## 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

#### 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 $ℝ 2 ∕ ℤ$ identified to $ℂ ∖ 0$, whose total cardinality is the Novikov–Betti number of the pair. The refinement $δ ̂ r f$ 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 $δ r f$. When $κ = ℂ$, 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

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

Dates
Revised: 12 March 2018
Accepted: 23 March 2018
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.agt/1535594431

Digital Object Identifier
doi:10.2140/agt.2018.18.3037

Mathematical Reviews number (MathSciNet)
MR3848408

Zentralblatt MATH identifier
06935829

#### Citation

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

#### References

• D Burghelea, Linear relations, monodromy and Jordan cells of a circle valued map, preprint (2015)
• D Burghelea, A refinement of Betti numbers and homology in the presence of a continuous function, I, Algebr. Geom. Topol. 17 (2017) 2051–2080
• D Burghelea, T K Dey, Topological persistence for circle-valued maps, Discrete Comput. Geom. 50 (2013) 69–98
• D Burghelea, S Haller, Topology of angle valued maps, bar codes and Jordan blocks, J. Appl. and Comput. Topology 1 (2017)
• T A Chapman, Lectures on Hilbert cube manifolds, CMBS Regional Conference Series in Mathematics 28, Amer. Math. Soc., Providence, RI (1976)
• M Farber, Topology of closed one-forms, Mathematical Surveys and Monographs 108, Amer. Math. Soc., Providence, RI (2004)
• S Friedl, L Maxim, Twisted Novikov homology of complex hypersurface complements, Math. Nachr. 290 (2017) 604–612
• S-T Hu, Theory of retracts, Wayne State Univ. Press, Detroit (1965)
• W Lück, Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$–invariants, Math. Ann. 309 (1997) 247–285
• W Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$–Betti numbers, I: Foundations, J. Reine Angew. Math. 495 (1998) 135–162
• L Maxim, $L^2$–Betti numbers of hypersurface complements, Int. Math. Res. Not. 2014 (2014) 4665–4678
• S P Novikov, Quasiperiodic structures in topology, from “Topological methods in modern mathematics” (L R Goldberg, A V Phillips, editors), Publish or Perish, Houston, TX (1993) 223–233
• A V Pajitnov, Circle-valued Morse theory, De Gruyter Studies in Mathematics 32, de Gruyter, Berlin (2006)