Given $n$ points $p\_1,\dots,p\_n$ in $S^2$ one gets a product link $L\_n=\{p\_1,\dots,p\_n\}\times S^1$ inside the closed 3-manifold $S^2\times S^1$, which can be looked at as a trivially framed link (by picking a tangent vector $v\_i$ at each $p\_i$ and dragging it along the factor $S^1$ to get the framing). I'm interested in the result of the surgery of $S^2\times S^1$ along $L\_n$. Is it diffeomorphic to some ``space with a name'' (e.g., $S^3$, $S^1\times S^1\times S^1$, a lens space, etc?)


I'm interested in the answer for an explict computation of a Reshetikhin-Turaev invariant I'm faced with. Yet, the problem is purely topological and I expect there is well known answer, but I haven't been able to find it or work it out myself so far.



Link surgery on $S^2\times S^1$
The manifold you're describing has a very natural open book decomposition (see e.g. John Etnyre's [lecture notes](http://arxiv.org/abs/math/0409402)) in which the pages are the $n$-punctured spheres $(S^2 \backslash \cup\_i N(p\_i)) \times \{\*\}$ and the monodromy is trivial. We can view this as a Murasugi sum of n-1 copies of the abstract open book $(S^1\times I, \mathrm{Id})$, which describes $S^1\times S^2$, and since Murasugi sums of open books correspond to connected sums of 3-manifolds, the manifold you want is $\#^{n-1}(S^1\times S^2)$.


