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\begin{document}
\title{Notes on Eliashberg's 1989 paper, ``Classification of overtwisted contact structures
on 3-manifolds"}
\author{Daniel Mathews}%
\maketitle
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\tableofcontents
\section{Introduction}
Eliashberg in 1989 in \cite{El89} triumphed over overtwisted
structures. They are now completely classified.
Let $M$ be an oriented connected $3$-manifold. Take a basepoint $p
\in M$ and an embedded disc $\Delta \subset M$ centred at $p$. Let
$\Distr(M)$ denote the space of all tangent 2-plane distributions on
$M$, with one extra condition: we fix the distribution at the point
$p$. We give $\Distr(M)$ the $C^\infty$ topology. We let:
\begin{itemize}
\item
$\Cont(M)$ be the subspace of $\Distr(M)$ which consists of
(positive) contact structures;
\item
$\Cont^{OT}(M)$ be the subspace of $\Cont(M)$ consisting of all
overtwisted structures which have the disc $\Delta \subset M$ as
the standard overtwisted disc.
\end{itemize}
There are then obvious inclusions
\begin{align*}
i: & \Cont(M) \To \Distr(M) \\
j: & \Cont^{OT}(M) \To \Distr(M).
\end{align*}
\begin{thm}
\label{homotopy_equivalence}
The inclusion $j: \Cont(M)^{OT} \To \Distr(M)$ is a homotopy
equivalence.
\end{thm}
For $M$ an open manifold this follows from an old (1969) theorem of
Gromov in \cite{Gr69}.
This theorem subsumed the theorem of Lutz \cite{Lutz77} that $i_*:
\pi_0 (\Cont(M)) \To \pi_0(\Distr(M))$ is surjective. Very much
subsumed it!
This basically means that the classification of overtwisted contact
structures is the same as the classification of 2-plane fields on
$M$. If you take a homotopy class of 2-plane fields, then there's an
overtwisted contact structure among them. And, if you have two
contact structures in the same homotopy class of 2-plane fields,
then they are homotopic through contact plane fields, i.e. they are
isotopic contact structures. It follows from Gray's theorem that
they're isomorphic.
\section{Sketch of the proof}
The proof has some nice ideas, but also some hard analysis. (Hard as
in many $\epsilon$'s and many things to keep control of, but still
elementary in nature.)
Theorem \ref{homotopy_equivalence} is really about an extension
problem. We need to find a homotopy inverse; from each 2-plane
distribution we need to find a contact structure. Their compositions
must be homotopic to the identity. In one direction this will be
easy, because homotopies of contact structures are obviously
homotopies of 2-plane distributions. But in the other direction we
have something to prove: given a homotopy of 2-plane distributions,
we need to find a homotopy of contact structures. This can be
considered an extension problem: in fact Eliashberg proves the
following theorem, which ``immediately" implies theorem
\ref{homotopy_equivalence}.
\begin{thm}
Let:
\begin{enumerate}
\item
$M$ be a compact 3-manifold;
\item
$A \subset M$ be a closed subset such that $M \backslash A$ is
connected (we are thinking of $A$ as a 1-skeleton of a
simplicial decomposition of $M$);
\item
$K$ be a compact space (parameter space for the homotopy)
\item
$L \subset K$ a closed subspace (smaller parameter space; we
want to extent from $L$ to $K$).
\item
$\xi_t$ be a family of 2-plane distributions on $M$ defined for
all $t \in K$. For $t$ in the smaller parameter space $L$, $\xi_t$ is
contact on $M$. And for all $t$ in the total parameter space $K$,
$\xi_t$ is contact on the closed subset $A$.
\end{enumerate}
(So we have a homotopy extension problem: the homotopy is defined on
$A$, and partially defined on $M$; we need to extent it.) Suppose
that there is an disc $\Delta$ in $M$ which is always a standard
overtwisted disc, for all $t \in K$: rigorously, suppose there
exists an embedded 2-disc $\Delta \subset M \backslash A$ such that
for all $t \in K$, $\xi_t$ is contact near $\Delta$ and $(\Delta,
\xi_t)$ is equivalent to the standard overtwisted disc.
