Given a $3$-manifold $M$, is $M\# S^3$ homeomorphic to $M$?

Question

Given a surface $N$, we have $N\# S^2$ homeomorphic to $N$, I wonder if given a $3$-manifold $M$, we have $M\# S^3$ homeomorphic to $M$?

Answer 1

Yes; you only need to identify the removed disk in $M$ with the disk in $S^3$.

By definition, $M\# S^3$ is given by taking $U\subset M$ and $V\subset S^3$ with $U\cong V\cong D^3$ and identifying their boundaries. You can write this with coordinates if you want, but it won't be much more enlightening than the verbal argument.