Non-k-equal configuration and immersion spaces

Abstract

We say the configuration space is the space of configurations of n distinct, labeled points in R d . We can imposea non-k-equal condition on the configurations that no k points coincide. One closely related space is the kth-component of the little discs operad, it is the configuration space of n open, labeled, distinct discs in the unit disc. Similarly, we can impose a non-k-overlapping condition such that no k discs share a common point. The homology of the little discs operad and the homology of the non-k-equal configuration spaces have both been known for several decades. Dobrinskaya and Turchin gave a geometric description of homology of non-k-overlapping discs using the operadic interpretation that is extensively used throughout the first four chapters. The first two chapters of this dissertation give the needed background on operads, modules, and bimodules, in general, and then more details about the little discs operad. There is also information given about symmetric sequences, including the homology of the non-k-overlapping discs. This leads to the third chapter where we give an explicit formula to compute the traces (or characters) of the symmetric group action on the homology of non-k-equal configuration spaces. This yields a generating function of these characters that is called the Cycle Index Sum. The fourth chapter defines the operad of overlapping discs, which is a filtered operad. The culmination of this chapter is a theorem that gives a description of an element in homology of non-k-overlapping discs that occurs when braces are composed with braces. In the final chapter of this dissertation, we define a cosimplicial model for the limit of the Taylor tower associated to the homotopy fiber of non-k-equal spaces of immersions of the disc of dimension 1 into the disc of dimension n over the space of all immersions of the disc of dimension 1 into the disc of dimension n.