From: "David L. Johnson" Newsgroups: sci.math.research Subject: Re: Are "Most" n-Manifolds Hyperbolic? Date: 21 Jun 1995 15:10:25 GMT asimov@nas.nasa.gov (Daniel A. Asimov) wrote: >It has often been stated that in a certain sense "most" 3-manifolds are >hyperbolic. > >This assertion seems easy to justify for surfaces as well. > >QUESTION: Is there any sense in which the same assertion -- that "most" >n-manifolds are hyperbolic (admit a Riemannian metric of constant sectional >curvature = -1) might be true for n >= 4 as well??? > >[A short answer is no, most closed 4-manifolds have signature. Even >among those with no signature, most have pi_2 (I think), most that have >no pi_2 are probably not negatively curved, and most that are >negatively curved are not hyperbolic. And you can bring other >invariants into the picture. Perhaps someone here can give a better >long answer. - Greg] That's certainly part of the answer. Negative curvature has all sorts of topological obstructions in higher dimensions (The Euler characteristic must be positive in dimension 4, etc.). There are even stronger restrictions on constant negative sectional curvature, because the whole characteristic algebra is determined. More to the point, the manifold is covered by a ball, in that the exponential map is a covering map, under negative curvature. But this occurs in dimension three as well, so perhaps the question should be looked at in the way it is for three-manifolds. That is, the space may be decomposed along certain `nice' submanifolds into pieces, each of which would be hyperbolic, perhaps. Then the question of topological obstructions becomes much more subtle, or may vanish completely. In this altered sense from what the moderator seems to have interpreted, I would guess that the question is open. -- David L. Johnson dlj0@lehigh.edu Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html Lehigh University 14 E. Packer Avenue (610) 758-3759 Bethlehem, PA 18015-3174 (610) 828-3708 ============================================================================== From: greg@ford.uchicago.edu (Greg Kuperberg) Newsgroups: sci.math.research Subject: Re: Are "Most" n-Manifolds Hyperbolic? Date: Wed, 21 Jun 1995 20:42:48 GMT In article , Daniel A. Asimov wrote: >It has often been stated that in a certain sense "most" 3-manifolds are >hyperbolic. My note as moderator attached to this posting was intended to recall well-known facts to put Dan's question in context; here are some more detailed comments about how one might form a precise interpretation of the above statement. Empirically, if you use the "monkey with a mouse" method to make a 3-manifold with SnapPea, then there is some chance that it will fail to be hyperbolic if it's reasonably small, but if it's complicated enough, it will almost always be hyperbolic. You have to be a little careful with how you monkey with the mouse for this to be true, though: If you draw knot or a link with many wide strokes across the screen and then do some random Dehn surgery with reasonably large coefficients, then you will get a hyperbolic manifold. But if the mouse exhibits Brownian motion and changes direction often, then there is a fair chance that the knot on which you do Dehn surgery has a trefoil summand somewhere, and the 3-manifold will not be hyperbolic but will rather have geometric pieces. Still, one might conjecture that just about any random process that yields a complicated 3-manifold would at least likely give you a huge hyperbolic piece plus a lot of small change attached to it. This would be akin to the likely topology of a large random graph (such as the World-Wide Web): It has one huge connected component and lots of little components, most of them trees and isolated vertices. Question: Suppose you take n vertices and you throw in tetrahedra one a time by choosing 4-tuples of the vertices at random. If the simplicial complex becomes a closed 3-manifold, you retain it and stop, otherwise keep going until it is not contained in any 3-manifold, at which point you throw it away and start over. As n goes to infinity, does the probability of hyperbolicity of the result go to 1? The answer in 4D is almost certainly "no"; it seems unlikely that the myriad invariants involved would all simultaneously allow hyperbolicity. Perhaps signature is the easiest invariant to use in a proof. For those who are wondering what signature has to do with hyperbolicity, the signature of a 4-manifold is proportional to the Pontryagin number, which is a chiral invariant that can be computed from an integral involving a Riemannian metric (a Chern-Weil integral). But a hyperbolic 4-manifold has an amphicheiral metric, and therefore the Chern-Weil integrand for the Pontryagin number must identically vanish. Alternatively, the Pontryagin number is an invariant of the stable tangent bundle, but the tangent bundle of a hyperbolic manifold is stably trivial for the same reason that that of a sphere is. See Milnor and Stasheff for details.