I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This is defined here Surgery theory (wikipedia)

It seems like Mayer-Vietoris is the way to go here (please let me know if this isn't or there are other ways, though). My question is about this map in the M-V sequence for $M'$, for $0<i<n$:

$H_i(S^p\times S^{n-p-1})\rightarrow H_i(M-\{S^p \times D^{n-p})) \oplus H_i(D^{p+1} \times S^{n-p-1} )$

I'd like to show that it is injective ($H_i(S^p\times S^{n-p-1})=\mathbb{Z}$ for $i=p$ and $n -p-1$, and $0$ for all other $0<i<n$). It looks like a cycle in $S^p\times S^{n-p-1}$ that is a boundary in $D^{p+1} \times S^{n-p-1}$, will not be a boundary in $M-\{S^p \times D^{n-p})$, using a few low-dimensional examples but I'm not sure how to show this more convincingly.