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

Dan Burghelea

Algebraic & Geometric Topology 17 (2017) 2051–2080

We propose a refinement of the Betti numbers and 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 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.

Betti numbers, homology, bar codes, configurations
Mathematical Subject Classification 2010
Primary: 55N35
Secondary: 46M20, 57R19
Received: 27 December 2015
Revised: 20 December 2016
Accepted: 8 January 2017
Published: 3 August 2017
Dan Burghelea
Department of Mathematics
The Ohio State University
Columbus, OH
United States