Locally toroidal polytopes and modular linear groups

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d≥2, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, here we develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.     

The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n$n$-polytopes with the smallest number of flags are found, for every rank n>1$n>1$. With a few small exceptions, the smallest regular n$n$-polytopes come from a family of ‘tight’ polytopes with 2⋅4ⁿ⁻¹$2\cdot 4^{n-1}$ flags, one for each n$n$, with Schläfli symbol {4∣4∣?∣4}$\{4\mid 4\mid ?\mid 4\}$. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.     

Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP ₁ andP ₂, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP ₁ and whose vertex-figures are isomorphic toP ₂? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.     

Spectra of regular polytopes

In this paper we compute the spectra with multiplicities of the adjancency graphs of all regular polytopes in ⁿ : it turns out that all such eigenvalues are algebraic intergers of degree no greater than 3.Both authors receive continuous support from FINEP and CNPq, Brazil.     

Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

Locally projective polytopes of type

This paper attempts to classify the locally projective section regular n-polytopes of type {4,3,…,3,p}, that is, to classify polytopes whose facets are cubes or hemicubes, and the vertex figures are spherical or projective polytopes of type {3,…,3,p}, with the facets and vertex figures being not both spherical. Spherical or projective (n−1)-polytopes of type {3,…,3,p} only exist when p4, or p=5 and n−14, or n−1=2. However, some existence and non-existence results are obtained for other values of p and n. In particular, a link is derived between the existence of polytopes of certain types, and vertex-colourability of certain graphs. The main result of the paper is that locally projective section regular n-polytopes exist only when p=4, or when p=5 and n=4 or 5.     

Mixing regular convex polytopes

The mixing operation for abstract polytopes gives a natural way to construct a minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case.     

Vertex embeddings of regular polytopes

The question of when one regular polytope (finite, convex) embedds in the vertices of another, of the same dimension, leads to a fascinating interplay of geometry, combinatorics, and matrix theory, with further relations to number theory and algebraic topology. This mainly expository paper is an account of this subject, its history, and the principal results together with an outline of their proofs. The relationships with other branches of mathematics are also explained.     

Lattices compatible with regular polytopes

In a recent paper, Karpenkov has classified the lattice polytopes (that is, with vertices in the integer lattice ) which are regular with respect to those affinities which preserve the lattice. An alternative approach is adopted in this paper. For each regular polytope P in euclidean space , those lattices Λ are classified which are compatible with P, in the sense that some translate of Λ contains the vertices of P, and this translate is preserved by the symmetries of P.     

All regular polytopes are Ramsey

In this paper it is shown that all regular polytopes are Ramsey. In the course of this proof all convex quasi-regular polyhedra are proved to be Ramsey.     

Realizations of regular polytopes,IV

In earlier papers, a rich theory of geometric realizations of an abstract regular polytope has been built up. More recently, a product was described, to add to blending and scaling as a way of combining realizations. This paper introduces an inner product of cosine vectors of normalized realizations, and shows that it has certain orthogonality properties; together with induced cosine vectors, these provide powerful new tools for investigating realizations. The enhanced theory is illustrated by revisiting the realization domains of several polytopes, including the 24-cell and 600-cell.     

Realizations of regular polytopes, III

There is a comprehensive theory of geometric realizations of a finite abstract regular polytope ${\mathcal{P}}$ in euclidean spaces. Identifying congruent realizations of ${\mathcal{P}}$ , their space forms a pointed convex cone, with scalar multiplication and blending the operations which combine different realizations. In this paper, a new way to combine realizations is introduced, that of the (tensor) product. Just as blending corresponds to sums of representations of groups, so the product corresponds to the usual product of representations. A range of examples is given to illustrate the new theory. As well as being of intrinsic interest, some of these examples lead to extra insight into already known realization spaces; more importantly, the structure of the realization cone of the regular 600-cell is here determined for the first time.     

The largest projections of regular polytopes

The projections of the regularn-dimensional simplex and crosspolytope intoR ^k with the largestk-volume are determined here for the casesk=2,n≧2 andk=3, 4≦n≦6. The proofs involve a combination of exterior algebra and computer gradient methods.     

Extensions of dually bipartite regular polytopes

This paper addresses the problem of finding abstract regular polytopes with preassigned facets and preassigned last entry of the Schläfli symbol. Using C-group permutation representation (CPR) graphs we give a solution to this problem for dually bipartite regular polytopes when the last entry of the Schläfli symbol is even. This construction is related to a previous construction by Schulte that solves the problem when the entry of the Schläfli symbol is 6.     

Manifold structures on abstract regular polytopes

Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS ³, connected sums of handlesS ¹ × S ² , euclidean and spherical space forms, and other examples with non-trivial fundamental group.     

Finite polytopes have finite regular covers

We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.