- #1

Jamma

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I was just wondering, is there a canonical relationship between the two, in the following sense (I only need to deal with products of elements of the first cohomology group here):

We know that elements of the first simplicial cohomology group can be represented as equivalence classes of maps into some representing space. I'm only bothered about H^1, for now, so we have that H^1(X) = [X,S^1].

We also know that there is a cup product on cohomology definied in a reasonable simple way on cochains.

My question is: given two functions f,g: X ->S^1 which define elements of H^1(X), is there a canonical way of defining the cup product of f and g in terms only of the function f and g? It seems that there should be, but I don't see a canonical way of doing so. Is there some way I should be embedding the torus into the infinite dimensional complex projective space?