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2018 A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map
Dan Burghelea
Algebr. Geom. Topol. 18(5): 3037-3087 (2018). DOI: 10.2140/agt.2018.18.3037

Abstract

For f : X S 1 a continuous angle-valued map defined on a compact ANR X , κ  a field and any integer r 0 , one proposes a refinement δ r f of the Novikov–Betti numbers of the pair ( X , ξ f ) and a refinement δ ̂ r f of the Novikov homology of ( X , ξ f ) , where ξ f denotes the integral degree one cohomology class represented by  f . The refinement δ r f is a configuration of points, with multiplicity located in 2 identified to 0 , whose total cardinality is the r  th Novikov–Betti number of the pair. The refinement δ ̂ r f is a configuration of submodules of the r  th Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of δ r f . When κ = , the configuration δ ̂ r f is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L 2 –homology of the infinite cyclic cover of X defined by f , which is an L ( S 1 ) –Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.

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Dan Burghelea. "A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map." Algebr. Geom. Topol. 18 (5) 3037 - 3087, 2018. https://doi.org/10.2140/agt.2018.18.3037

Information

Received: 21 November 2017; Revised: 12 March 2018; Accepted: 23 March 2018; Published: 2018
First available in Project Euclid: 30 August 2018

zbMATH: 06935829
MathSciNet: MR3848408
Digital Object Identifier: 10.2140/agt.2018.18.3037

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

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 5 • 2018
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