Presentation of Fundamental Group for a Seifert Fibred Space

Question

I am trying to understand how to make sense of the presentation for a seifert fibred space geometrically. I understand that for each exceptional fibre (with index a/b) you get a generator with torsion for the fund grp s.t. the index corresponds to its order. And a regular fibre(which is homotopic to all of the other regular fibres) is a generator which commutes with everything. What I would like to understand is the other generators and the relations between them and how to think about it visually if possible. I am a newcomer to this subject and geometric topology in general so the more details the better. Thanks!

Answer 1

Consider (for simplicity) the case when your Seifert manifold fibers over an orientable surface. Let $\Sigma$ be a genus $g$ orientable surface with $m$ boundary components, and $\frac{p_i}{q_i}$ for $i=1, \dots , m$ a collection of $m$ rational numbers. Let us compute the fundamental group of the Seifert Manifold $$M= \left( g ;\frac{p_i}{q_i}, \ i=1, \dots , m \right). $$

By definition, $M$ is obtained from $\Sigma \times S^1$ (a three-manifold with $m$ torus boundary components) by Dehn filling. So the fundamental group of $M$ is a quotient of $\pi_1(\Sigma \times S^1) = \pi_1(\Sigma) \times \pi_1(S^1)$ modulo the normal closure of the subgroup $H \subset \pi_1(\Sigma) \times \mathbb{Z}$ generated by the curves of slopes $\frac{p_i}{q_i}$ killed during the Dehn filling procedure (this is because Van Kampen's theorem). Thus in order to obtain a presentation $\pi_1(M)$ we only have to take a presentation of $\pi_1(\Sigma) \times \pi_1(S^1)$ and to add the relations $$ \text{filling curves} =1.$$

Let's do it. A presentation of $\pi_1(\Sigma)$ is given by $$\langle a_1, b_1, \dots a_g, b_g, c_1, \dots c_m \ | \ [a_1, b_1] \dots [a_g, b_g]= c_1 \dots c_m \rangle$$ where $c_1 \dots c_m$ are the homotopy classes for the boundary components. In order to obtain a presentation of $\pi_1(\Sigma) \times \mathbb{Z}$ we add to the presentation above one generator more $\gamma$ commuting with all the other generators $$\langle a_1, b_1, \dots a_g, b_g, c_1, \dots c_m , \gamma \ | \ [a_1, b_1] \dots [a_g, b_g]= c_1 \dots c_m, \ [\gamma, a_i]=[\gamma, b_i]=[\gamma, c_i]=1\rangle$$ Notice that in that presentation $\gamma$ represents the homotopy class of a non-singular fiber. Finally we add the relations $$c_i^{p_i} \cdot \gamma^{q_i} =1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1, \dots ,m$$ coming from the Dehn filling operation, and the presentation is done.