×

Integer homology 3-spheres admit irreducible representations in \(\mathrm{SL}(2,{\mathbb C})\). (English) Zbl 1407.57008

Much information about \(3\)-manifolds and their fundamental groups is hidden in the variety of their group representations to suitable Lie groups. Traditionally, representations to \(\mathrm{SU}(2)\) and \(\mathrm{SL}(2,{\mathbb C})\) have been the most studied ones.
It is however still an open question which \(3\)-manifold groups admit nontrivial representations to \(\mathrm{SL}(2,{\mathbb C})\). Such a representation clearly exists if the first Betti number is positive, because then there are nontrivial homomorphisms from the fundamental group to the integers and hence to \(\mathrm{SU}(2)\). This reduces the question of existence to rational homology spheres.
The paper under review proves existence of nontrivial \(\mathrm{SL}(2,{\mathbb C})\)-representations for all integer homology \(3\)-spheres. The analogue in higher dimensions is not true as the author shows via explicit examples.
A main reduction is that every integer homology \(3\)-sphere except \(S^3\) admits a degree \(1\) map (thus an epimorphism of fundamental groups) onto either a hyperbolic \(3\)-manifold, a Seifert fibration or the untwisted splicing of two nontrivial knots. By the latter one means that the complements of the knots in \(S^3\) are glued together along their boundaries via identifying the meridian of one knot with the longitude of the other one and vice versa. Hyperbolic \(3\)-manifolds have their monodromy representation to \(\mathrm{PSL}(2,{\mathbb C})\), which by a result of Culler lifts to \(\mathrm{SL}(2,{\mathbb C})\). Moreover, the author proves that fundamental groups of Seifert fibrations admit nontrivial representations to \(\mathrm{SU}(2)\). This reduces the claim to those integer homology spheres which are untwisted splicings of nontrivial knots. The proof in this case constitutes the main bulk of the paper. The author proves in this case the existence of nontrivial representations to \(\mathrm{SU}(2)\), thus if the corresponding result for hyperbolic \(3\)-manifolds were known, one would be able to even conclude that every integer homology \(3\)-sphere admits a nontrivial representation to \(\mathrm{SU}(2)\).
Let the untwisted splicing \(Y\) be constructed from the knot complements \(Y(K_1),Y(K_2)\) via identifications of their torus boundaries \(T\), interchanging meridians and longitudes. It is known that \(Y(K_1)\) and \(Y(K_2)\) admit nontrivial representations to \(\mathrm{SU}(2)\). The author’s idea is to look at the images of the two representation varieties in the representation variety of \(\pi_1T\) and to find a point of intersection, which then corresponds to a representation of the fundamental group of the splicing. The \(\mathrm{SU}(2)\)-representation variety of \(\pi_1T={\mathbb Z}^2\) is the pillowcase, i.e., the quotient of \({\mathbb R}^2/(2\pi{\mathbb Z})^2\) under the hyperelliptic involution \((\alpha,\beta)\sim(2\pi-\alpha,2\pi-\beta)\). The Main Theorem proved in this paper is the following. Let \(K\) be a nontrivial knot. Then any topologically embedded path from \((0,\pi)\) to \((\pi,\pi)\) in the pillowcase missing the line \(\left\{\beta=0 \bmod 2\pi{\mathbb Z}\right\}\) has an intersection point with the image of the \(\mathrm{SU}(2)\)-representation variety of \(Y(K)\) in the pillowcase. This theorem implies that images of the \(\mathrm{SU}(2)\)-representation varieties of \(Y(K_1)\) and \(Y(K_2)\) in the pillowcase will intersect after the transformation \((\alpha,\beta)\to(\beta,\alpha)\) (corresponding to the mapping class that interchanges longitude and meridian of the boundary torus) is applied to one of them. This intersection corresponds to a representation of the fundamental group of the untwisted splicing.
The proof of the Main Theorem uses holonomy perturbations of the Chern-Simons functional in an exhaustive way. (Recall that representations correspond to flat connections and thus critical points of the Chern-Simons functional.) The proof proceeds as follows. Assume by contradiction that there is an embedded path as in the Main Theorem which does not intersect the image of the representation variety of \(Y(K)\). Then (after going to the 2-fold branched cover of the pillowcase, i.e., the torus) there is a \({\mathbb Z}/2{\mathbb Z}\)-equivariant isotopy of the torus which moves the path in question to the straight line \(\left\{\beta =\pi \bmod 2\pi{\mathbb Z}\right\}\). (This line corresponds to critical points of the Chern-Simons functional, i.e., flat connections, on the \(0\)-surgered manifold \(Y_0(K)\)). The author shows that this isotopy can be realised by an area-preserving isotopy, that this area-preserving isotopy can be \(C^0\)-approximated by a sequence of so-called shearing isotopies, and that these shearing isotopies can be realised geometrically through holonomy perturbations of the flatness equation in a thickened torus. The latter result is considered by the author to be the main technical result of the paper. It finally allows to bring the machinery of Kronheimer-Mrowka into play.
