Examples
Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. First consider the Torus, T. One may check that if, then has identical cohomology groups. However, the multiplication of the cup product distinguishes the associated cohomology rings. In the case of X the multiplication of the cochains associated to the copies of is degenerate, whereas in T multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to Z (more generally M where this is the base module).
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