I'm trying to compute the fundamental group (using Van Kampen) of a space which appears when identifying the disjoint boundaries of a 3-manifold with boundaries. My knowledge of 3-manifolds is none and that's why I am open to approach the task from other perspective.
First of all, I have found this related question Fundamental group of a quotient on a solid torus.
but I don't understand quite well what it is explained there. And in order to solve my problems I think I should start by understanding this one.
My first problem is to understand this:
''imagine two solid torus one inside of the other one. Now consider the topological space that is the difference of the bigger one with the one that is inside. You get a torus with a hole that is a 3-manifold with borders. Now you identify the interior border with the exterior border (in an obvious way). ''
I have pictured one torus inside of the other:
So by ''you identify the interior border with the exterior border (in an obvious way)'' I imagine you just obtain a single torus. But I suppose I am wrong since in that case the fundamental group would be $\mathbb{Z} \times \mathbb{Z}$ and the user who asked that questions says that he obtains a different fundamental group.
Could you help me with that please? Where is my flaw?
Second,
Given a torus inside another one. Suppose the first winds twice. I mean, we have a 3-dimensional manifold such that the disjoint union of both torus is the boundary.
And then we identify both boundaries along each other. Suppose we name that final space $X$. I'm not sure I see $X$. Could you help me by pointing out a way to imagine it? I'm almost sure that i'm wrong but could we deformation retract $X$ to the following space?:
which would be a mapping torus from $S^1 \vee S^1$ Because in that case I think I could compute the fundamental group easily.
I also have tried to write down a CW complex but since I don't see the space it is hard to do so.
And do you know any reference: book, class notes or resource on the net which helps to develop the intuition to see that kind of spaces?
I hope that hen you help me to see that space it will be easier to tackle my problem.
Any help would be appreciated!! And sorry for the long question\s.