  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.