In [Math. Res. Lett. 11, No. 5–6, 741–754 (2004; Zbl 1084.57006)], P. B. Kronheimer and T. S. Mrowka proved that the \(0\)-surgery \(Y_0(K)\) of a nontrivial knot admits nontrivial representations to \(\mathrm{SU}(2)\). For this purpose they proved a nonvanishing result for Donaldson invariants as follows: if \(X\) is a closed, oriented \(4\)-manifold, and \(Y\subset X\) a connected, separating \(3\)-manifold for which a certain perturbed \(\mathrm{SU}(2)\)-representation variety of a holonomy perturbation is empty, then the Donaldson invariants of \(X\) are identically zero. (And they proved that \(Y_0(K)\) embeds into some \(X\) with nonvanishing Donaldson invariants, thus getting a contradiction.)
To apply this to the case under consideration, the author uses that the holonomy perturbation data on the thickened torus also determine holonomy perturbation data on \(Y_0(K)\), and he shows that the perturbed \(\mathrm{SU}(2)\)-representation variety would be empty under the made assumptions. (The line \(\left\{\beta=\pi \bmod 2\pi{\mathbb Z}\right\}\) corresponds to critical points of the Chern-Simons functional on \(Y_0(K)\), so the result of the holonomy perturbation, that is the original embedded path in the pillowcase, corresponds to critical points of the perturbed Chern-Simons functional. But by assumption this embedded path does not intersect the image of the representation variety.) Thus the non-vanishing result of Kronheimer and Mrowka for Donaldson invariants of a certain 4-manifold containing the 0-surgery of the knot as a splitting hypersurface yields again a contradiction. Hence no such holonomy perturbation can exist and the Main Theorem follows.
This is an impressive paper, which uses a variety of methods to solve a long-standing problem.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R57 Applications of global analysis to structures on manifolds

Citations:

Zbl 1084.57006
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Addison-Wesley, Reading, Mass., 1978. · Zbl 0393.70001
[2] E. Andersén, Volume-preserving automorphisms of \(\mathbb{C}^{n}\), Complex Variables Theory Appl. 14 (1990), 223-235.
[3] E. Andersén and L. Lempert, On the group of holomorphic automorphisms of \(\mathbb{C}^{n}\), Invent. Math. 110 (1992), 371-388. · Zbl 0770.32015
[4] M. Aschenbrenner, S. Friedl, and H. Wilton, 3-Manifold Groups, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2015. · Zbl 1326.57001
[5] J. A. Baldwin and S. Sivek, Instanton Floer homology and contact structures, Selecta Math. (N.S.) 22 (2016), 939-978. · Zbl 1344.53059
[6] J. A. Baldwin and S. Sivek, Stein fillings and \(SU(2)\) representations, to appear in Geom. Topol., preprint, arXiv:1611.05629v1 [math.SG].
[7] J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. (3) 12, Springer, Berlin, 1987. · Zbl 0633.14016
[8] M. Boileau, J. Rubinstein, and S. Wang, Finiteness of \(3\)-manifolds associated with non-zero degree mappings, Comment. Math. Helv. 89 (2014), 33-68. · Zbl 1290.57001
[9] F. Bonahon and L. Siebenmann, New geometric splittings of classical knots and the classification and symmetries of arborescent knots, preprint, http://www-bcf.usc.edu/ fbonahon/Research/Preprints/BonSieb.pdf.
[10] P. J. Braam and S. K. Donaldson, “Floer’s work on instanton homology, knots and surgery” in The Floer Memorial Volume, Progr. Math. 133, Birkhäuser, Basel, 1995, 196-256. · Zbl 0996.57516
[11] C. Campbell, E. Robertson, and P. Williams, “Efficient presentations for finite simple groups and related groups” in Groups-Korea 1988 (Pusan, 1988), Lecture Notes in Math. 1398, Springer, Berlin, 1989, 65-72.
[12] C. Cornwell, Character varieties of knot complements and branched double-covers via the cord ring, preprint, arXiv:1509.04962v1 [math.GT].
[13] M. Culler, A-polynomials and \(\operatorname{SU}(2)\) character varieties, conference lecture at “10th W. R. Hamilton Workshop,” Trinity College, Dublin, 2014.
[14] S. K. Donaldson, The orientation of Yang-Mills moduli spaces and \(4\)-manifold topology, J. Differential Geom. 26 (1987), 397-428. · Zbl 0683.57005
[15] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257-315. · Zbl 0715.57007
[16] S. K. Donaldson, Floer Homology Groups in Yang-Mills Theory, with the assistance of M. Furuta and D. Kotschick, Cambridge Tracts Math. 147, Cambridge Univ. Press, Cambridge, 2002.
[17] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Math. Monogr., Oxford Univ. Press, New York, 1990. · Zbl 0820.57002
[18] Y. M. Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004), 277-293. · Zbl 1067.53070
[19] Y. M. Eliashberg and W. P. Thurston, Confoliations, Univ. Lect. Ser. 13, Amer. Math. Soc., Providence, 1998.
