# Integer homology 3-spheres admit irreducible representations in SL(2,C)

@article{Zentner2018IntegerH3, title={Integer homology 3-spheres admit irreducible representations in SL(2,C)}, author={Raphael Zentner}, journal={Duke Mathematical Journal}, year={2018} }

We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition, and for Seifert fibered integer homology spheres this is well known. We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation. By work of Boileau, Rubinstein, and Wang, the general case follows. Using…

## 28 Citations

Stein fillings and SU(2) representations

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We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are…

A sheaf-theoretic model for SL(2,C) Floer homology

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Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth…

Toroidal homology spheres and SU(2)-representations

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We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations. Our methods use instanton Floer…

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We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be…

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A knot K Ă S is called SUp2q-abundant if for all but finitely many r P Qzt0u, there is an irreducible representation π1pS r pKqq Ñ SUp2q, and the slope r “ u{v ‰ 0 with no irreducible SUp2q…

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A knot is circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the…

On meridian-traceless SU(2)-representations of link groups

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- 2021

Suppose L is a link in S3. We show that π1(S − L) admits an irreducible meridian-traceless representation in SU(2) if and only if L is not the unknot, the Hopf link, or a connected sum of Hopf links.…

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We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An…

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It is shown that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth, then the Heegaard genus of M is at most 48(k+1) (resp. pathwidth) k, and it is not shown that there exists an infinite family of closed 3- manifolds not admitting triangulations of bounded pathwidth or trewidth.

Embeddability in the 3-Sphere Is Decidable

- Mathematics, Computer ScienceJ. ACM
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We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R3? By a known…

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