Work

Inducing a map on homology from a correspondence

Year: 2015

Type: article

Abstract: We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/12812

Authors: Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Paweł Pilarczyk

Institutions: Rutgers, The State University of New Jersey, Kyoto University, Institute of Science and Technology Austria

Language: English

Topic: Topological and Geometric Data Analysis

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: bronze