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）资助的。拟议的研究将扩展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子复形的拓扑和枚举性质，（B）了解这些多面体子复形与其他具有类似拓扑和枚举性质的对象之间的关系，（c）了解整个多面体的对称群对子复形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、枚举组合学和有限群表示论。 将使用GAP和Kenzo软件包来协助制定图表和检查结果。自古以来，人们就知道有五种正凸多面体：四面体，立方体，八面体，十二面体和二十面体。 每个多面体都有一个属性，它的所有面都是相同的类型：正方形的情况下，立方体，五边形的情况下，十二面体和三角形的其他情况。 还有其他高度对称多面体的例子，其中有两种或多种类型的面。 一个熟悉的例子是足球，它有12个五边形面和20个六边形面。 拟议的研究考虑了具有多种类型面的高维对称多面体（称为“多面体”），并研究了当某种类型的面被移除时这些对象的拓扑结构如何变化（以形成“多面体子复合体”）。 在足球的例子中，如果六边形面被移除，则获得12个不连通的面片，如果五边形面被移除，则获得具有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源表示，它描述了对象的对称性和由此产生的洞之间的关系。

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.