Banach Algebras and Rational Homotopy Theory

Authors

Gregory Lupton, Cleveland State University
N. Christopher Phillips, University of Oregon
Claude Schochet, Wayne State UniversityFollow
Samuel B. Smith, Saint Joseph's University

Document Type

Article

Abstract

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GL_n(A), the group of invertible n x n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lc_n(A) denote the space of "last columns" of GL_n(A). We construct a natural isomorphism

Ȟ^s(Max(A);ℚ) ≅ π_2n-1-s(Lc_n(A)) ⊗ ℚ

for n > ½s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain guage groups F(X,G) for G a Lie group or, more generally, a rational H-space.

Disciplines

Algebra | Analysis | Geometry and Topology

Comments

First published in Transactions of the American Mathematical Society in 361.1 (January 2009, http://www.ams.org/journals/tran/2009-361-01/S0002-9947-08-04477-2/), published by the American Mathematical Society.

The research of the second author was partially supported by NSF grant DMS 0302401.

The authors would like to thank Daniel Isaksen and Jim Stasheff for helpful discussions.

Recommended Citation

G. Lupton, N. C. Phillips, C. L. Schochet and S. B. Smith, Banach algebras and rational homotopy theory, Transactions of the American Mathematical Society 361.1 (2009), 267-295.