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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics


Authors: Mauricio Bustamante, Francis Thomas Farrell and Yi Jiang
Journal: Trans. Amer. Math. Soc. 373 (2020), 7225-7252
MSC (2010): Primary 19D10; Secondary 55N91
DOI: https://doi.org/10.1090/tran/8135
Published electronically: July 28, 2020
MathSciNet review: 4155206
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Abstract: We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $ K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $ S^{1}$-equivariant homology groups of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $ V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $ V$ have nontrivial rational homotopy groups.


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Additional Information

Mauricio Bustamante
Affiliation: Department of Pure Mathematics and Mathematical Sciences, University of Cambridge, United Kingdom
MR Author ID: 1164502
Email: bustamante@dpmms.cam.ac.uk

Francis Thomas Farrell
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing, People’s Republic of China
MR Author ID: 65305
Email: farrell@math.binghamton.edu

Yi Jiang
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing, People’s Republic of China
MR Author ID: 1079891
Email: yjiang117@mail.tsinghua.edu.cn

DOI: https://doi.org/10.1090/tran/8135
Received by editor(s): May 4, 2017
Received by editor(s) in revised form: August 8, 2019, and February 6, 2020
Published electronically: July 28, 2020
Additional Notes: The third author’s research was partially supported by NSFC 11571343 and NSFC 11801298.
Article copyright: © Copyright 2020 American Mathematical Society