Selected topics in combinatorial topology and geometry : Nested set complexes, equivariant trisp closure maps, rigid Rips complexes, and vector space partitions

This cumulative dissertation consists of four papers: Nested set complexes for posets and the Bier construction, Equivariant closure operators and trisp closure maps, Rigid Rips complexes and topological data analysis (joint work with Eva-Maria Feichtner and Dmitry Kozlov), and Some necessary conditions for vector space partitions (joint work with Olof Heden). Those got partially enhanced with some additional expository material. In the first part, we generalize the framework of combinatorial nested set complexes to the setting of posets, and demonstrate that some older proofs seamlessly generalize along, on the example of the topology of the Bier poset and the relationship between the complex of k-trees and the order complex of a certain subposet of the partition lattice. A trisp closure map is a compact certificate for collapsibility of a trisp onto a subtrisp. We discuss the relationship to closure operators (on posets) and the interaction of trisp closure maps with group operations on the trisp. We introduce Rigid Rips complexes as another filtration obtained from a finite metric space, which is sufficiently easy to calculate and has persistent homology provably different from the Vietoris-Rips filtration, showing features that are missed using the latter filtration. In the final part, we introduce a family of necessary conditions for the existence of a partition of a finite vector space into subspaces. We exploit these in the situation of a partition of a 2t-dimensional space with spaces of dimension at most t: We give bounds on the number of t-dimensional spaces in terms of the number and dimension of lower-dimensional spaces, and we remark on the relationship between the t-dimensional spaces in the partition and the same t-dimensional spaces seen as a partial t-spread. Two new constructions for vector space partitions are also given.