article
Inducing a map on homology from a correspondence
published
yes
Shaun
Harker
author
Hiroshi
Kokubu
author
Konstantin
Mischaikow
author
Pawel
Pilarczyk
author 3768D56A-F248-11E8-B48F-1D18A9856A87
HeEd
department
Persistent Homology - Images, Data and Maps
project
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.
American Mathematical Society2016
eng
Proceedings of the American Mathematical Society10.1090/proc/12812
14441787 - 1801
S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801.
Harker, S., Kokubu, H., Mischaikow, K., & Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>, <i>144</i>(4), 1787–1801. <a href="https://doi.org/10.1090/proc/12812">https://doi.org/10.1090/proc/12812</a>
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.
S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on homology from a correspondence,” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4, pp. 1787–1801, 2016.
Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1787–801, doi:<a href="https://doi.org/10.1090/proc/12812">10.1090/proc/12812</a>.
Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. 2016;144(4):1787-1801. doi:<a href="https://doi.org/10.1090/proc/12812">10.1090/proc/12812</a>
Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i> 144, no. 4 (2016): 1787–1801. <a href="https://doi.org/10.1090/proc/12812">https://doi.org/10.1090/proc/12812</a>.
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