Assume the embedding of $S^0$ in $S^3$ extends to an (orientation preserving) embedding of $S^0 \times D^3$ in $S^3$. Show that the manifold which is the result of surgery is diffeomorphic to $S^1\times S^2$.
This is somehow explained on Wikipedia, but the explanation there is incomprehensible to me (and moreover the dimension doesn't allow me to visualize what's going on). Could anyone show me why this is true in more detail?
I will show only that the result of the surgery is homeomorphic to $S^1 \times S^2$. Them being diffeomorphic follows from Moise's Theorem.
You can split $S^1 \times S^2$ into two manifolds $D^1 \times S^2$, where the boundary spheres are identified. One of these, by definition, is just the handle of the surgery. So all that remains to show is that $D^1 \times S^2$ is homeomorphic to $S^3$ with 2 balls removed. To see this, we can embed $[1,2] \times S^2$ into $\mathbb{R}^3$ by interpreting each point as spherical coordinates for $\mathbb{R}^3$, and then use stereographic projection to get to $S^3$. The image of this embedding is just $S^3$ with two balls around the poles removed.