- #1

Bacle

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A couple of things, please:

1) I am going over the Leray-Hirsch theorem in Hatcher's AT , which gives the conditions

under which we can obtain the cohomology of the top space of the bundle

from the tensor product of the cohomology of the fiber, and that of the base

( a sort of relative to Kunneth's theorem), and I see the statement, that

(paraphrase) the isomorphism:

H* (E;R)=H*(B;R)(x)H*(F;R)

where R is a ring, and (x) is the tensor product "is not always a ring homomorphism"

question: is this then an isomorphism of cochain complexes.?. If so, does

anyone know the def. of iso. of cochain complexes.?.

2)How do we tensor cochains.?. How do we tensor Cochain complexes

The isomorphism above is described explicitly, and uses the tensor product of chains.

Anyone know how to define this.?

How about the tensor product of cochain complexes C,C'.?. My naive guess would be:

H_n( C(x)C') = (+)(H_i(C;R)(x)H_(n-i)(C';R)) as a set

but I don't see how to define the coboundary. I tried to imitate the construction

of the tensor of chain complexes, but I am just going in circles.

Any Ideas.?

Thanks.