We propose a refinement of the Betti numbers and of the homology withcoefficients in a field of a compact ANR in the presence of a continuous realvalued function. The refinement of Betti numbers consists of finiteconfigurations of points with multiplicities in the complex plane whose totalcardinality are the Betti numbers and the refinement of homology consists ofconfigurations of vector spaces indexed by points in complex plane, with thesame support as the first, whose direct sum is isomorphic to the homology. Whenthe homology is equipped with a scalar product these vector spaces arecanonically realized as mutually orthogonal subspaces of the homology. Theassignments above are in analogy with the collections of eigenvalues andgeneralized eigenspaces of a linear map in a finite dimensional complex vectorspace. A number of remarkable properties of the above configurations arediscussed.
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