Stratifications indexed by partitions and combinatorial models for homology
Abstract: We shall consider several topological spaces equipped with stratifications indexed by integer partitions. In each case we consider the problem of studying homology groups of strata. We shall first describe how to construct various models for computing these groups and then present the following applications:
1) Determining the homology of resonance-free orbit arrangements (with the help of general lexicographic shellability), thereby settling a conjecture of Bjorner for this special case;
2) A combinatorial reproof of Arnol'd theorem regarding the rational homology of the space of monic complex polynomials with at least q roots of multiplicity k;
3) A counterexample to a conjecture by Sundaram and Welker;
4) A computation of the homology groups of the space of hyperbolic polynomials with at least q roots of multiplicity k.