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Proceedings of the American Mathematical Society

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Quantum extensions of ordinary maps


Author: Andre Kornell
Journal: Proc. Amer. Math. Soc. 148 (2020), 1971-1986
MSC (2010): Primary 46L85; Secondary 54C20
DOI: https://doi.org/10.1090/proc/14851
Published electronically: January 28, 2020
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Abstract: We define a loop to be quantum nullhomotopic if and only if it admits a nonempty quantum set of extensions to the unit disk. We show that the canonical loop in the unit circle is not quantum nullhomotopic, but that every loop in the real projective plane is quantum nullhomotopic. Furthermore, we apply Kuiper's theorem to show that the canonical loop admits a continuous family of extensions to the unit disk that is indexed by an infinite quantum space. We obtain these results using a purely topological condition that we show to be equivalent to the existence of a quantum family of extensions of a given map.


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Additional Information

Andre Kornell
Affiliation: Department of Mathematics, University of California, Davis, Davis, California 95616
Email: kornell@math.ucdavis.edu

DOI: https://doi.org/10.1090/proc/14851
Received by editor(s): December 30, 2018
Published electronically: January 28, 2020
Communicated by: Adrian Ioana
Article copyright: © Copyright 2020 American Mathematical Society