Homotopic Nerve Complexes with Free Group Presentations

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.