I am trying to calculate the fundamental group of the Brieskorn sphere $\Sigma(p,q,r)$. However, I have no clue how to do that. Is there an elementary way to calculate it?
$\Sigma(p,q,r)$ is defined for given positive integers $p,q,r$ in the following way:
Take the polynomial $f(z_1,z_2,z_3)=z_1^p+z_2^q+z_3^r$ defined $\mathbb C^3\to\mathbb C$ with an isolated singularity at the origin $(0,0,0)$. So, the zero set $V_f$ of this polynomial is a 2-dimensional complex manifold, and essentially a 4-manifold, away from the origin. When we (transversally) intersect $V_f$ with some 5-dimensional sphere $S^5_\epsilon\subset \mathbb C^3$ centered at the origin with radius $\epsilon$, we obtain a closed connected orientable 3-manifold. This manifold is denoted $\Sigma(p,q,r)$, and called a Brieskorn sphere, named after Egbert Brieskorn.
This is all done in Milnor's paper on the subject : http://www.maths.ed.ac.uk/~aar/papers/milnbries.pdf