TITLE: A question about Dehn surgery and Brieskorn homology 3-spheres
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: 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.