Subject: A question about classifying spaces
Date: Wed, 11 Jun 2003 15:27:32 +0200
From: "Boccellari" <boccellari@inwind.it>

I have a question about classifying spaces whose answer should be well
known, but I can't find it anywere.

Consider the classifying space BG where G is Top(n), SO(n) or SF(n), and let
U the Thom class of the universal fibration with section and fibre S^n on
BG.

Let K_n be the Eilenberg-MacLane space K(Z_2,n) and let L be a subset made
by homogeneous elements in the Steenrod algebra (1 is not allowed).
Let (E,q,K_n) the principal fibration obtained deleting all the
elements of L from the cohomology of K_n.

Let M be the subset of H^*(BG;Z_2) whose image through the Thom isomorphism
is the set of elements obtained applying the elements of L to U.
Let DG the total space of the principal fibration obtained deleting the
elements of M from the cohomology of BG.

Take the n-th loop of  (E,q,K_n) and let W the path component of
\Omega^n (E) on 1 \in Z_2 = \Omega^n (K_n).
W is a G space and so let consider the space B(W,G,*) obtained from the two
sided geometric bar construction.

What is the relation between B(W,G,*) and DG?
Are they homotopy equivalent?
As an exercize I found that their cohomologies have isomorphic Z_2-linear
structure, but I am interested in knowing more.

Thank you for your kind attention.

Yours faithfully.

Tommaso Boccellari