Subject: Re: two questions
From: Steve Halperin
Date: Fri, 23 Jun 2006 19:20:31 -0400
Re the formality question: Since the model of BG coincides with its
cohomology it seems to me that any map from a formal space to BG is
automatically formal
The problem of formality of a map arises in trying to factor a morphism
from the source model (== model of the target space) over a quasi-iso from
that model to the cohomology of the target space.
steve
Don Davis wrote:
> Two postings: Both new questions.............DMD
> ____________________________________________________________
>
> 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
>
> Claude Schochet
> Math Dept, Wayne State University
> Detroit, MI 48202
> claude@math.wayne.edu
Steve Halperin
Dean
College of Computer, Mathematical, and Physical Sciences
University of Maryland
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