Abstract:

We say that a 3-manifold is SU(2)-abelian if every SU(2)-representation of its fundamental group has abelian image. In this work, we classify all SU(2)-abelian graph manifold rational homology 3-spheres with a single JSJ torus. For a 3-manifold Y with torus boundary, we define the invariant T(Y, ∂Y ) that describes Hom(π1(Y ), SU(2)) up to conjugation. In particular, the invariant T(Y, ∂Y ) is a subspace of a torus. For a generic closed manifold Y1 ∪Σ Y2, we determine the SU(2)-abelian status of Y1 ∪Σ Y2 by studying the intersection of T(Y1, ∂Y1) with T(Y2, ∂Y2). Finally, we prove that if a graph manifold rational homology 3-sphere with a single JSJ torus is SU(2)-abelian, then it is an L-space for Heegaard Floer

homology.

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