  In this work we complement the description of the extreme rays of the $6
\times 6$ copositive cone with some topological structure. In a previous paper
we decomposed the set of extreme elements of this cone into a disjoint union of
pieces of algebraic varieties of different dimension. In this paper we link
this classification to the recently introduced combinatorial characteristic
called extended minimal zero support set. We determine those components which
are essential, i.e., which are not embedded in the boundary of other
components. This allows to drastically decrease the number of cases one has to
consider when investigating different properties of the $6 \times 6$ copositive
cone. As an application, we construct an example of a copositive $6 \times 6$
matrix with unit diagonal which does not belong to the Parrilo inner sum of
squares relaxation ${\cal K}^{(1)}_6$.
