Subject: Re: two questions
From: Steve Halperin <shalper@deans.umd.edu>
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 <claude@math.wayne.edu>
> 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

Office: 3400 A.V. Williams Building
       301- 405-2316