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Finding the fundamental group of the complement of a certain graph

The problem is in the picture.enter image description here

enter image description here

My question is: how to show $\pi_1(Y)$ has the presentation $\langle a,b,c \mid aba^{-1}b^{-1}cb^{\varepsilon}c^{-1} \rangle$ for $\varepsilon = \pm 1$, and how to prove $\pi_1(Y) = \pi_1(\mathbb{R}^3-Z)$?

Thank you in advance!

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Proton
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  • Maybe you can give a description of this manifold as a CW-complex with: one $0$ cell; three $1$ cells named "a," "b," and "c"; and one $2$-cell, attached along the given relation. – William Apr 14 '12 at 03:26
  • @you Hi,now I can give Y the CW-complex structure,but how to prove $\pi_1(Y)=\pi_1(\mathbb{R}^3-Z)$?I can image it,but cann't prove it. – Proton Apr 14 '12 at 12:45

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