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Adding a point to configurations in closed balls


Authors: Lei Chen, Nir Gadish and Justin Lanier
Journal: Proc. Amer. Math. Soc. 148 (2020), 885-891
MSC (2010): Primary 55M20, 55R80; Secondary 20F36
DOI: https://doi.org/10.1090/proc/14712
Published electronically: October 18, 2019
MathSciNet review: 4052223
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Abstract: We answer the question of when a new point can be added in a continuous way to configurations of $n$ distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of $n$ points if and only if $n \neq 1$. On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if $n = 2$. These results generalize the Brouwer fixed-point theorem, which gives the negative answer when $n=1$. We also show that when $n=2$, there is a unique solution to both the ordered and unordered versions of the problem up to homotopy.


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References
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Additional Information

Lei Chen
Affiliation: Department of Mathematics, California Institute of Technology, MC 253-37, Pasadena, California 91125
Email: chenlei1991919@gmail.com

Nir Gadish
Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
MR Author ID: 1211998
ORCID: 0000-0003-4479-0537
Email: nirg@math.uchicago.edu

Justin Lanier
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332
MR Author ID: 1280905
ORCID: 0000-0003-4483-8553
Email: jlanier8@gatech.edu

Received by editor(s): December 19, 2018
Received by editor(s) in revised form: May 6, 2019, and May 20, 2019
Published electronically: October 18, 2019
Additional Notes: The third author was supported by the NSF grant DGE-1650044.
Communicated by: David Futer
Article copyright: © Copyright 2019 American Mathematical Society