This paper introduces good covers for cell complexes in the form of what are known as path nerve complexes in a planar Whitehead CW space together with their Rotman free group presentations. A path is a mapping
h
:
[
0
,
1
]
→
K
h:[0,1]\to K
over a space
K
K
. Paths provide the backbone of homotopy theory introduced by J.H.C. Whitehead during the 1930s. A path nerve complex is a collection of sets of path-connected points (path images) that have nonempty intersection.
A space
X
X
has a good cover, provided
X
=
⋃
E
X=\bigcup E
for subsets
E
E
in
X
X
and the space is contractible. This form of good cover was introduced by K. Tanaka in 2021. The focus here is on intersecting path cycles (sequences of paths attached to each other in vortexes with nonempty interiors) that form a path nerve. A path nerve results from the nonvoid intersection of a collection of path cycles. The geometric realization of a path cycle is a 1-cycle. A 1-cycle is a finite sequence of path-connected 0-cells (vertexes) with no end vertex and with a nonvoid interior. A 1-cycle has the structure of a path cycle in which sequences of paths are replaced by edges.
A group
G
(
V
,
+
)
G(V,+)
containing a basis
B
\mathcal {B}
is free, provided every member of
V
V
can be written as a linear combination of elements (generators) of the basis
B
⊂
V
\mathcal {B}\subset V
. Let
△
\bigtriangleup
be the members
v
v
of
V
V
, each written as a linear combination of the basis elements of
B
\mathcal {B}
. A presentation of
G
(
V
,
+
)
G(V,+)
is a mapping
B
×
△→
G
(
{
v
∈
V
:=
∑
k
∈
Z
g
∈
B
k
g
}
,
+
)
\mathcal {B}\times \bigtriangleup \to G(\left \{v\in V:=\sum _{\substack {k\in \mathbb {Z}\\ g\in \mathcal {B}}}{kg}\right \},+)
. This form of presentation of structures was introduced by J.J. Rotman during the 1960s as part of a study of groups. The main results in this paper are (1) Every path triangle cluster has a free group presentation, (2) Every path triangle cluster has a free group presentation, (3) Every path vortex has a free group presentation, (4) Every path vortex nerve has a free group presentation, (5) A vortex nerve and the union of the sets in the nerve have the same homotopy type and (6) Every path triangulaton of a cell complex has a good cover.