Using Mayer-Vietoris sequence often one needs to know which map is induced in homology by an inclusion, let me try to explain with an example:
Let's take the real projective plane with a hole i.e. $X=\mathbb{PR}^2\setminus D$ where $D$ is an open disc. The inclusion $i:\partial D\hookrightarrow X$ induces a morphism in homology $i_*:H_1(\partial D)\to H_1(X)$. Both homology groups are $\mathbb{Z}$ because $\partial D\sim S^1$ and $X$ deformation retracts to $\mathbb{PR}^1\sim S^1$.
My question is: how to compute $i_*(1)$?
I've tried different approaches:
1) With simplicial homology: I can triangulate $X$ and "push" a fundamental cycle for $\partial D$ to two times a fundamental cycle for $\mathbb{PR}^1$ by summing some boundaries. This way is the most clear to me but i don't think is the most direct one.
2) With singular homology: I take a singular generator for $H_1(\partial D)$, i.e. $\sigma:[0,1]\to \partial D=S^1$ $\sigma(t)=e^{2i\pi t}$. By the deformation retraction of $X$ to $\mathbb{PR}^1$ we see that $i\circ\sigma:[0,1]\to X$ is homotopic to twice a generator for the singular homology group of $\mathbb{PR}^1$.
Is there and even more direct way to see this? Maybe using CW homology?
As an another example, I took this from Hatcher's book:
What's behind the sentence: "The map $\Phi$ is $\mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z}$, $1\to (2,-2)$ , since the boundary circle of a Mobius band wraps twice around the core circle"?