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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 $F$?

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    $\begingroup$ 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. $\endgroup$
    – Mark Grant
    Dec 2, 2011 at 8:48
  • $\begingroup$ @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. $S^3 = SU(2)$. $\endgroup$
    – Oscar Randal-Williams
    Dec 2, 2011 at 11:01
  • $\begingroup$ 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. $\endgroup$
    – fred137
    Dec 2, 2011 at 11:21
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    $\begingroup$ 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)? $\endgroup$
    – Johannes Ebert
    Dec 2, 2011 at 12:40
  • $\begingroup$ @Johannes sorry, my answer was quite wrong and I deleted it. $\endgroup$
    – Vitali Kapovitch
    Dec 2, 2011 at 19:45

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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

Lin, James P, Cohomology rings of multiplicative fibrations. Topology Appl. 156 (2009)

or

Browder, William, On differential Hopf algebras. Trans. Amer. Math. Soc. 107 (1963).

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