Polytopal Subcomplexes and Homology Representations

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这项研究将扩展PI在最近的工作中发起的计划，即(A)研究高度对称多面体的CW子复合体的拓扑和计数性质，(B)了解这些多面体子复合体与其他具有相似拓扑和计数性质的对象之间的关系，以及(C)了解整个多面体的对称群对子复合体的同调群的诱导作用。解决这些问题的方法包括组合拓扑学、计数组合学和有限群表示论。GAP和KEZO软件将被用来辅助猜想的提出和结果的检验。自古以来，人们就知道有五个正则凸多面体：四面体、立方体、八面体、十二面体和二十面体。每个多面体都具有其所有面都属于同一类型的属性：立方体为正方形，十二面体为五边形，其他情况下为三角形。还有其他高度对称多面体的示例，其中有两种或两种以上类型的面。一个熟悉的例子是足球，它有12个五角形的面和20个六角形的面。这项拟议的研究考虑了具有一种以上类型的面的高维对称多面体(称为“多面体”)，并研究了当移除某一类型的面时这些对象的拓扑如何变化(形成“多面体亚复合体”)。在足球的例子中，如果移除六角形的面，则得到12个不连通的面片，如果移除五角形的面，则得到具有12个二维孔的二维表面。删除面后，对象将保留其所有原始对称性。拟议的研究还将考虑同调表示，它描述了对象的对称性和所产生的空洞之间的关系。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The proposed research will extend the program initiated by the PI in recent work, namely (a) to study topological and enumerative properties of CW subcomplexes of highly symmetric polytopes, (b) to understand the relationships between these polytopal subcomplexes and other objects with similar topological and enumerative properties, and (c) to understand the induced actions of the symmetry groups of the whole polytope on the homology groups of the subcomplex. The methods used to solve these problems include techniques from combinatorial topology, enumerative combinatorics and representation theory of finite groups. The software packages GAP and Kenzo will be used to assist in the formulation of conjectures and the checking of results.It has been known since antiquity that there are five regular convex polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Each polyhedron has the property that all of its faces are of the same type: squares in the case of the cube, pentagons in the case of the dodecahedron and triangles in the other cases. There are other examples of highly symmetric polyhedra in which there are two or more types of faces. A familiar example is a soccer ball, which has 12 pentagonal faces and 20 hexagonal faces. The proposed research considers higher dimensional symmetric polyhedra (called ``polytopes'') with more than one type of face, and studies how the topology of these objects changes when faces of a certain type are removed (to form a ``polytopal subcomplex''). In the soccer ball example, one obtains 12 disconnected patches if the hexagonal faces are removed, and a two-dimensional surface with 12 two-dimensional holes if the pentagonal faces are removed. After the faces are removed, the object retains all its original symmetry. The proposed research will also consider homology representations, which describe the relationship between the symmetries of the object and the resulting holes.