[20] P. M. N. Feehan and T. G. Leness, “On Donaldson and Seiberg-Witten invariants” in Topology and Geometry of Manifolds (Athens, 2001), Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, 2003, 237-248. · Zbl 1042.57017
[21] R. Fintushel and R. J. Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. (3) 61 (1990), 109-137. · Zbl 0705.57009
[22] A. Floer, An instanton-invariant for \(3\)-manifolds, Comm. Math. Phys. 118 (1988), 215-240. · Zbl 0684.53027
[23] D. Freed and K. Uhlenbeck, Instantons and Four-Manifolds, Math. Sci. Res. Inst. Publ. 1, Springer, New York, 1984. · Zbl 0559.57001
[24] D. Gabai, Foliations and the topology of \(3\)-manifolds, III, J. Differential Geom. 26 (1987), 479-536. · Zbl 0639.57008
[25] C. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. · Zbl 0678.57005
[26] J. Hass and G. Kuperberg, “The complexity of recognizing the 3-sphere” in Triangulations, Oberwolfach Rep. 9, Eur. Math. Soc., Zürich, 2012, 1425-1426.
[27] C. M. Herald, Legendrian cobordism and Chern-Simons theory on \(3\)-manifolds with boundary, Comm. Anal. Geom. 2 (1994), 337-413. · Zbl 0854.58013
[28] C. M. Herald and P. Kirk, Holonomy perturbations in a cylinder, and regularity for traceless \({SU}(2)\) character varieties of tangles, preprint, arXiv:1511.00308v2 [math.GT]. · Zbl 1396.57022
[29] S. Kaliman and F. Kutzschebauch, “On the present state of the Andersén-Lempert theory” in Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, 2011, 85-122. · Zbl 1266.32028
[30] M. A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67-72. · Zbl 0187.20401
[31] E. Klassen, Representations of knot groups in \(\operatorname{SU}(2)\), Trans. Amer. Math. Soc. 326 (1991), no. 2, 795-828. · Zbl 0743.57003
[32] B. Kleiner and J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), 2587-2855. · Zbl 1204.53033
[33] P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 41 (1995), 573-734. · Zbl 0842.57022
[34] P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and \(\operatorname{SU}(2)\), Math. Res. Lett. 11 (2004), no. 5-6, 741-754. · Zbl 1084.57006
[35] P. B. Kronheimer and T. S. Mrowka, Witten’s conjecture and property P, Geom. Topol. 8 (2004), 295-310. · Zbl 1072.57005
[36] P. B. Kronheimer and T. S. Mrowka, Knots, sutures and excision, J. Differential Geom. 84 (2010), 301-364. · Zbl 1208.57008
[37] G. Kuperberg, Knottedness is in \(\mathsf{NP}\), modulo GRH, Adv. Math. 256 (2014), 493-506. · Zbl 1358.68138
[38] J. Lin, The A-polynomial and holonomy perturbations, Math. Res. Lett. 22 (2015), no. 5, 1401-1416. · Zbl 1348.57027
[39] J. Lin, \(\operatorname{SU}(2)\)-cyclic surgeries on knots, Int. Math. Res. Not. IMRN 2016, no. 19, 6018-6033. · Zbl 1404.57015
[40] J. Montesinos, Revêtements ramifiés de noeuds, espaces fibrés de Seifert et scindements de Heegaard, lecture notes, Orsay, France, 1976.
[41] K. Motegi, Haken manifolds and representations of their fundamental group in \(\operatorname{SL}(2,\mathbb{C})\), Topology Appl. 29 (1988), 207-212. · Zbl 0647.57007
[42] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv:math/0211159v1 [math.DG].
[43] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, arXiv:math/0307245v1 [math.DG].
[44] G. Perelman, Ricci flow with surgery on three-manifolds, preprint, arXiv:math/0303109v1 [math.DG].
[45] J. H. Rubinstein, “An algorithm to recognize the \(3\)-sphere” in Proceedings of the International Congress of Mathematics, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 601-611. · Zbl 0864.57009
[46] S. Schleimer, “Sphere recognition lies in NP” in Low-Dimensional and Symplectic Topology, Proc. Sympos. Pure Math. 82, Amer. Math. Soc., Providence, 2011, 183-213. · Zbl 1250.57024
[47] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250. · JFM 42.0154.02
[48] S. Sivek and R. Zentner, \(\operatorname{SU}(2)\)-cyclic surgeries and the pillowcase, preprint, arXiv:1710.01957v1 [math.GT].
[49] C. H. Taubes, Casson’s invariant and gauge theory, J. Differential Geom. 31 (1990), 547-599. · Zbl 0702.53017
[50] C. H. Taubes, Unique continuation theorems in gauge theories, Comm. Anal. Geom. 2 (1994), 35-52. · Zbl 0838.53028
[51] A. Thompson, Thin position and the recognition problem for \(S^{3}\), Math. Res. Lett. 1 (1994), no. 5, 613-630. · Zbl 0849.57009
[52] R. Zentner, A class of knots with simple \({SU}(2)\)-representations, Selecta Math. (N.S.) 23 (2017), 2219-2242. · Zbl 1398.57020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.