Then the homotopy problem can be solved! That is, there exists a
family $\xi'_t$ of contact structures on $M$ for all $t \in K$ such
that
\begin{enumerate}
\item
for all $t$ in the total parameter space $K$, $\xi'_t$ coincides with
$\xi_t$ near $A$;
\item
for all $t$ in the smaller parameter space $L$, $\xi'_t$
coincides with $\xi_t$ everywhere on $M$.
\item
the family $\xi'_t$, over all $t$ in the total parameter space $K$,
can be connected with $\xi_t$ through a homotopy, which is fixed
on $A \times K \cup M \times L$.
\end{enumerate}
\end{thm}
Because it's a homotopy extension problem, it is sufficient to
consider $M$ a compact subset of $\R^3$. $M$ can be covered by such
sets; and we just extend repeatedly over them. Then for any
two-plane field $\xi$, we have a Gauss map $M \To S^2$ and we define
the \emph{norm} $||\xi||$ of a 2-plane distribution to be the
maximum of the derivative of the Gauss map. The norm of the
distribution is the fastest speed at which the plane turns.
\section{Step I: Construct near a 2-skeleton}
The first step is to construct the contact structure (and homotopy!)
near the 2-skeleton of a general simplicial complex for $M$. We can
effectively take $M,A$ to be simplicial. We will take a very fine
subdivision so that the diameter goes to zero while all relevant
angles are bounded below, and while the minimal distance between
disjoint simplices (relative to the diameter) is bounded below. To
be precise! $P$ is a simplicial complex.
\begin{enumerate}
\item
$\alpha(P)$ is the minimal angle between non-incident 1- or
2-simplices which have a mutual vertex;
\item
$d(P)$ is the maximal diameter of a simplex of $P$;
\item
$\delta(P)$ is the minimal distance between two 0-, 1- or
2-simplices without mutual vertices.
\end{enumerate}
A lot of subdivision and a little perturbing and thought gives us
\begin{lem}
\label{subdivision}
There exists a sequence of general subdivisions $P_i$ of $P$ such that
$d(P_i) \To 0$ while $\delta(P_i)/d(P_0)$ and $\alpha(P_i)$ are
bounded below.
\end{lem}
In effect, we can make $d(P)$ arbitrarily small without worrying
about the other parameters.
Now, we will find a simplicial complex $P$ such that we can do the
extension in a neighbourhood of its 2-skeleton. To be precise:
\begin{lem}
Let $M,A,K,L, \xi_t$ be as above. (Although now $M$ is a compact
subset of $\R^3$.) Then there exists a general simplicial complex
$P$ containing $M$ and a family of distributions $\xi'_t$ (on $M$)
defined for all $t$ in the total parameter space $K$, such that:
\begin{enumerate}
\item
for all $t$ in the parameter space $K$, $\xi'_t$ is $C^0$-close
to $\xi_t$;
\item
$\xi'_t$ agrees with $\xi_t$ on $A \times K \cup M \times L$;
\item
$\xi'_t$ is contact on a neighbourhood on the 2-skeleton of $P$.
In fact it is contact in a ``uniform" way: there exists an
$\epsilon>0$ such that $\xi'_t$ is contact in an $\epsilon \cdot
d(P)$-neighbourhood of the 2-skeleton. Here $\epsilon$ depends
only on $\alpha$ and $\delta/d$.
\item
The norm $||\xi'_t||$ is bounded as $||\xi'_t|| \leq C || \xi_t
|| + D$, for some universal constants $C$ and $D$.
\end{enumerate}
\end{lem}
Note that $\xi'_t$ is actually defined on $M$, but we really only
require something of it on $A$ (once this is done, all we require is
to extend it to $M$ while remaining $C^0$-close to $\xi_t$; this is
easy).
