The lattice of periods of a group action and its topology

Abstract

In this thesis, we study the topology of the poset obtained by removing the greatest and least elements of lattice of periods of a group action. For a G-set X where G is a finite group, the lattice of periods is defined as the image of the map from the subgroup lattice of G to the partition lattice of X which sends a subgroup H of G to the partition of X whose blocks are the H-orbits of X. We study the homotopy type of the associated simplicial complex. When the group G belongs to one of the families dihedral group of order 2n , dihedral group of order 2p n where p is an odd prime, semi-dihedral group, or quaternion group and the set X is transitive, we find the homotopy type of the corresponding poset. If G is the dihedral group of order 2n or one of semidihedral and quaternion groups, we find that the homotopy type of the complex is either contractible or has the homotopy type of three points. In the case of dihedral group of order 2p n , the associated complex is either contractible or it has the homotopy type of p points or it has the homotopy type of p + 1 points.