Surgery and Euler Characteristic 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.
 A: Let $U = M\setminus(S^p\times D^{n-p})$ and $V = S^p\times D^{n-p}$. Then $M = U\cup V$ and $U\cap V = S^p\times S^{n-p-1}$, so 
\begin{align*}
\chi(M) &= \chi(U) + \chi(V) - \chi(U\cap V)\\ 
&= \chi(U) + \chi(S^p\times D^{n-p}) - \chi(S^p\times S^{n-p-1})\\ 
&= \chi(U) + \chi(S^p) - \chi(S^p)\chi(S^{n-p-1}).
\end{align*}
Now let $W = D^{p+1}\times S^{n-p-1}$ and note that $M' = U\cup W$ and $U\cap W = S^p\times S^{n-p-1}$, so 
\begin{align*}
\chi(M') &= \chi(U) + \chi(W) - \chi(U\cap W)\\
&= \chi(U) + \chi(D^{p+1}\times S^{n-p-1}) - \chi(S^p\times S^{n-p-1})\\
&= \chi(U) + \chi(S^{n-p-1}) - \chi(S^p)\chi(S^{n-p-1}).
\end{align*}
Using the fact that $\chi(U) = \chi(M) - \chi(S^p) + \chi(S^p)\chi(S^{n-p-1})$ we see that
\begin{align*}
\chi(M') &= \chi(M) - \chi(S^p) + \chi(S^p)\chi(S^{n-p-1}) + \chi(S^{n-p-1}) - \chi(S^p)\chi(S^{n-p-1})\\ 
&= \chi(M) - \chi(S^p) + \chi(S^{n-p-1})\\
&= \chi(M) - [1 + (-1)^p] + [1 + (-1)^{n-p-1}]\\
&= \chi(M) - (-1)^p + (-1)^{n-p-1}\\
&= \chi(M) + (-1)^{p+1}[1 + (-1)^n].
\end{align*}
That is, 
$$\chi(M') = \begin{cases}
\chi(M) & n\ \text{odd}\\
\chi(M) - 2 & n\ \text{even}, p\ \text{even}\\
\chi(M) + 2 & n\ \text{even}, p\ \text{odd}.
\end{cases}$$

In particular, surgeries do not change the parity of the Euler characteristic. Another way to see this is to use the fact that if $M$ and $N$ are related by a sequence of surgeries, then they are cobordant, so they have the same Stiefel-Whitney numbers. In particular, $\underline{w_n}(M) = \underline{w_n}(N)$, but this is nothing but the Euler characteristic mod two.
Moreover, any two cobordant manifolds are related by a sequence of surgeries. The above computation shows that if $M$ and $N$ are cobordant, then $\frac{1}{2}|\chi(M) - \chi(N)|$ gives a lower bound on the number of surgeries to obtain one from the other.
