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Let $E\to B$ be a fibre bundle. The Leray-Hirsch theorem states under suitable assumptions, the cohomology of $E$ is an $H^*(B)$-module generated by suitable cohomology classes in $E$.

Is there any analogous results for the homology of $E$? Any reference?

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    $\begingroup$ It's automatic that the cohomology of E is a module over the cohomology of B. The point of Leray-Hirsch is that it is sometimes free. $\endgroup$
    – Ben Webster ♦
    Commented Nov 5, 2010 at 21:17
  • $\begingroup$ Sorry. I was stupid to miss this condition. $\endgroup$
    – Guangbo Xu
    Commented Nov 5, 2010 at 21:43
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    $\begingroup$ A version of the Leray-Hirsch for homology is as follows: let $p:E\to B$ be a Serre fibration with $B$ connected and let $A$ be an abelian group. Assume that $p_\ast :H_\ast(F,A)\to H_\ast(E,A)$ is injective where $F$ is the fiber over some point of $B$, then $H_n(E,A)\cong \bigoplus_{i+j=n} H_i(B,H_j(F,A))$. The free module part is a bit tricky since, although the cohomology with coefficients in $A$ is always an $A$-algebra as long as $A$ is a commutative ring, the homology is not necessarily an $A$-coalgebra. $\endgroup$
    – algori
    Commented Nov 5, 2010 at 22:30
  • $\begingroup$ Is that isomorphism true or is just true up to group extension? $\endgroup$
    – Hari Rau-Murthy
    Commented Aug 31, 2016 at 21:39

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