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 $GL_n (A)$ 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 $Lc_n (A)$ denote the space of "last columns" of $GL_n (A).$ 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 $π_{2 n - 1 - s} (Lc_n (A)) \otimes Q,$ 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.