  We propose a refinement of the Betti numbers and of the homology with
coefficients in a field of a compact ANR in the presence of a continuous real
valued function. The refinement of Betti numbers consists of finite
configurations of points with multiplicities in the complex plane whose total
cardinality are the Betti numbers and the refinement of homology consists of
configurations of vector spaces indexed by points in complex plane, with the
same support as the first, whose direct sum is isomorphic to the homology. When
the homology is equipped with a scalar product these vector spaces are
canonically realized as mutually orthogonal subspaces of the homology. The
assignments above are in analogy with the collections of eigenvalues and
generalized eigenspaces of a linear map in a finite dimensional complex vector
space. A number of remarkable properties of the above configurations are
discussed.
