Low Dimensional Topology

August 27, 2008

Involutions of handlebodies

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 2:42 pm

An involution of a topological space X is a homeomorphism h from X to itself such that h is not the identity but the composition of h with itself is the identity.  A simple involution of a surface can be constructed by drawing the standard picture of a surface, then skewering the surface with a horizontal line (as if you’re making a shish kabob) and applying a 180 degree rotation around the skewer.  If you think of that surface as being in 3-space then it bounds a handlebody and the involution extends to this handlebody as well.  The set of fixed points on the surface consists of 2g+2 isolated points (where g is the genus of the surface).  The set of fixed points in the handlebody consists of g+1 arcs connecting the isolated points in the surface (the intersection of the skewer with the handlebody).

If you do this for a genus two surface, the resulting involution has some really nice properties.  First, this is the only involution of a genus 2 surface with exactly six fixed points.  (Note: an earlier version of this sentence was incorrect, but Saul sent me an e-mail pointing that out.)  Second, this involution sends each simple closed curve in the surface onto itself (up to isotopy), which implies that the involution extends to every handlebody bounded by the surface.  (It also means that this involution induces the identity map on the curve complex!)  If the genus two surface is a Heegaard splitting, then it bounds two handlebodies and the involution extends to both of them, i.e. to the ambient 3-manifold.  The quotient of each handlebody by this involution is a ball and the fixed point set descends to boundary parallel arcs in these balls.  In this way, every 3-manifold with a genus two Heegaard splitting can be written as a double branched cover of a three bridge knot in the 3-sphere, and this is one of the main links between the theory of Heegaard splittings and the theory of bridge positions of knots.  (See my previous post on branched covers and bridge positions.)

For higher genus handlebodies, there are more types of involutions than the one above.  However, a nice classification of them is given in a recent preprint by Pantaleoni and Piergallini [1].  They show that the involutions fall into two groups.  First, there are fixed point free involutions, which occur only in odd genus surfaces and look like you’ve taken a standard picture of the handlebody, skewered it through the middle hole (so the skewer misses the surface) and then rotated it 180 degrees.  (The paper has a nice picture of this which I won’t reproduce here.)  The second type of involution, one with fixed points, breaks down into some pieces where there are arcs of fixed points, some pieces with loops of fixed points, and some free pieces.  There is a very nice picture of this in the paper, which I won’t reproduce here, either.  The proof is nice and short and is based on finding a system of disks that behaves nicely under the involution.

The reason I’m writing about this here is, of course, because it’s related to mapping class groups of Heegaard splittings.  If there is an involution of a 3-manifold that takes a Heegaard surface to itself then this involution either swaps the two handlebodies or it extends to an involution of the two handlebodies.  In the case when the involution extends, that means that the involution of the surface commutes with the gluing map between the handlebodies.  It might be possible to use this fact to classify Heegaard splittings that admit such involutions, though that seems like a hard problem.

I should also mention that you can think of an involution as a group action by Z_2 on the space.  McCullough, Miller and Zimmerman [2] have proved some things about general group actions on handlebodies and Zimmerman has used ideas along these lines to proved results about group actions on hyperbolic 3-manifolds.

[2] D. McCullough, A. Miller and B. Zimmermann, Group actions on handlebodies, Proc. London Math. Soc. 59 (1989), 373-416.