Title: Banach Algebras and Rational Homotopy Theory


Authors: Gregory Lupton, N.Christopher Phillips, Claude L.~Schochet
and Samuel B. Smith

AMS MSC (2000): 46J05, 46L85, 55P62, 54C35, 55P15, 55P45


arXiv number: math.AT/0509269


Addresses:

Department of Mathematics, Cleveland State University, Cleveland
OH44115

Department of Mathematics, University of Oregon,
Eugene OR 97403-1222

Mathematics Department, Wayne State University, Detroit MI 48202

Department of Mathematics, Saint Joseph's University, Philadelphia
PA19131


e-mail addresses:

g.lupton@csuohio.edu

ncp@darkwing.uoregon.edu

claude@math.wayne.edu

smith@sju.edu


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 $n \times n$ 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
\[
{\check{H}}^s (Max(A); Q)
  \cong \pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q
\]
for $n > (1/2) s + 1$ 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 gauge groups F(X,G) for G a Lie group or, more
generally, a rational H-space.