UC Santa Barbara

The space of monic polynomials of degree $d$ with distinct roots is a well-known classifying space for the braid group on $d$ strands. Recently, Dougherty and McCammond have defined a compactification of this space using the Lyashko-Looijenga map, giving it a piece-wise Euclidean cell structure indexed by pairs of non-crossing partitions. The compactified space retracts onto another well-known classifying space of the braid group. Both the compactification and its spine are conjectured to be CAT(0).

In this dissertation, I investigate the subspaces and subcomplexes that result when attention is restricted to real polynomials. These subspaces have multiple components with well-known fundamental groups and I explicitly describe the corresponding subcomplexes for small values of $d$. The non-crossing partitions used to index the cells in the complexes for real polynomials belong two specific classes of non-crossing partition chains: reflection symmetric chains and palindromic chains. Although reflection symmetric non-crossing partitions have previously been defined, the streamlined techniques introduced here clarify many aspects of their structure. Palindromic non-crossing partition chains are defined here for the first time.