Although there are good articles about this theme like induced map homology example, I would like to get a more explicit answer.
I know that one way to find such a map is the following: $ f:X\to Y $, then $ f_\ast[x]=[f (x)] $. So we have to look at the generator of $ H_p(X) $ under $ f $ and express it in terms of generators of $ H_p (Y) $. I can easily imagine the generators of the case $ p=1$, since the homology group is then the abelization of the corresponding fundamental group, but for $ p\neq1$ I have no picture in mind. Could someone explain me how to find the generators of the other homology groups to calculate the induced map? How can I find the map in the other cases ($ p\neq1$)? An explicit example would be helpful.
An other way: We look at the CW-structure or at the simplicial structure and deduce the induced map on the homology.
I can't imagine how this way works. Could someone explain it by giving an explicit example?
This question often arises, when I use Mayer-Vietoris. Maybe an example in this context would be helpful. Thank you!