{"publication_status":"published","publisher":"American Mathematical Society","intvolume":"       144","title":"Inducing a map on homology from a correspondence","quality_controlled":"1","abstract":[{"text":"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.","lang":"eng"}],"date_updated":"2022-05-24T09:35:58Z","ec_funded":1,"date_published":"2016-04-01T00:00:00Z","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:50:57Z","issue":"4","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["1411.7563"]},"year":"2016","acknowledgement":"The authors gratefully acknowledge the support of the Lorenz Center which\r\nprovided 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\r\nproject FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008),\r\nas well as from the People Programme (Marie Curie Actions) of the European\r\nUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033 (supporting PP). The work of the first and third author has\r\nbeen partially supported by NSF grants NSF-DMS-0835621, 0915019, 1125174,\r\n1248071, and contracts from AFOSR and DARPA. The work of the second author\r\nwas supported by Grant-in-Aid for Scientific Research (No. 25287029), Ministry of\r\nEducation, Science, Technology, Culture and Sports, Japan.","main_file_link":[{"url":"https://arxiv.org/abs/1411.7563","open_access":"1"}],"oa_version":"Preprint","month":"04","article_type":"original","publication_identifier":{"issn":["1088-6826"]},"day":"01","author":[{"last_name":"Harker","first_name":"Shaun","full_name":"Harker, Shaun"},{"full_name":"Kokubu, Hiroshi","first_name":"Hiroshi","last_name":"Kokubu"},{"first_name":"Konstantin","full_name":"Mischaikow, Konstantin","last_name":"Mischaikow"},{"first_name":"Pawel","full_name":"Pilarczyk, Pawel","last_name":"Pilarczyk","id":"3768D56A-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","doi":"10.1090/proc/12812","_id":"1252","oa":1,"project":[{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","name":"Persistent Homology - Images, Data and Maps","call_identifier":"FP7","grant_number":"622033"}],"publication":"Proceedings of the American Mathematical Society","volume":144,"scopus_import":"1","citation":{"apa":"Harker, S., Kokubu, H., Mischaikow, K., &#38; Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>","chicago":"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>. American Mathematical Society, 2016. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>.","short":"S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801.","ieee":"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. American Mathematical Society, pp. 1787–1801, 2016.","mla":"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>.","ama":"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>","ista":"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."},"page":"1787 - 1801","publist_id":"6075","type":"journal_article","department":[{"_id":"HeEd"}],"status":"public"}