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Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(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 Lcn(A) denote the space of "last columns" of GLn(A). We construct a natural isomorphism

Ȟs(Max(A);ℚ) ≅ π2n-1-s(Lcn(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.


Algebra | Analysis | Geometry and Topology


First published in Transactions of the American Mathematical Society in 361.1 (January 2009,, 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.