Inducing a map on homology from a correspondence
Harker S, Kokubu H, Mischaikow K, Pilarczyk P. 2016. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 144(4), 1787–1801.
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Author
Harker, Shaun;
Kokubu, Hiroshi;
Mischaikow, Konstantin;
Pilarczyk, PawelISTA
Corresponding author has ISTA affiliation
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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.
Publishing Year
Date Published
2016-04-01
Journal Title
Proceedings of the American Mathematical Society
Publisher
American Mathematical Society
Acknowledgement
The authors gratefully acknowledge the support of the Lorenz Center which
provided an opportunity for us to discuss in depth the work of this paper. Research leading to these results has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) in the framework of the research
project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008),
as well as from the People Programme (Marie Curie Actions) of the European
Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033 (supporting PP). The work of the first and third author has
been partially supported by NSF grants NSF-DMS-0835621, 0915019, 1125174,
1248071, and contracts from AFOSR and DARPA. The work of the second author
was supported by Grant-in-Aid for Scientific Research (No. 25287029), Ministry of
Education, Science, Technology, Culture and Sports, Japan.
Volume
144
Issue
4
Page
1787 - 1801
ISSN
IST-REx-ID
Cite this
Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 2016;144(4):1787-1801. doi:10.1090/proc/12812
Harker, S., Kokubu, H., Mischaikow, K., & Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/12812
Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.” Proceedings of the American Mathematical Society. American Mathematical Society, 2016. https://doi.org/10.1090/proc/12812.
S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on homology from a correspondence,” Proceedings of the American Mathematical Society, vol. 144, no. 4. American Mathematical Society, pp. 1787–1801, 2016.
Harker S, Kokubu H, Mischaikow K, Pilarczyk P. 2016. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 144(4), 1787–1801.
Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” Proceedings of the American Mathematical Society, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1787–801, doi:10.1090/proc/12812.
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