I am reading Hatcher, section 3.2 and when he defines a cup product he says that we need to "consider cohomology with coefficients in a ring R" and I am just confused why we need the coefficients to be in a ring.
The reason I ask is because in section 2.2 under "Homology with Coefficients" he defines
Homology groups with coefficients in G, where G is a fixed abelian group. Why are we switching to rings and modules when we cross over into cohomology?
After some more reading I am understanding a little bit more:
From the very beginning of Chapter 3 in Hatcher: (Which I should have read to begin with but thought it was ok to skip over)
Homology groups are contravariant functors, while cohomology groups are covariant functors.
Contravariance leads to extra structure in cohomology: that's what the cup product is.
Why doesn't this happen for homology? Actually, we can define the cross product (discussed in section 3.B) but this relies on the map $X \times X \rightarrow X$. And, for a general X, the only way we can define a map like that is by using projection onto one of the factors, and since projections "collapse the other factor to a point, the resulting product is rather trivial"
For cohomology on the other hand, we need to rely on a map like this: $X \rightarrow X\times X$, for which we use the "diagonal map" $\Delta(x) = (x,x)$ which is actually waht the cup product is.
I am still a bit confused on this though, and would appreciate further clarification: How does the diagonal map lead to the cup product explicitly?
Is it because we somehow get to switch the arrows in the cross product map? In that case, it seems strange to define the cup product first and then the cross product later.