We propose a refinement of the Betti numbers and the homology with coefficients in a field of a
compact ANR
,
in the presence of a continuous real-valued function
on .
The refinement of Betti numbers consists of finite configurations of points with
multiplicities in the complex plane whose total cardinalities are the Betti numbers,
and the refinement of homology consists of configurations of vector spaces indexed by
points in the 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.
Keywords
Betti numbers, homology, bar codes, configurations
