A refinement of Betti numbers in the presence of a continuous function. ( I )

TL;DR

This paper introduces a novel refinement of Betti numbers and homology for compact ANRs with continuous functions, representing them as configurations of points and vector spaces in the complex plane, revealing new structural insights.

Contribution

It presents a new framework for refining Betti numbers and homology using configurations of points and vector spaces, analogous to eigenvalues and eigenspaces, for compact ANRs with continuous functions.

Findings

Refined Betti numbers as point configurations with multiplicities.

Homology as configurations of vector spaces indexed by complex plane points.

Properties of these configurations and their relation to classical invariants.

Abstract

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.