## Algebraic & Geometric Topology

- Algebr. Geom. Topol.
- Volume 17, Number 4 (2017), 2051-2080.

### A refinement of Betti numbers and homology in the presence of a continuous function, I

#### Abstract

We propose a refinement of the Betti numbers and the homology with coefficients in a field of a compact ANR $X$, in the presence of a continuous real-valued function on $X$. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinalities are the Betti numbers, and the refinement of homology consists of configurations of vector spaces indexed by points in the complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product, these vector spaces are canonically realized as mutually orthogonal subspaces of the homology.

The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite-dimensional complex vector space. A number of remarkable properties of the above configurations are discussed.

#### Article information

**Source**

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2051-2080.

**Dates**

Received: 27 December 2015

Revised: 20 December 2016

Accepted: 8 January 2017

First available in Project Euclid: 16 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.agt/1510841437

**Digital Object Identifier**

doi:10.2140/agt.2017.17.2051

**Mathematical Reviews number (MathSciNet)**

MR3685602

**Zentralblatt MATH identifier**

1378.55003

**Subjects**

Primary: 55N35: Other homology theories

Secondary: 46M20: Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.) [See also 14F05, 18Fxx, 19Kxx, 32Cxx, 32Lxx, 46L80, 46M15, 46M18, 55Rxx] 57R19: Algebraic topology on manifolds

**Keywords**

Betti numbers homology bar codes configurations

#### Citation

Burghelea, Dan. A refinement of Betti numbers and homology in the presence of a continuous function, I. Algebr. Geom. Topol. 17 (2017), no. 4, 2051--2080. doi:10.2140/agt.2017.17.2051. https://projecteuclid.org/euclid.agt/1510841437