Subject: formal bundles
From: Claude Schochet
Date: Thu, 22 Jun 2006 07:24:47 -0400 (EDT)
for toplist
Suppose that G is a compact Lie group (G = PU_k in my situation) , X is a
finite complex (smooth manifold if you like) and f; X \to BG . Now
suppose that X is formal in the sense of rational homotopy theory. (The
space BG is automatically formal.) I am looking at fibre bundles
classified by f. (In the case at hand the universal fibre bundle has the
form U_k \to W \to BPU_k.)
Question--- when is the map f formal? I am looking for geometrically
meaningful statements about the pullback bundle or its associated
principal G-bundle that imply that f is formal. For instance, is the
formality of f equivalent to some conditions on the rational Chern classes
of the bundle? I do not want to assume further conditions on X other than
its being connected and nilpotent.
Thanks!
Claude
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Claude Schochet
Math Dept, Wayne State University
Detroit, MI 48202
claude@math.wayne.edu