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