Stanford Holomorphic Curves and Low-dimensional topology

Over the last week I’ve been at Stanford for a workshop. The speakers and abstracts are here http://www.math.umn.edu/~akhmedov/Stanford2012.html. There were 6 minicourses, so I’ll post on some of them over the next few days.

One of the minicourses that finished up yesterday was given by Akbulut on studying 4-manifolds via handlebody decompositions. There is a long set of notes covering a lot more than was discussed in the 3 lectures, that may eventually turn into a book: http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf. I’ll just summarize some of the things that were discussed in the lectures that I found interesting.

1. Suppose you have 2 handlebody decompositions for 4-manifolds X, X’ with diffeomorphic boundaries, and you think the 4-manifolds may be diffeomorphic. Looking for the right sequence of Kirby moves from one diagram to the other can be an intractable problem, so here is another way to try to extend the diffeomorphism from the boundary to the interior. Consider the cocores of each of the 2-handles in X, and look at their circle boundaries in $\partial X$. See where those circles are sent under the diffeomorphism of $\partial X \to \partial X'$. Check if the image of those circles bound disks in X’. If they do we can extend the diffeomorphism on the boundary over a neighborhood of these cocores. Then we are left with just extending a diffeomorphisms of $\#_n S^1\times S^2$ over $\natural_n S^1\times D^3$. In the case that n=0, we are trying to extend a diffeomorphism of $S^3$ to itself over $D^4$. The diffeomorphism probably extends, since if it doesn’t you get a counterexample to the smooth Poincare conjecture in dimension 4, so either way you should be happy.

2. The guiding principle during these talks was essentially that you can answer most questions about a 4-manifold by drawing the correct handlebody diagram. In Akbulut’s notes he describes a bunch of ways to get a handlebody diagram from various descriptions of 4-manifolds that I’ve seen before in the literature (e.g. Gompf-Stipsicz 4-manifolds and Kirby Calculus), but he also discussed some techniques for producing diagrams he has been using more recently that I hadn’t seen before.

One such technique is the “cylinder method” which allows one to take two 4-manifolds with diffeomorphic boundary, and glue them together without having to turn one of the manifolds upside-down. Turning a 4-manifold upside-down can get pretty complicated and it is hard to recognize the two pieces you started with when everything is glued up. This cylinder method allows you to keep the original handlebody diagrams for the two 4-manifolds with diffeomorphic boundary, and just adds on more handles to build a mapping cylinder (diffeomorphic to one boundary cross an interval) connecting the two diffeomorphic boundary components.

Conceptually it is a very natural idea, but there are some things to keep track of to draw the new handles correctly. Suppose X and Y are 4-manifolds with boundaries M and N respectively, and $f: M\to N$ is a diffeomorphism. We glue the boundary of X to the boundary of Y handle by handle. First to glue the 0-handle of X to the 0-handle of Y requires an additional 1-handle connecting the two 0-handles. Since a standard diagram assumes a unique 0-handle in the background, we can cancel the new 1-handle with one of the 0-handles. To glue a 1-handle of X to a 1-handle of Y, we need to add a 2-handle that passes through each 1-handle. This would be straightforward if the diffeomorphism $f: M\to N$ identifying the boundaries were simply the identity map, but in general the diffeomorphism may move the 1-handles around. To keep track of this Akbulut drops down “ropes” from some point up above the boundaries where the ropes are looped around the cores of the 1-handles that show up on the boundary. Then apply the diffeomorphism f and track how the ropes get tangled up. Now draw the attaching circles for the new 2-handles so they hook through the core of the 1-handle of X, both strands go straight up and then together follow the path through the tangle described by the ropes after applying f, and finally hook around the core of the 1-handle of Y to close up the circle. Akbulut has a good picture describing this in the notes on page 29. To connect 2- and 3-handles requires adding 3- and 4-handles but the attaching maps of 3- and 4-handles are canonically determined for a closed 4-manifold.

Akbulut said he used this cylinder method as part of producing useful pictures of the Akhmedov-Park manifolds, which are exotic copies of $\mathbb{CP}^2 \# 3 \overline{\mathbb{CP}^2}$ and $\mathbb{CP}^2 \# 2 \overline{\mathbb{CP}^2}$. Such diagrams apparently made it easy to read off the fundamental group of the manifolds, to confirm that they were simply connected and the constructions were actually the exotic manifolds they were suspected of being.

3. Here is another interesting application of drawing pictures: the construction of infinitely many homeomorphic but non-diffeomorphic Stein fillings of a contact 3-manifold. Take the elliptic surface E(1) which is a torus fibration over $S^2$ with 12 singular fibers, and do a p-log transform on one regular fiber and a q-log transform on another parallel regular fiber. This gives a 4-manifold $E(1)_{p,q}$ which is an exotic copy of E(1). Akbulut constructs diagrams for E(1) where you can see the regular torus fibers explicitly and performs the log transforms then notices that much of the diagram for $E(1)_{p,q}$ is unaffected by changing the values of p and q. Find a domain which contains all the handles that change with p and q. Then the boundary of the domain is constant since the complement of this domain you chose is independent of choice of p and q. If the domain you chose was Stein, you get infinitely many Stein fillings of a 3-manifold. There are only finitely many contact structures on a 3-manifold which are Stein fillable, so there is some contact structure with infinitely many fillings. With some extra work Akbulut and Yasui can show that these fillings obtained by varying p and q are all homeomorphic but non-diffeomorphic.