From: baez@math.ucr.edu (John Baez) Subject: This Week's Finds in Mathematical Physics (Week 164) Date: 14 Jan 2001 04:35:36 GMT Newsgroups: sci.physics.research,sci.physics,sci.math Summary: Homology spheres and exotic spheres Also available at http://math.ucr.edu/home/baez/week164.html [deletia -- physics material available from source above. --djr] Careful readers of This Week's Finds will remember Diarmuid Crowley from "week151". This week he visited U. C. Riverside and talked about the topology of 7- and 15-dimensional manifolds. He also told me the following cool things. You may recall from "week163" that the Poincare homology 3-sphere is a compact 3-manifold that has the same homology groups as the ordinary 3-sphere, but is not homeomorphic to the 3-sphere. I explained how this marvelous space can be obtained as the quotient of SU(2) = S^3 by a 120-element subgroup - the double cover of the symmetry group of the dodecahedron. Even better, the points in S^3 which lie in this subgroup are the centers of the faces a 4d regular polytope with 120 dodecahedral faces. That's pretty cool. But here's another cool way to get the Poincare homology sphere: E8 is the biggest of the exceptional Lie groups. As I explained in "week64", the Dynkin diagram of this group looks like this: o----o---o----o----o----o----o | | o Now, make a model of this diagram by linking together 8 rings: /\ /\ /\ /\ /\ /\ /\ / \ / \ / \ / \ / \ / \ / \ / \ \ \ \ \ \ \ / / \ / \ / \ / \ / \ / \ \ \ \ / \ / \ / \ / \ / \ / / \ \ \ \ \ /\ \ \ / \ / \ / \ / \ / \ \ \ / \ / \/ \/ \/ \/ / \/ \ \/ \/ / \ \ / \ / \ / \/ Imagine this model as living in S^3. Next, hollow out all these rings: actually delete the portion of space that lies inside them! We now have a 3-manifold M whose boundary dM consists of 8 connected components, each a torus. Of course, a solid torus also has a torus as its boundary. So attach solid tori to each of these 8 components of dM, but do it via this attaching map: (x,y) -> (y,-x+2y) where x and y are the obvious coordinates on the torus, numbers between 0 and 2pi, and we do the arithmetic mod 2pi. We now have a new 3-manifold without boundary... and this is the Poincare homology sphere. We see here a strange and indirect connection between E8 and the dodecahedron. This is not the only such connection! There's also the "McKay correspondence" (see "week65") and a way of getting the E8 root lattice from the "icosians" (see "week20"). Are these three superficially different connections secretly just different views of the same grand picture? I'm not sure. I think I'd know the answer to part of this puzzle if I better understood the relation between ADE theory and singularities. But Diarmuid Crowley told me much more. The Poincare homology sphere is actually the boundary of a 4-manifold, and it's not hard to say what this 4-manifold is. I just gave you a recipe for cutting out 8 solid tori from the 3-sphere and gluing them back in with a twist. Suppose we think of 3-sphere as the boundary of the 4-ball D^4, and think of each solid torus as part of the boundary of a copy of D^2 x D^2, using the fact that d(D^2 x D^2) = S^1 x D^2 + D^2 x S^1. Then the same recipe can be seen as instructions for gluing 8 copies of D^2 x D^2 to the 4-ball along part of their boundary, getting a new 4-manifold with boundary. If you ponder it, you'll see that the boundary of this 4-manifold is the Poincare homology 3-sphere. Now, this is actually no big deal, at least for folks who know some 4-dimensional topology. But Crowley like higher-dimensional topology, and what he told me is this: the whole story generalizes to higher dimensions! Instead of starting with this picture of linked 1-spheres in the 3-sphere: /\ /\ /\ /\ /\ /\ /\ / \ / \ / \ / \ / \ / \ / \ / \ \ \ \ \ \ \ / / \ / \ / \ / \ / \ / \ \ \ \ / \ / \ / \ / \ / \ / / \ \ \ \ \ /\ \ \ / \ / \ / \ / \ / \ \ \ / \ / \/ \/ \/ \/ / \/ \ \/ \/ / \ \ / \ / \ / \/ start with an analogous pattern of 8 n-spheres linked in the (2n+1)-sphere. Do all the same stuff, boosting the dimensions appropriately... and you'll get an interesting (2n+1)-dimensional manifold dM which is the boundary of a (2n+2)-dimensional manifold M. When n is *odd* and greater than 1, this manifold dM is actually an "exotic sphere". In other words, it's homeomorphic but not diffeomorphic to the usual sphere of dimension 2n+1. Now, exotic spheres of a given dimension form an abelian group G under connected sum (see "week41"). This group consists of two parts: the easy part and the hard part. The easy part is a normal subgroup N consisting of the exotic spheres that bound parallelizable smooth manifolds. The size of this subgroup can be computed in terms of Bernoulli numbers and stuff like that. The hard part is the quotient group G/N. This is usually the cokernel of a famous gadget called the "J-homomorphism". I say "usually" because this is