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

Burghelea, D. (2017). A refinement of Betti numbers in the presence of a continuous function, I. Algebraic & Geometric Topology, 17(4), 2051-2080. doi:10.2140/agt.2017.17.2051.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0004-6721-3 Version Permalink: https://hdl.handle.net/21.11116/0000-0004-6722-2
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 Creators:
Burghelea, Dan1, Author           
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Algebraic Topology
 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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Language(s): eng - English
 Dates: 2017
 Publication Status: Issued
 Pages: 30
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 Rev. Type: Peer
 Identifiers: arXiv: 1501.01012
DOI: 10.2140/agt.2017.17.2051
URI: http://arxiv.org/abs/1501.01012
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Title: Algebraic & Geometric Topology
  Abbreviation : Algebr. Geom. Topol.
Source Genre: Journal
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Pages: - Volume / Issue: 17 (4) Sequence Number: - Start / End Page: 2051 - 2080 Identifier: -