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Thank you very much for answering again and adding so many details! But there is still one point which I can't get my head around. If you assume that $\mathcal{M}$ is prime and the conclusion (in the compressible case) is that $\mathcal{M}=\mathcal{N}\#\hat{T}$ for some $3$-manifold different from the $3$-sphere, isn't that a contradiction? I mean, prime means that every splitting of the form $\mathcal{M}=N_{1}\# N_{2}$ implies that either $N_{1}$ or $N_{2}$ has to be a $3$-sphere, right?....
After some time, I cam back to this problem and I still do not fully understand. In the paragraph starting with "Such a manifol" you assume that $\mathcal{M}$ is non prime, right? (or as explained above, equivalently, non boundary-prime). But why is the compressing disk then "necessarily non-seperating"?
@MoisheKohan Do understand the refined result correctly, is it correct that (2) can be divided into manifolds of the form $\hat{T}\#\mathcal{M}$ (compressible boundary and non-prime) and into non-prime manifolds with incompressible boundary? Furthermore, the solid torus is prime and has compressible boundary, hence the list basically just contains the four possible combinations of prime vs. non-prime an compressible vs. incompressible.
Thank you very much for your advice, this is indeed a very usefull website! I know that finding a phd position takes some time, for that reason I already start looking yet. I will finish my Master's around April and my plan was to start with a phd sometimes after, so some time after finishing my Masters and the start of the new winterterm 2022/23
Thank you very much for your answer and your advice. Indeed, the terminology "hot topic" was badly chosen and I agree that it is not an advisable criterion for a PhD. My question was more in the direction about some current topics in this area. :-)