We build up the extension from $A$ to $M$, simplex by simplex. What
makes the extension difficult is if $\xi_t(x)$ varies too much, over
$t$, or over $x$. The variation over $t \in K$ can be dealt with
directly: taking a subdivision of $K$ now, we can assume that for
any $t,t' \in K$ the planes $\xi_t$ and $\xi_{t'}$ are always close.
(We will have to make sure that we can keep extending over all parts
of $K$!) And variation over $x \in M$ is dealt with by a division
into cases. A 1-simplex $\sigma$ is called \emph{special} if at some
point $x$ of $\sigma$ and for some $t \in K$, $\xi_t$ is too close
to being perpendicular to $\sigma$. A 2-simplex $\sigma$ is called
\emph{special} if at some point $x$ of $\sigma$ and for some $t \in
K$, $\xi_t$ is too close to being parallel to $\sigma$.
By the above lemma \ref{subdivision}, we can make the diameter $d$
arbitrarily close to zero while keeping $\alpha$ bounded below,
which makes special simplices isolated. ($\xi$ doesn't change much
in $t$ since we subdivided $K$; $\xi$ doesn't change much on a
simplex $x$ since $d$ is small; other nearby simplices differ a
definite angle since $\alpha$ is bounded below; so a nearby simplex
will not be special.)
The trick is to consider a 2-dimensional foliation $\F_\sigma$ near
each simplex $\sigma$ over which we wish to extend. $\F_\sigma$ is a
foliation of a neighbourhood of $\sigma$ (not depending on any $t$,
but of course varying with $x$ by planes which are perpendicular to
for $\xi_t(x)$, for one (random) value of $t$. If $\sigma$ is a
1-simplex, we require $\F_\sigma$ to be parallel to $\sigma$; if
$\sigma$ is a 2-simplex, we require $\F_\sigma$ to be perpendicular
to $\sigma$. (This is sufficient to define a plane at each point;
and clearly it's integrable. There is ambiguity when $\sigma$
parallel/perpendicular to $\xi_t$; but this only occurs at special
points in special simplices!) On each (2-dimensional) leaf of
$\F_\sigma$, for each $t \in K$ we may obtain a 1-dimensional
foliation by intersecting with the 2-plane field $\xi_t$. (By
construction $\xi_t$ is always transverse to $\F_\sigma$.) We
perturb $\xi_t$ along these 1-dimensional leaves. It turns out that
this can be done provided that for each 1-dimensional leaf $l$ we
have $\pi_1(l, A \cap l) = 0$. Why is this? The idea is that along a
tangent curve $l$, a plane field has a standard form and if you can
make it ``always twist in the same direction", it's contact. The
$\pi_1$ condition allows you to put a twist in continuously. (We
will not yet worry that $\xi'_t$ is $C^0$-close!) The $\pi_1$
condition is satisfied for special simplices because special
simplices are isolated, and have at most one face belonging to $A$.
Next we turn to neighbourhoods of non-special 1- and 2-simplices.
For sufficiently small neighbourhoods, the $\pi_1$ condition will be
satisfied; there are less complications since the angle between
$\xi_t$ and $\sigma$ is never too close (for any $t$ or $x$). Again
we can perturb $\xi_t$ to $\xi'_t$ using the foliation $\F_\sigma$.
Provided we take $d$ sufficiently small, we can get $\xi'_t$
sufficiently $C^0$-close to $\xi_t$. The norm $||\xi'_t||$ may
increase, but only linearly. So we can continue extending over all
the simplices until we are done. And then we can continue over
different subdivisions of $K$. By the end our neighbourhood will be
very small indeed; but it is still a subdivision. However, if we
keep track of the geometry, it only depends on the geometry of the
simplices, namely it is of the form required.
\section{Step II: A contactization with holes}
Now we've done our extension on a neighbourhood of a two-skeleton.
