Banach Algebras and Rational Homotopy Theory

Abstract

Let $A$ be a unital commutative Banach algebra with maximal ideal space $X.$ We determine the rational H-type of the group $G{L}_n(A)$ of invertible n by n matrices with coefficients in A, in terms of the rational cohomology of $X.$ We also address an old problem of J. L. Taylor. Let $L{c}_n(A)$ denote the space of "last columns" of $G{L}_n(A).$ For $n>1+s/2,$ we construct a natural isomorphism from the rational Cech cohomology group $H^s(X;Q)$ to the rational homotopy group ${\pi }_{2n-1-s}(L{c}_n(A))\otimes Q,$ which shows that the rational cohomology groups of $X$ 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.

29 pages, AMSLaTeX2e, uses xypic. Version 2 contains various minor corrections and improvements (including additional references), and is to appear in Trans. AMS