I have been learning about Brieskorn homology 3-spheres $\Sigma(a\_1,...,a\_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres." 


If I understand correctly, it is explained that one can realize the homology three spheres by considering a trivial bundle $S^2 \times S^1$ and remove small discs $D\_i$ about marked points $x\_i ,..., x\_n$ on the two sphere. One can then perform a rational Dehn surgery with surgery slopes $a\_i/b\_i$ along the links given by $\partial D\_i \times pt$ to obtain the Brieskorn homology spheres. 


More precisely Saveliev shows that there exist $b\_i$ that make this Dehn surgery construction into a rational homology sphere and then states that it follows from Kirby calculus that the resulting 3 manifolds are independent of $b\_i$. I should say that, unfortunately, I don't know anything about Kirby calculus. Clearly, in this construction the $a\_i$ and $b\_i$ do not play symmetric roles. Nevertheless, I am interested in the following:


A general question is: To what extent can we determine the homology three spheres from the $b\_i$? 


A specific question/motivation: I am not at all a low dimensional topologist. However, a Brieskorn homology sphere naturally arises as a surgery of the type above in a construction I am working on. I know that $n=4$ and all $b\_i= \pm 1$, or $n=3$ and $b\_1, b\_2= \pm 1.$ It is natural to ask: what manifold am I possibly looking at?



A question about Dehn surgery and Brieskorn homology 3-spheres
Seifert manifolds that are homology spheres can be determined as follows. Let 
$$ M =(S^2, (p\_1,q\_1), \ldots, (p\_h,q\_h)) $$
denote the Seifert manifold obtained by filling the $h$ tori in $D\_h \times S^1$ (where $D\_h$ is the 2-sphere minus $h$ discs) with coprime parameters $(p\_i, q\_i)$. Since we consider Seifert manifolds, we ask $p\_i \neq 0$. We may say that $M$ fibers over the orbifold
$$(S^2, p\_1, \ldots, p\_h).$$


The  *Euler number*  of $M$ is an important invariant, defined as:
$$ e = \sum\_{i=1}^h \frac {q\_i}{p\_i}.$$
A simple homology calculation shows that the integral homology $H\_1(M)$ is finite if and only if $e\neq 0$, and in that case we have
$$|H\_1(M)| = ep\_1\cdots p\_h = \sum\_{i=1}^h q\_ip\_1\cdots \widehat{p\_i}\cdots p\_h.$$
This implies that $M$ is an integral homology sphere if and only if
$$\sum\_{i=1}^h q\_ip\_1\cdots \widehat{p\_i}\cdots p\_h = \pm 1.$$
This in turn easily implies the following:



> 
> For every set $p\_1, \ldots, p\_h\geqslant 2$, of $h \geqslant 3$ pairwise coprime integers there is a unique homology sphere $\Sigma(p\_1,\ldots,p\_h)$, which fibers over the orbifold $(S^2,p\_1, \ldots, p\_h)$. Every Seifert homology sphere distinct from $S^3$ arises in this way. 
> 
> 
> 


Proof: it is easy to see that the $p\_i$'s must be coprime. On the other hand, if they are, one can find easily $q\_1,\ldots, q\_h$ that give $|H\_1(M)|=1$. Two different strings of $q\_1,\ldots, q\_h$ are related by ``moves'' that do not affect $M$. (The same $M$ can be described by different parameters.)


This theorem is probably written somewhere, like in Orlik's book or Seifert's original paper, but I can't remember. 


