of vertices in X form an edge.

The Fano plane PG(2,F2) is just one of a family of finite projective planes PG(2,Fq) for prime powers q. Hence we ask:

Question 1.1

Does the 7-vertex triangulation of T2, 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 (k+1)-configuration C (see Definition 2.8), and produces a simplicial complex X(C) with π1(X(C))Zk. The complex X(C) is not Tk in general, but can be made homeomorphic to Tk by adding vertices and faces.

For example, if C is PG(2,Fq), then we obtain the following:

Corollary 6.6

Let q be a prime power. Then there exists a connected q-dimensional simplicial complex Xq with π1(Xq)Zq, such that:

  • Xq has exactly q2+q+1 vertices.

  • Xq contains two copies of PG(2,Fq), each consisting of q2+q+1 facets of Xq. These two copies fully describe Xq, in that these 2(q2+q+1) facets are all of the facets of Xq, and every face of Xq is contained in a facet.

  • Xq is cyclic; the cyclic group Zq2+q+1 acts freely on Xq.

  • Xq is 2-neighborly.

In the general construction, the complex X(C) contains one copy of C, and one copy of its dual C, which is obtained from C by switching points and lines. Since PG(2,Fq) is isomorphic to its dual, in this case we obtain two copies of PG(2,Fq). See Section 6 for details.

We make no claim of vertex-minimality. However, we note that the smallest known triangulations of Tk use 2k+11 vertices ([15]; see [16]). Our complex Xq from Corollary 6.6 uses fewer vertices but lacks the full structure of Tq. We conjecture that Xq is vertex-minimal in the following sense:

Conjecture 9.4

Suppose X is a simplicial complex on n vertices, such that π1(X)Zk, and such that X admits a free Zn-action. Then nk2+k+1, 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 Ak={xZk+1:ixi=0}. 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:

  • (1)

    Pairs (G,B), where G is an abelian group with |G|=n, and B is a Sidon set of order 2 in G with |B|=k+1.

  • (2)

    Linear codes L with radius 1 in Ak, with |Ak/L|=n.

  • (3)

    Colored (k+1)-configurations C with n points and n lines.

The relationship between the first two structures above is known [14]; our contribution is to introduce the third. This raises the possibility of topological obstructions to the existence of Sidon sets and linear codes, via the simplicial complex X(C) given by Theorem 5.3.

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 C be a colored k-configuration which is also a projective plane. Then C 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 kN):

Definition 2.1

A k-configuration consists of finite sets P,L (whose elements are called “points” and “lines,” respectively) and an incidence relation RP×L, satisfying the following conditions:

  • (1)

    There do not exist distinct p1,p2P and distinct l1,l2L with (pi,lj)R for all i,j{1,2}. (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

[8]

A projective plane consists of sets P,L (whose elements are called “points” and “lines,” respectively), and an incidence relation RP×L, satisfying the following conditions:

  • (1)

    For distinct p1,p2P, there exists a unique lL

The lattice An and the simplicial complex Wn

We first introduce the lattice An, which is well known from the sphere packing literature (see [4]):

Definition 4.1

For n0, the lattice An is defined byAn={(x1,,xn+1)Zn+1:i=1n+1xi=0}.

For example, A2 is the hexagonal lattice, and A3 is the face-centered cubic lattice. Locally, An has the structure of the expanded simplex (see [5]). We may consider An a metric space by using a scaled 1-metric, d(x,y)=xy1/2. (This metric d can be viewed as a graph metric, where we consider x,y to be adjacent if x

Construction of simplicial complexes from colored k-configurations

We now give a construction taking a colored (k+1)-configuration and giving a labeling of the complex Wn defined in Section 4. After taking a quotient of Wn according to this labeling, we obtain a simplicial complex with properties analogous to the 7-vertex triangulation of T2 (Theorem 5.3).

Lemma 5.1

Let C be a colored (k+1)-configuration. Then there exists a surjective labeling :V(Wk)P(C) of the vertices of Wk, such that:

  • (1)

    If u,vV(Wk) are adjacent, then (u)(v).

  • (2)

    Given uV(Wk) and a point pP(C),

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 T2 (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 AG, such that for each gG 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

[12]; see [22]

A semifield consists of a set S and operations (+),(), such that:

  • S forms a group under addition; we call the additive identity 0.

  • If a,bS, a0, then there exists unique xS with ax=b.

  • If a,bS, a0,

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

[18]; see [14]

Let G be an abelian group, written additively. A set B={b0,b1,,bk}G is a Sidon set of order h if the sums bi1++bih, 0i1ihk, are all different.

For h=2, this condition is equivalent to the condition that the differences bibj for ij are all distinct, so |G|k2+k+1. Moreover:

Remark 8.2

Let G be an abelian group with |G|=k2+k+1. 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 C be a colored k-configuration which is also a projective plane. Then C admits a free cyclic group action.

The conjecture that all planar difference sets are Desarguesian becomes:

Conjecture 9.2

Let C be a colored k-configuration which is also a projective plane. Then C is Desarguesian (that is, isomorphic to PG(2,Fq)).

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 T2 consists of two copies of PG(2,Fq), and for many helpful conversations throughout.

References (22)

  • E. Reynaud

    Algebraic fundamental group and simplicial complexes

    J. Pure Appl. Algebra

    (2003)
  • T. Beth et al.

    Design Theory. Vol. I

    (1999)
  • R.C. Bose et al.

    Theorems in the additive theory of numbers

    Comment. Math. Helv.

    (1962/63)
  • R.H. Bruck et al.

    The nonexistence of certain finite projective planes

    Can. J. Math.

    (1949)
  • J.H. Conway et al.

    Sphere Packings, Lattices and Groups

    (1999)
  • H.S.M. Coxeter

    The derivation of Schoenberg's star-polytopes from Schoute's simplex nets

  • F. Frick et al.

    Vertex numbers of simplicial complexes with free abelian fundamental group

  • B. Grünbaum

    Configurations of Points and Lines

    (2009)
  • M. Hall

    Projective planes

    Trans. Am. Math. Soc.

    (1943)
  • A. Hatcher

    Algebraic Topology

    (2002)
  • Y. Huang et al.

    Uniqueness of some cyclic projective planes

    Des. Codes Cryptogr.

    (2009)
  • Cited by (0)

    Supported by NSF grant DMS 1855591.

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