= Euler characteristic of a connected sum
{c}
{title2=$\chi(M\#N)=\chi(M)+\chi(N)-\chi(S^d)$}
For closed connected oriented $d$-manifolds with $d\geq2$,
$$
\chi(M\#N)=\chi(M)+\chi(N)-\chi(S^d).
$$
Deleting an open ball changes the <Euler characteristic> by $-1+\chi(S^{d-1})$, and gluing subtracts $\chi(S^{d-1})$. Since $\chi(S^{d-1})+\chi(S^d)=2$, the formula follows. Thus a <connected sum of oriented manifolds> subtracts two from the sum of Euler characteristics in even dimension and subtracts zero in odd dimension.
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