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Given fibration sequences $F\rightarrow E\rightarrow B$ and $F'\rightarrow E'\rightarrow B'$, consider the homology Serre spectral sequence $S$ for the product of fibrations $F\times F'\rightarrow E\times E'\rightarrow B\times B'$.

Under the ideal conditions on our spaces (torsion-free homology to be safe) I would guess that $S$ has the expected behaviour, namely with differentials being derivations given in terms of the differentials in the spectral sequences for each of the individual fibrations in the product.

I haven't been able to find a single reference, even in McCleary's book. Unless I didn't look hard enough, or this is obvious from the construction of the spectral sequence. Anybody have any idea?

Peter
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    Why do you need it ? Usually one computes the (co)homology of the product of spaces by the Künneth formula. – tj_ Mar 11 '16 at 22:37
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    Remark 15.4 in Switzer's Algebraic topology: Homology and homotopy talks about constructing the internal product on a Serre spectral sequence by passing first through the external product you describe. (He also doesn't finish the job, which he decries as extremely tedious.) There might also be something in the references of this recent MO question: http://mathoverflow.net/questions/225579/multiplicative-structure-of-the-atiyah-hirzebruch-leray-serre-spectral-sequence#comment556933_225579 . – Eric Peterson Mar 12 '16 at 12:49
  • You're right tj, not much to do with computing the homology of the total space. More to do with comparing spectral sequences via a diagram of fibrations. Thanks Eric. – Peter Mar 13 '16 at 00:34

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