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Inducing a map on homology from a correspondence


Authors: Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow and Paweł Pilarczyk
Journal: Proc. Amer. Math. Soc. 144 (2016), 1787-1801
MSC (2010): Primary 55M99; Secondary 55-04
DOI: https://doi.org/10.1090/proc/12812
Published electronically: August 12, 2015
MathSciNet review: 3451254
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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.


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Shaun Harker
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghusen Road, Piscataway, New Jersey 08854-8019
Email: sharker@math.rutgers.edu

Hiroshi Kokubu
Affiliation: Department of Mathematics / JST CREST, Kyoto University, Kyoto 606-8502, Japan
Email: kokubu@math.kyoto-u.ac.jp

Konstantin Mischaikow
Affiliation: Department of Mathematics and BioMaPS, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghusen Road, Piscataway, New Jersey 08854-8019
Email: mischaik@math.rutgers.edu

Paweł Pilarczyk
Affiliation: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: pawel.pilarczyk@ist.ac.at

DOI: https://doi.org/10.1090/proc/12812
Keywords: Homology, homomorphism, continuous map, time series, combinatorial approximation, grid, multivalued map, correspondence, acyclicity
Received by editor(s): November 18, 2014
Received by editor(s) in revised form: April 15, 2015
Published electronically: August 12, 2015
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2015 American Mathematical Society