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Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it :

1)Basically, understanding how/why the (co)homology functor can/does fail to agree with the product of spaces, when we do not work with coefficients in a field i.e.,

## H^k(X \times Y) \neq \bigoplus_{i+j=k} H^i (X) \otimes H^j(X) ##

is not satisfied when we do not work with coefficients in a field; I am tooignorant at this point to understand why/how field coefficients matter ( I was considering the issue of torsion, but there was no mention that the fields had to be of characteristic zero). Also,

2)What kind of correction terms do we need when we consider (co)chain groups on product spaces, i.e., what is the relationship between## C_p(X \times Y)## and ##C_p(X), C_p(Y) ## and how the differential/boundary operator works on the (co)homology of products, i.e., if we have ## \delta ## boundary operator on different (co)chain complexes, how/when can we define a boundary operator on the product ## X \times X ##. Does it make sense to consider a product of (co)chain complexes with different boundary operators?

Basically, we work with inclusion , diagonal and projection diagrams/operations, i.e., ## x \rightarrow (x,x)##, etc.

Thanks.

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# Ref. Please: Homology Functor on Prod. Spaces ; Chain Groups, Differentials on Product spaces

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