known to be true in most dimensions, but in certain dimensions it remains an open question. Anyway: the easy part N is always a finite cyclic group, and this is *generated* by the exotic sphere dM that I just described! For example: In dimension 7 we have G = N = Z/28, so there are 28 exotic spheres in this dimension (up to orientation-preserving diffeomorphism), and they are all connected sums of the exotic 7-sphere dM formed by the above construction. In dimension 11 we have G = N = Z/992, so there are 992 exotic spheres, and they are all connected sums of the exotic 11-sphere dM formed by the above construction. In dimension 15 we no longer have G = N. Instead we have N = Z/8128 and G = Z/8128 + Z/2. There are thus 16256 exotic spheres in this dimension, only half of which are connected sums of the exotic 15-sphere dM formed by the above construction. And so on. While we're on the subject of exotic 15-spheres, I can't resist mentioning this. I explained in "week141" how to construct a bunch of exotic 7-spheres (24 of them, actually) using the quaternions. Once you understand this trick, it's natural to wonder if you can construct exotic 15-spheres the same way, but using octonions instead of quaternions. Well, you can: 3) Nobuo Shimada, Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. Jour. (12) 1957, 59-69. I should also explain what I really like about the above stuff. In topological quantum field theory, people like to get 3-manifolds by "surgery on framed links". The idea is to start with a framed link in the 3-sphere, use the framing to thicken each component to an embedded solid torus, cut out these solid tori, and reattach them "the other way", using the fact that S^1 x S^1 is the boundary of both S^1 x D^2 and D^2 x S^1. We can get any compact oriented 3-manifold this way. The above construction of the Poincare homology sphere was just an example of this, where the link was /\ /\ /\ /\ /\ /\ /\ / \ / \ / \ / \ / \ / \ / \ / \ \ \ \ \ \ \ / / \ / \ / \ / \ / \ / \ \ \ \ / \ / \ / \ / \ / \ / / \ \ \ \ \ /\ \ \ / \ / \ / \ / \ / \ \ \ / \ / \/ \/ \/ \/ / \/ \ \/ \/ / \ \ / \ / \ / \/ and each component had two twists in the framing as we go around, as compared to the standard "blackboard" framing. This is why there was that mysterious number "2" in my formula for the attaching map. Whenever we describe a 3-manifold using "surgery on framed links" this way, there's an important matrix where the entry in the ith row and jth column is the linking number of the ith component and the jth component of our framed link, with the diagonal entries standing for the "self-linking" numbers of the components, that is, the number of twists their framings have. This matrix is important because it also describes the "intersection form" on the 2nd homology group of a simply-connected 4-manifold M whose boundary dM is the 3-manifold we're describing. For example, in the case of the Poincare homology sphere, this matrix is called the E8 Cartan matrix: 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 -1 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 -1 0 -1 2 The Dynkin diagram simply summarizes this matrix in pictorial form. I already described the 4-manifold M whose boundary is the Poincare homology sphere; now you know its intersection form. Anyway, what I find exciting is that all this stuff generalizes to higher dimensions if we restrict attention to manifolds that have trivial homotopy groups up to a certain point! For example, it works for compact oriented smooth 7-manifolds that have trivial pi_1 and pi_2. Any such manifold can be obtained by doing surgery on some framed 3-spheres embedded in S^7. Just as 1-spheres can link in 3d space since 1+1 is one less than 3, 3-spheres can link in 7d space since 3+3 is one less than 7. We again get a matrix of linking numbers. As before, this matrix is also an intersection form: namely, the intersection form on the 4th homology group of an 8-manifold M whose boundary dM is the 7-manifold we're describing. Moreover, this matrix is symmetric in both the 3-manifold example and the 7-manifold example, since it describes an intersection pairing on an *even-dimensional* homology group. Even better, all the same stuff happens in manifolds with enough trivial homotopy groups in dimension 11, and dimension 15... and all dimensions of the form 4n-1. And what's *really* neat is that these higher-dimensional generalizations are in some ways simpler than the 3d story. The reason is that a 1-sphere can be knotted in 3-space in really complicated ways, but the higher-dimensional generalizations do not involve such complicated knotting. The framing aspects can be more complicated, since there's more to framing an embedded sphere than just an integer, but it's not all *that* complicated. So maybe I can learn some more 3d topology by first warming up with the simpler 7d case.... [deletia --djr