Pseudospheres: combinatorics, topology and distributed systems

Abstract

Pseudospheres are simplicial complexes defined in the late 1990s to model some aspects of distributed systems. Since then, combinatorial properties of pseudospheres combined with topological properties have been very useful to derive distributed computability results. The goal of this paper is to study pseudospheres in more depth as mathematical objects and to give an overview of the properties that have been used in distributed computing. In this work we focus in combinatorial and topological aspects of pseudospheres. While doing so, the paper shows that these structures can be viewed from different perspectives, in addition to models of distributed computing. We show that the properties of pseudospheres that have been proved in distributed computing, as well as new ones, can be derived using combinatorial topology techniques and other combinatorial techniques taken from matroids and partial orders. A subclass of pseudospheres is related to universal bundles, and the Borsuk–Ulam theorem can be extended to apply to them.