# Inducing a map on homology from a correspondence

S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801.

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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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Date Published

2016-04-01

Journal Title

Proceedings of the American Mathematical Society

Volume

144

Issue

4

Page

1787 - 1801

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### 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/12812Harker, 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. https://doi.org/10.1090/proc/12812Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.”

*Proceedings of the American Mathematical Society*144, no. 4 (2016): 1787–1801. 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, 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.### Export

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arXiv 1411.7563