Glue two spheres

Question

Taking a small open neighbourhood around a point $p$ in $S^3$ gives a new domain $M=S^3\setminus U(p)$, then gluing $M$ with itself along the boundary will give a new space $\tilde{M}\sim S^3$. I am just curious if I do the similar thing to a circle instead of point $p$. what will the new space look like ? Or what topology does this new space has ? for example, how do fundamental groups change $\pi_1,\pi_2$?

Answer 1

What you're asking about is called "Dehn surgery" of 3-manifolds. You can get an incredible number of different 3-manifolds, with very different fundamental groups, by taking a knot in $S^3$, deleting a neighborhood, then gluing it back in by a diffeomorphism of the boundary; the best way to talk about what $\pi_1$ is of the result is to talk about $\pi_1$ the knot complement.

It's a big subject, you might enjoy reading about it.