From harelb Thu Nov 7 01:03 EST 1996 Date: Thu, 7 Nov 1996 01:03:29 -0500 From: Harel BarzilaiDear Professor [----],To: [----] Subject: thesis >>> Dear Harel, >>> [----] told me of your work on mapping class groups of >>> handlebodies. If you have any preprints I'ld like to look at >>> them. --Best, [----]
Thank you for your interest in my thesis. I just gave a talk at the Topology Seminar here about some of my work. Unfortunately, I have only certain parts Tex'd, and those are still, many hours away from being in 'preprint' form clean enough to share with others. I will try however to outline, below, the key results which will form my thesis, and would be happy to elaborate, give references to papers, send a few pictures by postal mail, etc, if you are interested. [What follows is a bit long. I could Tex it and send you a hardcopy, but perhaps using [your] printers would be simplest; there aren't lots of complicated formulas].
A good survey article on surface mapping class groups is N.V. Ivanov's "Complexes of curves and the Teichmuller modular group" in Russian Math Surveys 42:3 (1987), 55-107. In more recent years there has been interest and work on mapping class groups (MCG) of manifolds other than surfaces that has produced analogous results, in particular Mapping Class Groups of 3-Manifolds, by Darryl McCullough, Journal of Differential Geometry 33 (1991) pp. 1-65.
McCullough uses some general techniques (characteristic submanifold theory of Johannson, and Jaco and Shalen that I'm not familiar with) to obtain certain results, in particular that for a compact, irreducible, orientable, sufficiently large 3-manifold, the MCG is finitely presented and is of type VFL (an algebraic condition; for a finitely presented group G, it is of type FL iff there is a K(G,1) which is a finite complex. A reference is the "finiteness conditions" chapter of Ken Brown's Cohomology of Groups for example. G is VFL if it has a finite index subgroup which is of type FL).
While McCullough's section on MCGs of products with handles doesn't use the general Johannson etc theories above, for the case of M = a handlebody, it seems that there should still be a more concrete way of studying the MCG and in particular for obtaining such results, directly resulting from a complex analogous to those used to study surface mapping class groups. The Disc Arc Complex (DAC) which I construct does this in several respects.
A vertex is a pair (\D,\A) where \D is a simple system of isotopy classes of essential discs having disjoint representatives (where "simple" means each complementary component is simply connected), and \A is a system of isotopy classes of arcs, having disjoint representatives with endpoints lying in boundary(P), where P is a "patch", a 2-disc lying entirely in the boundary surface.
\D and \A are required to be "compatible" in two ways: (i)The arcs "connect up" in each component of [(handlebody) - \D], the leftover 'imprints' of the cutting discs (I can draw a picture which makes this clear. The idea is that each connected complementary component of the handlebody minus the discs, is a 3-ball, and one wants that subset of its boundary-2-sphere, which comes from the discs and arcs, to be connected) and (ii) given a representative disc D_i and arc A_j, their intersection must be either empty, a single point, or two points 'in opposite directions'.
This is reminiscent of Ivanov's hybrid complex which he used to study the MCG of surfaces, which has both "white" vertices corresponding to (isotopy classes of) circles, and "blue" vertices corresponding to isotopy classes of arcs.
DAC enjoys the following properties:
DAC is acted on by \Gamma^{V}_{g,1}, the mapping class group of the handlebody of genus g with one "patch" (a disc lying entirely in the boundary surface, which is preserved), such that:
(1)All simplex stabilizers are finite (2)The quotient is compact (the action is cocompact) (3)DAC is contractible (4)DAC is locally finite
From (1)-(3) it follows that this 'patched' mapping class group is
finitely presented and of type VFL (by a short exact sequence, 1 --> pi_1(Surface_{g}) --> \Gamma^{V}_{g,1} --> \Gamma^{V}_{g,0} relating the patched to the unpatched handlebody mapping class group (when g>1), one obtains another proof of a special case of McCullough's result, namely that the unpatched handlebody mapping class group is finitely presented).
It might be worth adding that from a theorem in Martin Bridson's thesis it follows that DAC is a complete geodesic metric space; since it is locally finite and hence locally compact, it follows from a 'famous fact' that DAC is proper (lemma cited in a paper of Ghys that a geodesic metric space is proper (closed balls are compact) iff it is complete and locally compact). I have to confess that I don't have any great corollary of this, but it does show that DAC is a pretty nice space. One possible future path is to try to strengthen these results, e.g. looking for CAT(0) type conditions, with which my feet have only recently been wet)
Another avenue of research that I only recently found out about includes "connectedness at infinity" properties for groups. In the thesis of Brad Jackson that was recently brought to my attention, he proves:
Given a short exact sequence of finitely presented infinite groups 1 --> K --> G --> Q --> 1 where either K or Q is 0-connected at infinity, it follows that G is LC-connected ("1-connected") at infinity.
It follows from the short exact sequence above therefore that my "patched" variants of handlebody MCGs are 1-connected at infinity.
