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let us suppose that we are computing homology of a product where none of the requirements of k√ľnneth theorem are valid : is there a general way to compute the homology of such products?

Many thanks

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Which Kunneth theorem do you mean? There are increasingly general versions.. – Aaron Mazel-Gee May 19 '11 at 10:17
What Kunneth formula are you using? There's a general Kunneth formula which holds for products of CW complexes, with coefficients in a PID, which (as I see it) is a fairly inclusive condition. See Hatcher section 3.B for more (the explicit formula is on p275). – Alex May 19 '11 at 10:18
You should probably describe what kind of situation you have in mind: I am pretty sure there are cases where nothing will work... – Mariano Suárez-Alvarez May 19 '11 at 12:20
What coefficients did you have in mind? As long as you're using a commutative ring, there's a Kunneth spectral sequence. If you've never used a SS before, they can be intimidating. If you give more detail, maybe someone here can help further. – jd.r May 19 '11 at 12:31
Thanks to all of you for your comments. I was talking about the basic Künneth formula when the ring is a field or one of the homology module of one the spaces is a free R-module. I was thinking about computing the (co)homology of $RP^{n}xRP^{m}$ – El Moro May 19 '11 at 16:55

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