Why we need Cup product on different topological spaces.

Question

I know the meaning of cohomology groups. These groups give some information about the topology of the space like connectedness. Also these groups give information about the different dimensional holes in topological spaces. I want to know that way we need extra operation on these groups like cup product.

Answer 1

Because there are spaces with identical cohomology (group) structure, but whose cup-product structures are different.

Thus, the cup product allows us to algebraically distinguish more spaces from one another.

As an example, the cohomology structure of $\Bbb RP^2$ (with $\Bbb Z/2\Bbb Z$ coefficients) is $\Bbb Z/2\Bbb Z, \Bbb Z/2\Bbb Z, \Bbb Z/2\Bbb Z$. That happens to also the structure that arises from attaching a circle to a sphere at the north pole. (i.e., $S^1 \vee S^2$).

But the cohomology RING structures are different: in the first case, letting $a$ be the generator of $H^1$, we have $a \cup a \ne 0$, but in the second, the generator $b$ of $H^1$ has $b \cup b = 0$. Hence these two spaces cannot be homotopy-equivalent.