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

Burghelea, Dan¹

¹ Department of Mathematics, The Ohio State University, Columbus, OH, United States

Algebraic and Geometric Topology, Tome 18 (2018) no. 5, pp. 3037-3087

Résumé

For f : X → S1 a continuous angle-valued map defined on a compact ANR X, κ a field and any integer r ≥ 0, one proposes a refinement δrf of the Novikov–Betti numbers of the pair (X,ξf) and a refinement δ̂rf of the Novikov homology of (X,ξf), where ξf denotes the integral degree one cohomology class represented by f. The refinement δrf is a configuration of points, with multiplicity located in ℝ2∕ℤ identified to ℂ ∖ 0, whose total cardinality is the r th Novikov–Betti number of the pair. The refinement δ̂rf 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 δrf. When κ = ℂ, the configuration δ̂rf is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2–homology of the infinite cyclic cover of X defined by f, which is an L∞(S1)–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.

Détail

DOI : 10.2140/agt.2018.18.3037

Classification : 46M20, 55N35, 57R19
Keywords: Novikov–Betti numbers, angle-valued maps, barcodes