Algebraic & Geometric Topology

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

Dan Burghelea

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • B,J Ball, Alternative approaches to proper shape theory, from “Studies in topology” (N,M Stavrakas, K,R Allen, editors), Academic Press (1975) 1–27
  • 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, preprint (2013)
  • G Carlsson, V de Silva, D Morozov, Zigzag persistent homology and real-valued functions, from “Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry”, ACM, New York (2009) 247–256
  • T,A Chapman, Lectures on Hilbert cube manifolds, CBMS 28, Amer. Math. Soc., Providence, RI (1976)
  • R,J Daverman, J,J Walsh, A ghastly generalized $n$–manifold, Illinois J. Math. 25 (1981) 555–576
  • S-t Hu, Theory of retracts, Wayne State Univ. Press, Detroit (1965)
  • J Milnor, On spaces having the homotopy type of a ${\rm CW}$–complex, Trans. Amer. Math. Soc. 90 (1959) 272–280