I sent an email inquiry to Mike Mihalik at Vanderbilt who I was told is an expert on these matters, as to whether results were known about the Q in such a sequence, or whether I might not hope that G is nicer. He wrote that he knows of no general results from which one could conclude that Q too is 1-connected at infinity; in fact, he suggested a "reasonable conjecture" and said that (what he generously called my intuition) is probably correct that G, here \Gamma^{V}_{g,1}, is "nicer" than the unpatched handlebody mapping class group, \Gamma^{V}_{g,0}.
It follows also from (1)-(4) that DAC is quasi-isometric to the (patched) handlebody MCG. The main application that I know of for such spaces is that they share the same isoperimetric inequality with the group. In fact my advisor Allen Hatcher wrote a paper with Karen Vogtmann here, on the isoperimetric inequality for Aut(F_n) the free group on n generators, using such a space.
This was actually my starting point, looking for something similar for handlebodies. However, in running into difficulties with my original space, the Truncated Disc Complex (TDC), the definition underwent several mutations, arriving finally at the DAC. I have (1)-(4) proven about the DAC, but the techniques of Hatcher/Vogtmann require a particularly nice contraction, while the contractibility of DAC is proven by showing it "fibers" over a base space which is shown to be contractible, and that the "Quillen fibers" are contractible, from which it follows that DAC is contractible.
So while I have these other results that I hadn't originally expected, it would be nice to go back, understand the TDC and the Hatcher/Vogtmann paper better, and possibly try to prove an isoperimetric inequality. I had proved (1) and (4) for the TDC, as well as computing the TDC in the case g=1 which was indeed contractible so that (3) holds in that case. Maybe if it wasn't for all the calculus reform and other math education I was involved with I wouldn't have given up so quickly! (I didn't write down a proof of (2), and didn't try to when I got discouraged about (3), but I remember at the time, that looking ok).
The other result is homological stability for handlebody MCGs. This is a couple of years old so my memory is quite rusty, but basically this did not involve any huge amount of originality on my part. The main ingredients were techniques of Allen Hatcher and John Harer, algebra, and a sufficient amount of coffee, so to speak.
I should mention in this regard, however, that that result lends not only to the idea of obtaining similar results for handlebody MCGs of what is known for surface MCGs (John Harer first proved homological stability for surface MCGs in 1985 in the Annals), but to these "patches" which act for the handlebody very much like the punctures do for a surface.
In Ivanov's survey article, he also refines several proofs and known results, among them, he streamlined Harer's proof of homological stability for surface MCGs using a two-part stabilization exploiting the punctures: If S_{g,s} denotes a surface of genus g with s punctures, then by gluing a pair of pants (i.e., 3-punctured sphere S_{0,3}) along one puncture one obtains S_{g,s+1}, while in the next step, if we glue a pair of pants along two boundary components, we obtain from S_{g,s+1} an S_{g+1,s}. As I recall the same idea worked in the handlebody MCG case. When working on the final writeup of my thesis in the spring I will certainly try to clean up this patchwork of different techniques into something more elegant. My advisor proved homological stability for Aut(F_n) (again with action on complexes being a key ingredient) a couple of years ago.
Maybe I should mention that during the earliest stage of working here under Hatcher I found some cohomology classes in the MCG of the handlebody, using Chern classes of representations and ideas from a paper by Glover and Mislin. (Here too I put these preliminary results aside, since I had originally aimed for looking analyzing or finding new classes in the co/homology of the surface mapping class group, while my representation turned out to give nontrivial classes in the handlebody case instead -- little did I know I would later completely return to handlebodies..!) I will at some point look back over these result and check what is known, if anything, of this kind, about the cohomology of the handlebody MCG, to see whether it's worth including, and how much of, these results in my thesis.
Although I haven't fully checked the details, the DAC should work with s>1 (two or more patches), and similarly there is a reduced DAC (rDAC) which is almost as nice as the DAC in its properties and whose dimension is less. I think rDAC is of dimension 4g+constant, which is better than DAC (at least 5g+constant). McCullough proved that the virtual cohomological dimension (VCD) for these MCGs is 3g+constant (actually he proved 3g+constant <= VCD <= 4g+constant and my advisor wrote to McCullough after noticing that McCullough's own techniques, pushed farther, show VCD is in fact 3g+constant), and it would be nice to have the complex be of exactly the same dimension as the VCD.
Finally, I should mention the connection with Aut(F_n). An element of the handlebody MCG gives an element of Out(F_n) by looking at the action on the fundamental group, or in the patched case, one actually gets an element of Aut(F_n), the automorphism group. In some sense the handlebody is "in between" the surface and Aut(F_n), through this connection.
In fact Allen was recently sent a preprint of a paper by Lustig and Oertel where they try to classify automorphisms of the handlebody. This is a not (directly) related area, of course, but the point is that they too see (in their words in the preprint) the handlebody "as a sort of bridge between" surfaces and free groups. There is the classical Thurston theory (pseudo-Anosov etc) classifying surface automorphism. Recently Bestvina and Handel proved a similar type of classification theorem for free group automorphisms. This preprint, which I just glanced at, apparently tries to take the same philosophical approach to handlebodies. All this is a sort of unrelated footnote, but suggests a 'context' for thinking about handlebodies.
Harel