This really amounts to the whole manifold, minus a few holes. But we
are worried about extending over those holes, so we need to keep
track of their geometry. Our neighbourhood of the 2-skeleton may be
small, but still we can take a ball inside each 3-cell, containing
the hole, and its curvatures (which may be very flat along the
faces) will be bounded below. This is a ``simple assertion":
\begin{lem}
Let $\sigma$ be a 3-simplex of diameter $d$. For any $\lambda > 0$,
there exists and embedded ball $B \subset \sigma$ such that its
boundary is contianed in a $\lambda$-neighbourhood of $\delta
\sigma$ and the normal curvatures of $\partial B$ are everywhere
$\geq 8\lambda/(4\lambda^2 + d^2)$.
\end{lem}
But because we have such a fine subdivision, $\xi_t$ doesn't change
very much. And since we can bound $||\xi'_t||$ in terms of
$||\xi_t||$, $\xi'_t$ doesn't change very much either. So the
characteristic foliation on the boundary $\partial B$ of our ball
$B$ will turn out to be rather simple.
How simple? Well, let us digress for a minute and consider
one-dimensional foliations $\F$ on $S^2$, in particular those with
precisely two elliptic singular points, at the poles. (Our situation
will have tow such poles.) Such a foliation is \emph{simple} if all
its limit cycles are isolated and placed on parallels between the
foci. It is \emph{almost horizontal} if there is a transversal to
$\F$ connecting the poles. (When you draw a picture, an almost
horizontal foliation ``never turns around" between limit cycles.)
Almost horizontal is very nice, because an almost horizontal
foliation gives a holonomy map $h(\F): I \to I$ which is a
diffeomorphism of the interval. (Consider a transversal and a return
map.) Almost horizontal is very nice also, because the present
situation is almost horizontal!
Why? Another ``simple assertion":
\begin{lem}
Let $S \subset \R^3$ be an embedded 2-sphere with all normal
curvatures $\geq K > 0$. Let $\xi$ be a contact structure near $S$
with $||\xi|| < K$. Then $\xi$ is almost horizontal near $S$.
\end{lem}
With a sufficiently fine subdivision, we have normal curvatures
everywhere arbitrarily high while keeping our neighbourhood
sufficiently small for the previous part of the proof to work(if you
check the dependencies in the previous part of the proof, the
neighbourhood width $\lambda$ was of the form $\epsilon d$, where
$\epsilon$ depended only on $\delta/d$ and $\alpha$). So this lemma
applies. As to why the lemma is true, well, being not almost
horizontal (that is, having a ``turn around" in the foliation)
implies the contact structure turning quite fast; faster than the
curvature, it seems.
It turns out that the topological type of the characteristic
foliation on the sphere is all that matters for our purposes.
\begin{lem}
Let $\xi$ be a simple contact structure near the boundary $S =
\partial B$ of the 3-ball $B$. The extendability of $\xi$ as a
contact structure to $B$ depends only on the topological type of
the foliation $S_\xi$.
\end{lem}
Why is this true? It's (what later became) a standard perturbation
argument. Take two contact structures $\xi_t$, $\xi'_t$ near $S$. We
take $L \subset S$ to be a union of transversals and limit cycles of
both characteristic foliations. [Taking transversals seems to imply
we're talking about almost horizontal, rather than simple, contact
structures near $S$. Hmm.] Let $N$ be a tubular neighbourhood of $L$
(in $\R^3$). We can get a contactomorphism $g$ on $S \backslash N
\To S \backslash N$, since these are just disks with standard
foliations. We extend this diffeomorphism to $S$, remaining constant
on $L$. Now $g$ will not necessarily be a contactomorphism on all of
$S$; but the characteristic foliation will be $C^1$-close to what is
required. We now perturb $C^0$-perturb $g$, to embed $S$ in $B$, and
$C^1$-adjust the characteristic foliation.
