## 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

#### Abstract

For $f:X\to {\mathbb{S}}^{1}$ a continuous angle-valued map defined on a compact ANR $X$, $\kappa $ a field and any integer $r\ge 0$, one proposes a refinement ${\delta}_{r}^{f}$ of the Novikov–Betti numbers of the pair $\left(X,{\xi}_{f}\right)$ and a refinement ${\widehat{\delta}}_{r}^{f}$ of the Novikov homology of $\left(X,{\xi}_{f}\right)$, where ${\xi}_{f}$ denotes the integral degree one cohomology class represented by $f$. The refinement ${\delta}_{r}^{f}$ is a configuration of points, with multiplicity located in ${\mathbb{R}}^{2}\u2215\mathbb{Z}$ identified to $\u2102\setminus 0$, whose total cardinality is the ${r}^{\text{th}}$ Novikov–Betti number of the pair. The refinement ${\widehat{\delta}}_{r}^{f}$ is a configuration of submodules of the ${r}^{\text{th}}$ Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of ${\delta}_{r}^{f}$. When $\kappa =\u2102$, the configuration ${\widehat{\delta}}_{r}^{f}$ 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 ${L}^{\infty}\left({\mathbb{S}}^{1}\right)$–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**

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**

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

**Subjects**

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

**Keywords**

Novikov–Betti numbers angle-valued maps barcodes

#### 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