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。本研究将扩展PI在最近的工作中所启动的项目，即(a)研究高度对称多体的CW子配合物的拓扑和枚举性质，(b)了解这些多体子配合物与具有类似拓扑和枚举性质的其他物体之间的关系，以及(c)了解整个多体的对称群对子配合物的同调群的诱导作用。用于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的表示理论。软件包GAP和Kenzo将用于协助制定推测和检查结果。自古以来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。每个多面体的所有面都是相同的类型：立方体是正方形，十二面体是五边形，其他情况下是三角形。还有其他高度对称多面体的例子，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个五边形面和20个六边形面。提出的研究考虑具有多种类型面的高维对称多面体（称为“多面体”），并研究当移除某种类型的面时这些对象的拓扑结构如何变化（形成“多面体亚复合体”）。在足球的例子中，如果去掉六边形的面，就会得到12个不相连的补丁，如果去掉五边形的面，就会得到一个有12个二维孔的二维表面。去除面后，物体保留其所有原始对称性。提出的研究还将考虑同调表征，它描述了物体的对称性和产生的孔之间的关系。

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.