Late night math thoughts April 14, 2010

David Gay

These are some thoughts on contact filling and packing questions, based on conversations with Olga Buse.

First observation: rather than embedding balls, as in the symplectic case, we try to embed solid tori (or higher-dimensional equivalents), i.e. $ S^1 \times B^{2n}(r)$ (where $ B^{2n}(r)$ means the open ball of radius $ r$ in $ \mathbb{R}^{2n}$ ).

So first we define the standard contact structure $ \xi_0$ on $ S^1
\times \mathbb{R}^{2n}$ by $ \xi_0 = \ker \alpha_0$ and

$\displaystyle \alpha_0 = d\lambda + r_1^2 d \theta_1 + \ldots + r_n^2 d
\theta_n $

where $ \lambda$ is the $ S^1$ coordinate (it stands for ``longitude'') and $ (r_i,\theta_i)$ are polar coordinates on the $ i$ 'th $ \mathbb{R}^2$ in $ \mathbb{R}^{2n}$ .

Then, given a contact $ (2n+1)$ -manifold $ (M,\xi)$ we are interested in finding embeddings $ \phi \colon\thinspace (S^1 \times B^{2n}(r), \xi_0) \ensuremath{\hookrightarrow}(M,\xi)$ for large values of $ r$ . As we shall see when looking at the case $ n=1$ , i.e. $ \dim(M)=2n+1=3$ , this question probably only makes sense when we fix the smooth isotopy class of $ \phi$ , and perhaps we also need to fix the ambient contact isotopy class of $ K=\phi(S^1 \times
\{0\})$ . (In dimension $ 3$ , fixing the ambient contact isotopy class of $ K$ and the smooth isotopy class of $ \phi$ means fixing a transverse isotopy class of framed transverse knots.)

Given $ (M,\xi)$ , we can attempt some definitions as follows:

  1. The supremum over all $ r$ such that there exists an embedding $ \phi \colon\thinspace (S^1 \times B^{2n}(r), \xi_0) \ensuremath{\hookrightarrow}(M,\xi)$ . I claim that, at least in dimension $ 3$ , this is stupid because this value is always $ \infty$ for the following stupid reason: Every transverse knot $ K$ in $ (M^3,\xi)$ has a neighborhood contactomorphic to $ (S^1
\times B^2(\epsilon),\xi_0)$ for some small $ \epsilon$ , and there exists the following embedding $ \Phi \colon\thinspace (S^1 \times \mathbb{R}^2,\xi_0)
\ensuremath{\hookrightarrow}(S^1 \times \mathbb{R}^2,\xi_0)$ :

    $\displaystyle \Phi \colon\thinspace (\lambda,r,\theta) \mapsto (\lambda,\frac{r}{\sqrt{1+
r^2}}, \theta-\lambda) $

    The image is $ S^1 \times B^2(1)$ , and if you iterate this embedding you can embed $ (S^1 \times
\mathbb{R}^2,\xi_0)$ into $ (S^1
\times B^2(\epsilon),\xi_0)$ for any $ \epsilon
> 0$ . Also note that this definition would be stupid even if we fixed the ambient contact isotopy class of $ K = \phi(K_0)$ . But the point is that this embedding is not isotopic to the identity, since it changes the framing of $ K_0 = S^1 \times \{0\}$ .

  2. Fix a transverse isotopy class of framed transverse knots $ (K,F) \subset (M^3,\xi)$ , with $ K$ being the knot and $ F$ being the framing, and let $ c(M,\xi,K,F)$ be the supremum over all $ r$ such that there exists an embedding $ \phi \colon\thinspace (S^1 \times B^{2}(r),
\xi_0) \ensuremath{\hookrightarrow}(M,\xi)$ with $ \phi(K_0)$ transverse isotopic to $ K$ and with the framing coming from $ \phi$ equal to $ F$ . I claim that this is a little stupid, but not as stupid, for the following reason: First, recall that two framings of the same knot differ by an integer, but that in general there is no preferred 0 framing; i.e. the set of framings is an affine $ \mathbb{Z}$ . Then observe that the observations in the preceding item show that either:
    1. For all framings $ F$ , $ c(M,\xi,K,F) = \infty$ , or
    2. There exists some framing $ F_K$ such that $ 1 \leq c(M,\xi,K,F_K) <
\infty$ and such that, for all $ F < F_K$ , $ c(M,\xi,K,F) = \infty$ .
    Furthermore, in the second case, given any framing $ F > F_K$ , $ c(M,\xi,K,F)$ is completely determined by $ c(M,\xi,K,F_K)$ according to some formula which can be derived from the explicit description of the embedding $ \phi$ above.

  3. Thus a better capacity is, given a transverse isotopy class of transverse knots $ K$ , let $ c(M,\xi,K) = (F_K,c(M,\xi,K,F_K))$ , which is in $ \{\mathrm{framings}\} \times [1,\infty)$ .
  4. What is the analog in higher dimensions? There are no framings now, and there is the funny contact squeezing that is isotopic to the identity. There does exist a definition, but I haven't looked at it carefully or internalized it.

So now on to packing questions:

  1. Full Packing: Given closed $ (M^{2n+1},\xi)$ and $ p \in
\mathbb{N}$ , does there exist a collection of radii $ \{r_1, \ldots, r_p\}
\in (0,\infty)$ and a collection of embeddings $ \{\phi_1, \ldots,
\phi_p\}$ , with $ \phi_i \colon\thinspace (S^1 \times B^{2n}(r_i),\xi_0) \ensuremath{\hookrightarrow}
(M,\xi)$ , with disjoint images, such that $ M$ is the closure of the union of the images. First case should be $ p=1$ .
  2. Full Packing fixing isotopy classes: Same question as above, but fix the smooth isotopy classes of the embeddings in advance, and perhaps fix the ambient contact isotopy class of the images of the cores $ K_0$ . This is presumably harder than the general full packing problem. But it would be interesting if, for example, there exist full $ 1$ -packings in dimension $ 3$ for some transverse knot types but not for others.
  3. Packings without worrying about fully filling: This one I can only make sense of in dimension $ 3$ , I think. Pick a collection of transverse knots $ K_1, \ldots, K_p$ for which these framings $ F_{K_i}$ mentioned above do exist, with $ c(M,\xi,K_i) > 1$ for each $ K_i$ . Can you disjointly embed $ p$ copies of $ (S^1 \times
B^2(1),\xi_0)$ as neighborhoods of the $ K_i$ 's realizing framings $ F_{K_i}$ ? By choosing radius $ 1$ , we sidestep the individual capacity issues.

OK, its late now, so next time I'll write about possible strategies to either construct or obstruct, in analogue with blow-ups, etc. in the symplectic world. (Surgeries and handle attachments.)

Also, I want to write down the idea about possibly getting interesting symplectic embeddings of $ B^2(a) \times B^2(b)$ in $ B^4(1)$ by using knottedness, or by using $ 2$ -handle attachments.

David Gay 2010-04-15