Suppose $Y_f=D^n \cup_fY$ is the space obtained from $Y$ by attaching an n-cell via the map $f: S^{n-1} \rightarrow Y$, $n \geq 3$.
I'm trying to understand the homology groups $H_{n+1}(Y_f),H_{n}(Y_f),H_{n-1}(Y_f)$ from the long exact sequence induced via mayer-vietoris sequence.
However, I am confused:
$...\rightarrow H_{n+1}(Y_f) \rightarrow H_{n}(S^{n-1})\rightarrow H_{n}(Y) \oplus H_{n}(D^n) \rightarrow H_{n}(Y_f) \rightarrow H_{n-1}(S^{n-1}) \rightarrow H_{n-1}(Y) \oplus H_{n-1}(D^n) \rightarrow H_{n}(Y_f) \rightarrow...$
Specifically on the map $H_{n-1}(S^{n-1}) \rightarrow H_{n-1}(Y) \oplus H_{n-1}(D^n)$. The Mayer Vietoris sequence is defined here by taking the generator, $<1>$, of $H_{n-1}(S^{n-1}) \cong \mathbb{Z}$ to $(1,-1)$.
My main question is this: What roll does the degree of the attaching map $f$ play in all of this? Shouldn't the degree of the attaching map show up in this homology sequence somehow? If not, why? If so, where? Thanks!