Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 5 (2018), 3037-3087.
A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map
For a continuous angle-valued map defined on a compact ANR , a field and any integer , one proposes a refinement of the Novikov–Betti numbers of the pair and a refinement of the Novikov homology of , where denotes the integral degree one cohomology class represented by . The refinement is a configuration of points, with multiplicity located in identified to , whose total cardinality is the Novikov–Betti number of the pair. The refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of . When , the configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the –homology of the infinite cyclic cover of defined by , which is an –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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.agt/1535594431