I'm trying to figure out how to actually compute maps in the Mayer-Vietoris sequence as it's quite useless unless you actually know some maps. Hatcher computes the homology for the Klein bottle $K$ on page 151. I still can't seem to understand the logic he uses to determine the map
$H_1(A\cap B)\to H_1(A)\oplus H_1(B)$,
where $A$ and $B$ are the usual Möbius bands. There are at least two steps here:
Finding the actual cycle in $A\cap B$ that generates $H_1(A\cap B)$.
Figuring out where this goes in $H_1(A)$.
I can't seem to be able to visualize what a cycle looks like. For homotopy this is easy, but I've usually just used homology based on the Eilenberg-Steenrod axioms and for MVS problems it actually looks like you need to look at the actual geometry.
I was also wondering if I can actually think about singular $n$-simplices under homotopy equivalences. So if two singular $n$-simplices are homotopic as maps, do they necessarily generate the same element in the homotopy group?
I would appreciate any help regarding ways to visualize this and how people generally think and construct the homomorphisms in the MVS.