So this now gives:
\begin{lem}
Let $M,A,K,L, \xi_t$ be as above. There exist disjoint 3-balls $B_1,
\ldots, B_N$ which avoid $A \cup \Delta$ and distributions $\xi'_t$
on $M$, defined for all $t$ in the parameter space $K$, such that
\begin{enumerate}
\item
$\xi'_t$ and $\xi_t$ agree on $(A \cup \Delta) \times K \cup M
\times L$;
\item
for all $t \in K$, $\xi'_t$ is contact everywhere except the interiors of the
$B_i$;
\item
for all $t \in K$, $\xi'_t$ is almost horizontal near every $\partial B_i$;
\item
for all $t \in K$, $\xi'_t$ is $C^0$-close to $\xi_t$;
\end{enumerate}
\end{lem}
Notice $A$ sneakily became $A \cup \Delta$ here, but that is no big
deal. We just treat the overtwisted disk $\Delta$ as part of $A$.
\section{Step III: Making one hole and filling it}
We can connect sum simple foliations on spheres by cutting off
neighbourhoods of poles and gluing and smoothing the glued
foliations. We order our balls arbitrarily $B_1, \ldots, B_N$ and
take for $B_0$ a small ball containing the overtwisted disk
$\Delta$. Now we connect the north pole of each $B_i$ with the south
pole of the next, by disjoint embedded curves $l_t^i$.
We need all of the $l_t^i$ to be transverse, else our connected
balls will not have standard characteristic foliations. So we need
to be able to perturb curves to be transverse; in fact, we need to
be able to perturb families of curves to be families of transverse
curves. It is not so difficult to do this for one curve: just take
enough wavefronts with the right gradients at the right points. It's
not so difficult to do this in families.
Having done that, we have connected the balls, so we have a contact
structure with one hole. If we can extend the contact structure over
the ball, and extend homotopies of plane fields to homotopies of
contact structures over the ball, we are done. But this is not
difficult. We know that the topological type of the almost
horizontal foliation is all that matters. In fact, given an almost
horizontal foliation, we can construct an explicit model of a solid
of revolution in $\R^3$ which gives that almost horizontal
foliation. (There's a picture in \cite{El89}.) These can be
homotoped, no problem, and remaining constant on $A$.
\section{Corollaries and related results}
The classification of overtwisted contact structures has an
interesting corollary about extending contact embeddings of $B^3 \To
S^3$ to $S^3 \To S^3$, Eliashberg's 1.6.2 in \cite{El89}.
\begin{thm}
There exists an overtwisted structure $\xi$ on $S^3$, a ball $B
\subset S^3$ and a contact embedding $\psi: (B, \xi) \To (S^3,
\xi)$ such that $\psi$ cannot be extended to a contact
diffeomorphism $S^3 \To S^3$. Hence $\psi$ cannot be connected
with the inclusion $B \To S^3$ by a contact isotopy. (If it
could, the isotopy would easily extend to an isotopy of
diffeomorphisms of $S^3$ and one end of this would be an
extension of $\psi$.)
\end{thm}
A very sketchy idea of the proof is to do Lutz twisting twice to get
a contact structure that has been made overtwisted: but the Lutz
twisting once (about $B$, giving $\zeta$) or twice (about $B$ and
$B'$, giving $\zeta'$) remains in the same homotopy class, and hence
$\zeta, \zeta'$ are isotopic, so also contactomorphic via $h: S^3
\To S^3$. The two balls $B,B'$ containing the Lutz twistings are
clearly contactomorphic. Mapping one to the other, and then applying
the contactomorphism $h$ gives a contact embedding $\psi$. But then
outside $B$ and its image, the contact structure is tight on one
side, and not on the other.
\begin{thebibliography}{99}
\bibitem{El89}
Y. Eliashberg, Classification of overtwisted contact structures on
3-manifolds
\bibitem{Gr69}
M. Gromov, Stable mappings of foliations into manifolds, Izv. Acad.
Naur SSSR, Ser. mat. 33, 1206--1209 (1969).
\bibitem{Lutz77}
R. Lutz, Structures de contact sur les fibr\'{e}s principaux en
cercles de dimension 3. Ann. Inst. Fourier, 3, 1--15 (1977).
\end{thebibliography}
\end{document}