Künneth theorem for fibred products

Question

Given the fibred product of two manifolds over a base space $X\times_Y Z$ , is there an analogue of Künneth theorem that allows one to compute the cohomology of the fibred product?

Answer 1

There is a Künneth theorem in sheaf cohomology, which takes the form of an isomorphism between (complexes of) sheaves on $Y$. More precisely let $f : X \to Y$, $g : Z \to Y$ and $h : X \times_Y Z \to Y$, let $F$ and $G$ be sheaves on $X$ and $Z$ and denote by the same name their pullbacks to $X \times_Y Z$. Then there is an isomorphism in the derived category of $Y$ $$ Rf_\ast F \otimes^L Rg_\ast G \cong Rh_\ast (F \otimes^L G) $$ under some mild hypotheses that I'm not going to worry about (if you're forming arbitrary fibered products of manifolds then you're not worrying about precise hypotheses). So there is no Künneth theorem directly for the cohomology of $X \times_Y Z$, but if you compute the cohomology of $X \times_Y Z$ via the Leray spectral sequence for $X \times_Y Z \to Y$ then you see a Künneth formula. For details I highly recommend Dimca's book "Sheaves in topology".

Answer 2

The Eilenberg-Moore spectral sequence can be understood as a parametrized K\"unneth theorem. There is a monograph (also a paper) by L. Smith giving this approach (Lectures on the EMSS, Springer LNM 134, 1970). Section 22.7 of http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf gives a new conceptual construction and a long list of relevant references.