Algebraic & Geometric Topology

A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map

Dan Burghelea

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 5 (2018), 3037-3087.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46M20: Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.) [See also 14F05, 18Fxx, 19Kxx, 32Cxx, 32Lxx, 46L80, 46M15, 46M18, 55Rxx] 55N35: Other homology theories 57R19: Algebraic topology on manifolds

Novikov–Betti numbers angle-valued maps barcodes


Burghelea, Dan. 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 (2018), no. 5, 3037--3087. doi:10.2140/agt.2018.18.3037.

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