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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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Banach algebras and rational homotopy theory
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by Gregory Lupton, N. Christopher Phillips, Claude L. Schochet and Samuel B. Smith PDF
Trans. Amer. Math. Soc. 361 (2009), 267-295 Request permission

Abstract:

Let $A$ be a unital commutative Banach algebra with maximal ideal space $\operatorname {Max}(A).$ We determine the rational H-type of $\operatorname {GL}_n (A),$ the group of invertible $n \times n$ matrices with coefficients in $A,$ in terms of the rational cohomology of $\operatorname {Max} (A).$ We also address an old problem of J. L. Taylor. Let $\operatorname {Lc}_n (A)$ denote the space of “last columns” of $\operatorname {GL}_n (A).$ We construct a natural isomorphism \[ {\check {H}}^s (\operatorname {Max} (A); \mathbb Q ) \cong \pi _{2 n - 1 - s} (\operatorname {Lc}_n (A)) \otimes \mathbb Q \] for $n > \frac {1}{2} s + 1$ which shows that the rational cohomology groups of $\operatorname {Max} (A)$ are determined by a topological invariant associated to $A.$ As part of our analysis, we determine the rational H-type of certain gauge groups $F (X, G)$ for $G$ a Lie group or, more generally, a rational H-space.
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Additional Information
  • Gregory Lupton
  • Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
  • MR Author ID: 259990
  • Email: G.Lupton@csuohio.edu
  • N. Christopher Phillips
  • Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
  • Email: ncp@darkwing.uoregon.edu
  • Claude L. Schochet
  • Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
  • MR Author ID: 191627
  • ORCID: 0000-0002-6177-2392
  • Email: claude@math.wayne.edu
  • Samuel B. Smith
  • Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
  • MR Author ID: 333158
  • Email: smith@sju.edu
  • Received by editor(s): April 17, 2006
  • Received by editor(s) in revised form: December 19, 2006
  • Published electronically: August 14, 2008
  • Additional Notes: The research of the second author was partially supported by NSF grant DMS 0302401.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 267-295
  • MSC (2000): Primary 46J05, 46L85, 55P62, 54C35, 55P15, 55P45
  • DOI: https://doi.org/10.1090/S0002-9947-08-04477-2
  • MathSciNet review: 2439407