Banach algebras and rational homotopy theory

Lupton, Gregory; Phillips, N.; Schochet, Claude; Smith, Samuel

doi:10.1090/S0002-9947-08-04477-2

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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