Flat SU(2) bundles over hyperbolic 3-manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:55:53Z http://mathoverflow.net/feeds/question/8890 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8890/flat-su2-bundles-over-hyperbolic-3-manifolds Flat SU(2) bundles over hyperbolic 3-manifolds Joel Fine 2009-12-14T17:16:26Z 2009-12-15T18:39:18Z <p>Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?</p> <p>The literature on such bundles over 3-manifolds is huge and my naive searches don't seem to turn up specific examples. </p> <p>Roughly speaking, the Casson invariant counts flat bundles over 3-manifolds, so in principal I suppose I would be happy with an example of a hyperbolic 3-manifold with non-zero SU(2) Casson invariant (and surely such things are known). In practice, I would really like to see the non-trivial bundle (or corresponding representation) more-or-less explicitly. </p> <p>Finally, I would also be happy with just being told precisely where I should go and look in the literature!</p> http://mathoverflow.net/questions/8890/flat-su2-bundles-over-hyperbolic-3-manifolds/8897#8897 Answer by Ian Agol for Flat SU(2) bundles over hyperbolic 3-manifolds Ian Agol 2009-12-14T18:54:58Z 2009-12-15T18:39:18Z <p>Many (compact orientable) hyperbolic 3-manifolds have non-trivial $SU(2)$ representations. </p> <p>By Mostow rigidity, the representation of the fundamental group $\Gamma$ of a closed hyperbolic 3-manifold into $SL(2,\mathbb{C})$ (lifted from $PSL(2,\mathbb{C})$) may be conjugated so that it lies in $SL(2,K)$, for $K$ a number field (because transcendental extensions have infinitesimal deformations in $\mathbb{C}$). In particular, the traces of elements will always lie in a number field. One may take different Galois embeddings of $K$ into $\mathbb{C}$, and get new representations of $\Gamma$ into $SL(2,\mathbb{C})$. Sometimes, this representation is just conjugate to the original (e.g. if $K$ was chosen too large), but in other cases the new representation of $\Gamma$ lies in $SL(2,\mathbb{R})$ or in $SU(2)$. A nice class of examples of this type are arithmetic hyperbolic 3-manifolds. In fact, they are characterized by the fact that all traces of elements are algebraic integers, and non-trivial Galois embeddings lie in $SU(2)$ (you have to be a bit careful about what this means). Some arithmetic manifolds will only have the complex conjugate representation this way (basically, if squares of the traces lie in a quadratic imaginary number field), but otherwise you get a non-trivial $SU(2)$ representation. The simplest example is the Weeks manifold, with trace field a cubic field. I suggest the book by <a href="http://www.ams.org/mathscinet-getitem?mr=1937957" rel="nofollow">MacLachlan and Reid</a> as an introduction to arithmetic 3-manifolds. The description I've given though is encoded in terms of quaternion algebras and other algebraic machinery. Another characterization of arithmeticity is in <a href="http://www.ams.org/mathscinet-getitem?mr=1433117" rel="nofollow">this paper</a>. The nice thing about these representations is that they are faithful. There is a very explicit way to see these representations for hyperbolic reflection groups (studied by Vinberg in the arithmetic case). Basically, given a hyperbolic polyhedron with acute angles of the form $\pi/q$, sometimes one can form a spherical polyhedron with corresponding angles which are $p\pi/q$, and get a representation into $O(4)$. Passing to finite index manifold subgroups, one can obtain $SU(2)$ reps. (since $SO(4)$ is essentially $SU(2)\times SU(2)$). </p> <p>There are other ways one has $SU(2)$ representations, but they are less explicit. <a href="http://www.ams.org/mathscinet-getitem?mr=2106239" rel="nofollow">Kronheimer and Mrowka</a> have shown that any non-trivial integral surgery on a knot has a non-abelian $SU(2)$ representation. Also, any hyperbolic 3-manifold with first betti number positive or a smooth taut orientable foliation has non-abelian $SU(2)$ representations. </p> <p><strong>Addendum:</strong> Another observation relating $SU(2)$ representations to hyperbolic geometry is via the observation that the binary icosahedral group (a $\mathbb{Z}/2$ extension of $A_5$) is a subgroup of $SU(2)$. By an <a href="http://www.ams.org/mathscinet-getitem?mr=1459136" rel="nofollow">observation of Long and Reid</a>, every hyperbolic 3-manifold group has infinitely many quotients of the form $PSL(2,p)$, $p$ prime. These groups always contain subgroups isomomorphic to $A_5 &lt; SO(3)$, so one may find a finite-sheeted cover which has a non-abelian $SO(3)$ and therefore $SU(2)$ representation. I have no idea though whether these representations are detected by the Casson invariant or Floer homology. </p> http://mathoverflow.net/questions/8890/flat-su2-bundles-over-hyperbolic-3-manifolds/8898#8898 Answer by Steven Sivek for Flat SU(2) bundles over hyperbolic 3-manifolds Steven Sivek 2009-12-14T19:08:52Z 2009-12-14T19:08:52Z <p>The figure eight knot is hyperbolic, so by Thurston all but finitely many 1/n-surgeries on it yield hyperbolic homology spheres. The Casson invariant of the 1/n surgery is (n/2)&Delta;''(1), where &Delta;(t) = -t + 3 - t<sup>-1</sup> is the Alexander polynomial of the figure eight, so it should be -n and therefore there are at least n distinct SU(2) representations.</p> <p>In practice you should be able to work these out explicitly from a presentation of the knot group: figure out the SU(2) representation variety of the knot group (see Klassen, "Representations of knot groups in SU(2)" for the figure eight), impose the relation &mu;&lambda;<sup>n</sup>=1, and compute what's left.</p> http://mathoverflow.net/questions/8890/flat-su2-bundles-over-hyperbolic-3-manifolds/8950#8950 Answer by Paul Kirk for Flat SU(2) bundles over hyperbolic 3-manifolds Paul Kirk 2009-12-15T04:05:13Z 2009-12-15T04:05:13Z <p>There is a huge literature on this. I second starting with Klassen's article. You should also go back to Riley's old article and then look at Burde for 2-bridge knots. But I suspect that noone knows an "explicit" description in the sense that "this loop goes to this matrix" in general the representations are given by the real points of a variety defined over Z. </p>