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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics
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by Mauricio Bustamante, Francis Thomas Farrell and Yi Jiang PDF
Trans. Amer. Math. Soc. 373 (2020), 7225-7252 Request permission

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
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 7225-7252
  • MSC (2010): Primary 19D10; Secondary 55N91
  • DOI: https://doi.org/10.1090/tran/8135
  • MathSciNet review: 4155206