A refinement of Betti numbers and homology in the presence of a continuous function. I

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR X, in the presence of a continuous real valued function on X.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.