REGENTS OF THE UNIVERSITY OF COLORADO, THE
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.