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.References
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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