Volume 5 (2022)
Baldwin, John A.;  Sivek, Steven
Instanton L-spaces and splicing
Annales Henri Lebesgue, Volume 5 (2022), pp. 1213-1233.

Metadata

DOI10.5802/ahl.148
Keywords Instanton Floer homology, L-spaces, incompressible tori

Abstract

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 an instanton L-space. This recovers the recent theorem of Lidman–Pinzón-Caicedo–Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial SU(2)-representation, and consequently Zentner’s earlier result that the fundamental group of every closed, oriented 3-manifold besides the 3-sphere admits a nontrivial SL(2,ℂ)-representation.


References

[ABDS20] Alfieri, Antonio; Baldwin, John A.; Dai, Irving; Sivek, Steven Instanton Floer homology of almost-rational plumbings (2020) (https://arxiv.org/abs/2010.03800)

[BS16] Baldwin, John A.; Sivek, Steven Instanton Floer homology and contact structures, Sel. Math., New Ser., Volume 22 (2016) no. 2, pp. 939-978 | DOI | MR | Zbl

[BS18] Baldwin, John A.; Sivek, Steven Stein fillings and SU(2) representations, Geom. Topol., Volume 22 (2018) no. 7, pp. 4307-4380 | DOI | MR | Zbl

[BS21] Baldwin, John A.; Sivek, Steven Framed instanton homology and concordance, J. Topol., Volume 14 (2021) no. 4, pp. 1113-1175 | DOI | MR

[BS22a] Baldwin, John A.; Sivek, Steven Framed instanton homology and concordance, II (2022) (https://arxiv.org/abs/2206.11531)

[BS22b] Baldwin, John A.; Sivek, Steven Khovanov homology detects the trefoils, Duke Math. J., Volume 171 (2022) no. 4, pp. 885-956 | DOI | MR | Zbl

[CS83] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. Math., Volume 117 (1983) no. 1, pp. 109-146 | DOI | MR | Zbl

[Eft15] Eftekhary, Eaman Floer homology and splicing knot complements, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3155-3213 | DOI | MR | Zbl

[Eft20] Eftekhary, Eaman Correction to the article Floer homology and splicing knot complements, Algebr. Geom. Topol., Volume 20 (2020) no. 6, pp. 3205-3218 | DOI | MR | Zbl

[Flo90] Floer, Andreas Instanton homology, surgery, and knots, Geometry of low-dimensional manifolds, 1 (Durham, 1989) (London Mathematical Society Lecture Note Series), Volume 150, Cambridge University Press, 1990, pp. 97-114 | MR | Zbl

[FS90] Fintushel, Ronald; Stern, Ronald J. Instanton homology of Seifert fibred homology three spheres, Proc. Lond. Math. Soc., Volume 61 (1990) no. 1, pp. 109-137 | DOI | MR | Zbl

[HL16] Hedden, Matthew; Levine, Adam Simon Splicing knot complements and bordered Floer homology, J. Reine Angew. Math., Volume 720 (2016), pp. 129-154 | DOI | MR | Zbl

[HRW17] Hanselman, Jonathan; Rasmussen, Jacob; Watson, Liam Bordered Floer homology for manifolds with torus boundary via immersed curves (2017) (https://arxiv.org/abs/1604.03466)

[Kirby97] Kirby, Rob Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) (Kirby, Rob; Kazez, William H., eds.) (AMS/IP Studies in Advanced Mathematics), Volume 2, American Mathematical Society, 1997, pp. 35-473 | MR | Zbl

[KLT21] Karakurt, Çağri; Lidman, Tye; Tweedy, Eamonn Heegaard Floer homology and splicing homology spheres, Math. Res. Lett., Volume 28 (2021) no. 1, pp. 93-106 | DOI | MR | Zbl

[KM04] Kronheimer, Peter B.; Mrowka, Tomasz S. Witten’s conjecture and property P, Geom. Topol., Volume 8 (2004), pp. 295-310 | DOI | MR | Zbl

[KM10] Kronheimer, Peter B.; Mrowka, Tomasz S. Knots, sutures, and excision, J. Differ. Geom., Volume 84 (2010) no. 2, pp. 301-364 | MR | Zbl

[KM11a] Kronheimer, Peter B.; Mrowka, Tomasz S. Khovanov homology is an unknot-detector, Publ. Math., Inst. Hautes Étud. Sci. (2011) no. 113, pp. 97-208 | DOI | Numdam | MR | Zbl

[KM11b] Kronheimer, Peter B.; Mrowka, Tomasz S. Knot homology groups from instantons, J. Topol., Volume 4 (2011) no. 4, pp. 835-918 | DOI | MR | Zbl

[Lek09] Lekili, Yankı Broken Lefschetz fibrations, Lagrangian matching invariants and Ozsváth–Szabó invariants, Ph. D. Thesis, Massachusetts Institute of Technology, USA (2009)

[LPCZ21] Lidman, Tye; Pinzón-Caicedo, Juanita; Zentner, Raphael Toroidal homology spheres and SU(2)-representations (2021) (https://arxiv.org/abs/2101.02621)

[Sav02] Saveliev, Nikolai Invariants for homology 3-spheres, Encyclopaedia of Mathematical Sciences, 140, Springer, 2002 (Low-Dimensional Topology, I) | DOI | MR | Zbl

[Sca15] Scaduto, Christopher W. Instantons and odd Khovanov homology, J. Topol., Volume 8 (2015) no. 3, pp. 744-810 | DOI | MR | Zbl

[SZ22] Sivek, Steven; Zentner, Raphael A menagerie of SU(2)-cyclic 3-manifolds, Int. Math. Res. Not., Volume 2022 (2022) no. 11, pp. 8038-8085 | DOI | MR | Zbl

[Zen18] Zentner, Raphael Integer homology 3-spheres admit irreducible representations in SL (2,ℂ), Duke Math. J., Volume 167 (2018) no. 9, pp. 1643-1712 | DOI | MR | Zbl