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2017 A refinement of Betti numbers and homology in the presence of a continuous function, I
Dan Burghelea
Algebr. Geom. Topol. 17(4): 2051-2080 (2017). DOI: 10.2140/agt.2017.17.2051

Abstract

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.

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Dan Burghelea. "A refinement of Betti numbers and homology in the presence of a continuous function, I." Algebr. Geom. Topol. 17 (4) 2051 - 2080, 2017. https://doi.org/10.2140/agt.2017.17.2051

Information

Received: 27 December 2015; Revised: 20 December 2016; Accepted: 8 January 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1378.55003
MathSciNet: MR3685602
Digital Object Identifier: 10.2140/agt.2017.17.2051

Subjects:
Primary: 55N35
Secondary: 46M20, 57R19

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 4 • 2017
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