Homotopic Nerve Complexes with Free Group Presentations

TL;DR

This paper explores homotopic nerve complexes in planar CW spaces, establishing their free group presentations and demonstrating that for convex set collections, the nerve and union share the same homotopy type.

Contribution

It introduces homotopic nerve complexes with free group presentations and proves their homotopy equivalence with unions of convex sets.

Findings

Homotopic vortex nerves have free group presentations.

Nerves of convex set collections are homotopy equivalent to their unions.

Abstract

This paper introduces homotopic nerve complexes in a planar Whitehead CW space and their Rotman free group presentations. Nerve complexes were introduced by P.S. Alexandrov during the 1930s and recently given a formal structure from a computational topology perspective by H. Edelsbrunner and J.L. Harer in 2010. A homotopic nerve results from the nonvoid intersection of a collection of homotopic 1-cycles. Briefly, a 1-cycle is a finite sequence of path-connected vertexes with no end vertex and with a nonvoid interior. A homotopic 1-cycle has the structure of a 1-cycle in a CW space in which cycle edges are replaced by homotopic maps. A group $G(V,+)$ containing a basis $\mathcal{B}$ is {\em free}, provided every member of $V$ can be written as a linear combination of elements (generators) of the basis $\mathcal{B}\subset V$. Let $\bigtriangleup$ be the members $v$ of $V$, each written as a linear combination of the basis elements of $\mathcal{B}$. A presentation of $G(V,+)$ is a mapping $\mathcal{B}\times \bigtriangleup\to G(\left\{v\in V:=sum_{k\in \mathbb{Z}\atop g\in \mathcal{B}}{kg}\right\},+)$. The main results in this paper are (1) Every homotopic vortex nerve has a free group presentation and (2) For a vortex nerve that consists of a finite collection of closed, convex sets in Euclidean space, the nerve and union of sets in the nerve have the same homotopy type.