Generalizations of no k-equal spaces

We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the n^th no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of \N^m. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of \R^d. Next, we consider the relation between no $k$-equal spaces of $\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the n^th no $k$-equal space of $\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in \R^d.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Kosar, Nicholas J

Contributors dc:contributor

• Baryshnikov, Yuliy
• Hirani, Anil
• Schenck, Hal
• Yong, Alexander

• Polychromatic configuration spaces
• k-spanning trees
• discriminantal arrangements

Statement dc:rights

• Copyright 2018 Nicholas Kosar

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101142

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101142