Geodesic
A refinement of Betti numbers and homology in the presence of a continuous function, I
Burghelea, Dan1
1Department of Mathematics, The Ohio State University, Columbus, OH, United States
Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2051-2080
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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.

DOI : 10.2140/agt.2017.17.2051
Classification : 55N35, 46M20, 57R19
Keywords: Betti numbers, homology, bar codes, configurations

Burghelea, Dan  1

1 Department of Mathematics, The Ohio State University, Columbus, OH, United States 
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Burghelea, Dan. A refinement of Betti numbers and homology in the presence of a continuous function, I. Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2051-2080. doi: 10.2140/agt.2017.17.2051

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