# Presentation of Fundamental Group for a Seifert Fibred Space

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!

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.