##
**A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map.**
*(English)*
Zbl 1402.55004

In a previous paper [Algebr. Geom. Topol. 17, No. 4, 2051–2080 (2017; Zbl 1378.55003)] the author described how, given a continuous real-valued function on a compact ANR \(X\), one can get a refinement of the Betti numbers and homology of \(X\). The refinement creates a configuration of points in complex space, whose total cardinalities are the Betti numbers. The refinement of the homology consists of a set of vector spaces indexed by points in the complex plane with the same support as that of the point configuration. The direct sum of these vector spaces is the homology of \(X\). In this paper the author starts with an “angle-valued” map which is simply a map into the circle \(f: X \;\rightarrow \;S^{1}\). By considering the lift \(\tilde{X} \;\rightarrow {\mathbb R}\) and applying similar, but more complex, reasoning the author uses the deck transformations to obtain a \(\kappa [t,t^{-1}]\) structure (where \(\kappa\) is a field) on the homology and obtains results parallel to those of the first paper. As in the first paper a PoincarĂ© duality result is shown for the case where \(X\) is a closed smooth manifold.

Reviewer: Jonathan Hodgson (Swarthmore)

### MSC:

55N35 | Other homology theories in algebraic topology |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

57R19 | Algebraic topology on manifolds and differential topology |

### Citations:

Zbl 1378.55003
PDF
BibTeX
XML
Cite

\textit{D. Burghelea}, Algebr. Geom. Topol. 18, No. 5, 3037--3087 (2018; Zbl 1402.55004)

### References:

[1] | 10.2140/agt.2017.17.2051 · Zbl 1378.55003 |

[2] | 10.1007/s00454-013-9497-x · Zbl 1275.55009 |

[3] | 10.1007/s41468-017-0005-x · Zbl 1404.55009 |

[4] | ; Chapman, Lectures on Hilbert cube manifolds. CMBS Regional Conference Series in Mathematics, 28, (1976) |

[5] | 10.1090/surv/108 |

[6] | 10.1002/mana.201600242 · Zbl 1372.14037 |

[7] | 10.1007/s002080050112 · Zbl 0886.57028 |

[8] | 10.1515/crll.1998.015 · Zbl 0921.55016 |

[9] | 10.1093/imrn/rnt093 · Zbl 1302.32023 |

[10] | ; Novikov, Topological methods in modern mathematics, 223, (1993) |

[11] | 10.1515/9783110197976 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.