Subject: Question for the Algebraic Topology List
From: Greg Landweber
Date: Sun, 15 Jan 2006 10:43:23 -0800
I was looking through McCleary's book "A User's Guide to Spectral
Sequences" and I came across his exercise 8.12. The problem supposes
that you have a path-loop fibration \Omega B --> PB --> B and a map
f : B_0 --> B. The statement is that the Eilenberg-Moore spectral
sequence for the cohomology of f^*(PB) collapses at E_2 if and only
if the corresponding Leray-Serre spectral sequence has all
differentials arising from transgressions.
I am interested in this problem in the special case where we are
computing the cohomology over the rationals, reals, or in general any
field with characteristic 0.
Also, I am interested in the cases where B = BG or B = G for G a
compact, connected, simply connected Lie group. I am further
interested in B = K(Z,3).
I managed to convince myself that the statement of this problem is
true. However, it raises an interesting question which I have not yet
been able to answer: precisely when does the the Leray-Serre spectral
sequence have all differentials arising from transgressions? I would
very much like the Eilenberg-Moore spectral sequence to collapse at
E_2 in these cases I mentioned above, and I was wondering what if any
conditions could be put on B_0 or the map f to insure the collapse.
On the other hand, I would also like to see an explicit example of a
case where E --> M is a principal G-bundle for which the rational
cohomology of E is not completely determined by the rational
cohomology of M and the characteristic classes of E.
Thank you very much for sharing any thoughts you have on these subjects.
-- Greg Landweber