Homology of connected sum

Statement
Suppose $$M_1$$ and $$M_2$$ are the connected manifolds of dimension $$n$$ whose connected sum is being taken. Assume both We have:

Euler characteristic
The Euler characteristics are related by the following formula when both $$M_1$$ and $$M_2$$ are compact connected manifolds:

$$\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)$$