I have given a lens space L(p,q). What compact three-manifold with boundary do we get if we remove the interior of a neighbourhood of the core curve of one of the two solid tori that make up the lens space?
I figured out that I have to remove a solid torus from L(p,q) but I can not visulaize or proof what do I get if do that.
It looks like the definition of $L(p,q)$ you are using is that you take two solid tori and identify their boundaries along a slope of $p/q$.
A solid torus minus its core is homeomorphic to $T\times I$, where $T=S^1\times S^1$ and $I=[0,1]$. Every homeomorphism $T\to T$ extends to a homeomorphism $T\times I\to T\times I$ that is unique up to isotopy, with $T$ being identified with $T\times \{0\}$. Thus, there is only one way (up to homeomorphism) to identify the boundary of a solid torus with one of the boundary components of $T\times I$. The result is homeomorphic to a solid torus.