Here is the question I am trying to compare two solutions of it:
Using the cup product $H^k(X, A;R) \times H^l(X,B;R) \to H^{k+l}(X, A\cup B;R),$ show that if $X$ is the union of contractible open subsets $A$ and $B,$ then all cup products of positive-dimensional classes in $H^*(X;R)$ are zero.
Here is the first solution:
My concerns about it is as follows:
I think I understand this solution, and nothing bothers me in it (Hopefully it is correct, let me know if there are typos that I missed in it please)
Here is the second solution:
My questions about it are:
What exactly this author is doing? Why using excision? Why he is proving using cochains?
Could someone help me answer these questions please?