  For compact regions Omega in R^3 with generic smooth boundary B, we consider
geometric properties of Omega which lie midway between their topology and
geometry and can be summarized by the term "geometric complexity". The
"geometric complexity" of Omega is captured by its Blum medial axis M, which is
a Whitney stratified set whose local structure at each point is given by
specific standard local types.
  We classify the geometric complexity by giving a structure theorem for the
Blum medial axis M. We do so by first giving an algorithm for decomposing M
using the local types into "irreducible components" and then representing each
medial component as obtained by attaching surfaces with boundaries to 4--valent
graphs. The two stages are described by a two level extended graph structure.
The top level describes a simplified form of the attaching of the irreducible
medial components to each other, and the second level extended graph structure
for each irreducible component specifies how to construct the component.
  We further use the data associated to the extended graph structures to
compute topological invariants of Omega such as the homology and fundamental
group in terms of the singular invariants of M defined using the local standard
types and the extended graph structures. Using the classification, we
characterize contractible regions in terms of the extended graph structures and
the associated data.
