Coverings by open and closed hemispheres

TL;DR

This paper investigates the topological properties of nerves formed by open and closed hemispherical coverings of spheres, revealing their homotopy types and connections to graph spaces, with implications for topological data analysis and machine learning.

Contribution

It characterizes the homotopy types of nerves from hemispherical coverings and links them to graph spaces, providing new insights and applications in topological data analysis and machine learning.

Findings

Nerve of open hemispheres is homotopy equivalent to a sphere.

Nerve of closed hemispheres is a wedge of spheres with a M"obius invariant.

Connections established between nerve spaces and graph posets, with applications in ML.

Abstract

In this paper we study the nerves of two types of coverings of a sphere $S^{d-1}$: (1) coverings by open hemispheres; (2) antipodal coverings by closed hemispheres. In the first case, nerve theorem implies that the nerve is homotopy equivalent to $S^{d-1}$. In the second case, we prove that the nerve is homotopy equivalent to a wedge of $(2d-2)$-dimensional spheres. The number of wedge summands equals the M\"{o}bius invariant of the geometric lattice (or hyperplane arrangement) associated with the covering. This result explains some observed large-scale phenomena in topological data analysis. We review the particular case, when the coverings are centered in the root system $A_d$. In this case the nerve of the covering by open hemispheres is the space of directed acyclic graphs (DAGs), and the nerve of the covering by closed hemispheres is the space of non-strongly connected directed graphs. The homotopy types of these spaces were described by Bj\"{o}rner and Welker, and the incarnation of these spaces appeared independently as "the poset of orders" and "the poset of preorders" respectively in the works of Bouc. We study the space of DAGs in terms of Gale and combinatorial Alexander dualities, and propose how this space can be applied in automated machine learning.