A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1

Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-skeleton is the graph Qn, the n-fold graph Cartesian product of K2. These are the three families of Platonic polytopes. For any n-dimensional polytope A and nonnegative integer k, let A denote the k-skeleton of A; see, e.g., [2]. If L and K are polytopal l-complexes, appeared in Geombinatorics, XXXI(1)(2021) 5–9. let L|K mean that L factors K; that is, K and L are of equal dimension l and K is the union of subcomplexes L1, . , Lr, each isomorphic to L, such that every l-face of K is contained in exactly one of the Li. An l-complex is even if each (l−1)-face is in a positive even number of l-faces. For X ∈ {∆,O,Q} we study the factorization of K = X n by L = X l l+1 for l ≤ n−1. We call L the canonical l-sphere as it is the boundary of a type X polytope of dimension l+1. For l ≥ 1, such factorizations always exist for O as the proof of the first theorem below shows explicitly. Observe, however, that these canonical spheres might not be composed of the minimum number of l-faces. Indeed, by Theorem 1 below, the 1-skeleton of O3 is factored into three 4-cycles, rather than four 3-cycles. Clearly, X n even is a necessary condition for a factorization into spheres. The l-skeleton of an n-simplex (or an n-cube) is an even complex if and only if n−l+1 (the number of l-faces containing an (l−1)-face) is positive and even; for the cross-polytope On, the l-skeleton is even for all 1 ≤ l < n. Evenness is automatic for O n as each (l−1)-face is a simplex and is part of two l-faces (also simplexes) one for each vertex in a copy of K2. The following extends [5] for l = 2; cf. Spreer [13], which partitions O n into face-disjoint surfaces that contain the 1-skeleton O n; such surfaces are called 1-Hamiltonian. Observe that our result does not depend on [11]. Theorem 1. If 1 ≤ l < n, then O l+1 | O l n. Proof. For each k ≥ 1, ∂Ok is the iterated topological join of k copies of the 2-point complex K2, which is a (k−1)-sphere. The l-skeleton of On is the l-cell-disjoint union of the ( n l+1 ) copies of O l+1 formed by the iterated join of each (l+1)-element subset of the set nK2. For the simplex, we use the theory of Steiner-type configurations (v, k, l), which consist of a v-element set X endowed with a family of k-element subsets of X (called blocks) such that each l-subset of X is in exactly one block. The case k = l+1 will be what we apply to factor ∆ v−1 by ∆ l−1 l . The translation from design to combinatorial factorization works for special cases, such as the Steiner triple systems as we noted in [6]. But by Keevash [11] (see also Gowers [4] and Kalai [10]), this sort of factorization must occur almost always for the l-skeleta of simplexes and, hence, of cubes, as we shall explain. Write a | b to mean b is an integer multiple of a. Define the divisibility set (e.g., [8]) to be the set of all feasible values