Combinatorial Topology of Quotients of Posets

Abstract

In this thesis we study the topology of quotients of posets. By the topology of a poset we mean the topology of its order complex called nerve in this thesis. An action of some group on a poset induces an action on its nerve.

The posets we consider are partition lattices of finite sets. It is well-known that the nerve of a partition lattice is homotopy equivalent to a wedge of spheres of equal dimension. The symmetric group acts on a partition lattice in a natural way. We consider quotients of such a nerve by subgroups of the symmetric group.

Especially we consider subgroups which fix at least one element. It turns out that quotients by such subgroups are also homotopy equivalent to wedges of spheres of equal dimension. Furthermore we consider sublattices of the partition lattice where certain block sizes are forbidden.

For the proofs we use Discrete Morse Theory as well as Equivariant Discrete Morse Theory. We use the notion of an acyclic matching. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory. There exists an adaption of Discrete Morse Theory to free chain complexes. In this thesis we develop an adaption for the equivariant case.