# Find a 4-manifold bounding a 3-torus with any abelian representation in SL_2

Fix $$x,y,z\in \mathbb{C}^*$$ and let $$M=S^1\times S^1\times S^1$$ with $$\rho:\pi_1(M)\to \operatorname{SL}_2(\mathbb{C})$$ mapping the three generators to diagonal matrices with entries $$(x,x^{-1})$$, $$(y,y^{-1})$$, $$(z,z^{-1})$$ respectively. The question is can you construct a (smooth) 4-manifold $$W$$ bounding $$M$$ with a representation $$\tilde{\rho}:\pi_1(W)\to \operatorname{SL}_2(\mathbb{C})$$ extending $$\rho$$?

1) Such a 4-manifold exists by a non-constructive argument based on the fact that the map $$H_3(\mathbb{C}^*,\mathbb{Z})\to H_3(\operatorname{SL}_2(\mathbb{C}),\mathbb{Z})$$ induced by the inclusion of diagonal matrices in $$\operatorname{SL}_2(\mathbb{C})$$ is almost 0.
2) I have a homological argument saying that the extension $$\tilde{\rho}$$ cannot be abelian.
3) By standard arguments, the same manifold should work for $$(x,y,z)$$ in a Zariski open subset of $$\mathbb{C}^*\times\mathbb{C}^*\times \mathbb{C}^*$$.