##
**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.

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.

Reviewer: Thilo Kuessner (Augsburg)

### 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 |

### Keywords:

instanton gauge theory; holonomy perturbations; knots; representation varieties; \(\mathrm{SU}(2)\); \(\mathrm{SL}(2,\mathbb{C})\); 3-sphere recognition; 3-manifold groups### Citations:

Zbl 1084.57006
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\textit{R. Zentner}, Duke Math. J. 167, No. 9, 1643--1712 (2018; Zbl 1407.57008)

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