Alpha Shape Topology of the Cosmic Web
  We study the topology of the Megaparsec Cosmic Web on the basis of the Alpha
Shapes of the galaxy distribution. The simplicial complexes of the alpha shapes
are used to determine the set of Betti numbers ($\beta_{\rm k},k=1,...,D$),
which represent a complete characterization of the topology of a manifold. This
forms a useful extension of the geometry and topology of the galaxy
distribution by Minkowski functionals, of which three specify the geometrical
structure of surfaces and one, the Euler characteristic, represents a key
aspect of its topology. In order to develop an intuitive understanding for the
relation between Betti numbers and the running $\alpha$ parameter of the alpha
shapes, and thus in how far they may discriminate between different topologies,
we study them within the context of simple heuristic Voronoi clustering models.
These may be tuned to consist of a few or even only one specific morphological
element of the Cosmic Web, ie. clusters, filaments or sheets.
