# Spectral sequence for H-space bundles

Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle.

One may use the Leray-Serre spectral sequence to compute the (co)homology of $E$. But it does not make use of the H-space structure of $F$.

Is there a spectral sequence or some other tool which one can use in order to compute the cohomology or homology of $E$ by using the H-space structure of $F$?

• In order for the question to admit a positive answer, I think you would need the $H$-space structure on $F$ to be compatible with the fibre bundle structure in some way. Otherwise, you could take, say, any bundle with fibre $S^1$ and ask whether the complex multiplication can help you compute (co)homology of the total space, which seems absurd. Dec 2, 2011 at 8:48
• @Mark: Actually, that's a bad example, as every $S^1$-bundle admits the structure of a principal $U(1)$-bundle. But the point stands with e.g. $S^3 = SU(2)$. Dec 2, 2011 at 11:01
• I am sorry for asking a question that is not very precise. But I am still interested if there is a theory with some extra conditions imposed on the compatibility of the H-space structure and bundle structure. Dec 2, 2011 at 11:21
• Just having a fibre that admits an H-space structure does not yield anything. Maybe you assume that there is a multiplication $E \times_B E \to E$ which is fibre-preserving (and hence turns every fibre into an H-space)? Dec 2, 2011 at 12:40
• @Johannes sorry, my answer was quite wrong and I deleted it. Dec 2, 2011 at 19:45

Following up on Johannes' comment, let's say you have a multiplicative fibration, meaning that $E$ and $B$ are $H$-spaces with multiplications $\mu\colon E\times E\to E$ and $\mu'\colon B\times B\to B$ and that the projection $p\colon E\to B$ is multiplicative in the sense that $$p\circ\mu=\mu'\circ(p\times p).$$ (If you only have this relation up to homotopy you can always deform one of the multiplications so that it holds on the nose.)
Then the fibre $F$ becomes an $H$-space, and the Leray-Serre spectral sequence is a spectral sequence of Hopf algebras. This extra structure can sometimes facilitate computations. See for example