arXiv 0806.14882008-06math.COmath.AT

Polytopal complexes: maps, chain complexes and... necklaces

Frédéric Meunier

备注：: Presented at the TGGT 08 Conference, May 2008, Paris. The definition of a polytopal map has been modified

英文摘要：

The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Affine hom-complexes

For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom. This extends the corresponding facts from single polytopes, systematic study of which was initiated in [6,12]. Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomple…

2016-03-30math.ATmath.CO

Ehrhart f*-coefficients of polytopal complexes are non-negative integers

The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart $δ$-vector), which is the vector of coefficients of $L_P$ with respect to a certain binomial basis and which coincides with the $h$-vector of a regular unimodular triangulation of $P$ (if one exists). One important result by Stanley about $h^*$-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory …

2012-03-06math.CO

Angle sums on polytopes and polytopal complexes

We will study the angle sums of polytopes, listed in the $α$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $α$-vector are analogous to the Euler and Dehn-Sommerville relations on the f-vector. First we describe the spaces spanned by the the alpha-vector and the $α$-f-vectors of certain classes of polytopes. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations. This shows that the dimension of the affine span of…

2006-07-19math.COmath.MG

Bottom complexes

The bottom complex of a finite polyhedal pointed rational cone is the lattice polytopal complex of the compact faces of the convex hull of nonzero lattice points in the cone. The algebra, associated to the bottom complex of a cone, defines a flat deformation of the affine toric variety, associated to the polyhedral cone, set-theoretically. We describe three explicit infinite families of abstract polytopal complexes, defining such flat deformations scheme-theoretically.

2022-06-07math.ACmath.CO

An Ehrhart Theory For Tautological Intersection Numbers

We discover that tautological intersection numbers on $\bar{\mathcal{M}}_{g, n}$, the moduli space of stable genus $g$ curves with $n$ marked points, are evaluations of Ehrhart polynomials of partial polytopal complexes. In order to prove this, we realize the Virasoro constraints for tautological intersection numbers as a recursion for integer-valued polynomials. Then we apply a theorem of Breuer that classifies Ehrhart polynomials of partial polytopal complexes by the nonnegativity of their $f^*$-vector. In dimensions 1 and 2, we show that the polytopal complexes that arise are \emph{inside-o…

2022-09-28math.AGmath.CO

Illumination complexes, Δ-zonotopes, and the polyhedral curtain theorem

Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A particularly nice example arises if Q is a Δ-zonotope (generalized rhombic dodecahedron), known also as the dual of the difference body Δ - Δ of a simplex Δ, or the dual of the convex hull of the root system A_n. We demonstrate that the illumination complexes and their relatives can be used as 'configuration spaces', leading to new 'fair division theorems'. Among…

2013-07-19math.COmath.MG

更多相关论文收起更多论文

Connectivity of pseudomanifold graphs from an algebraic point of view

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings.

2014-11-11math.ACmath.CO