Continuing towards a discussion of the proof of existence and classification of overtwisted contact structures in higher dimensions, here I want to talk about h-principles, and contact structures on open manifolds.
Given a partial differential equation or partial differential relation (like the contact condition ), one can formally replace the derivatives of the variables with independent formal variables (i.e. for a 2-form ). Solving this new problem where the derivatives are replaced by independent formal variables is purely an algebraic topology problem. If the algebraic topology problem has no solution then certainly the partial differential relation has no solution. However, it is generally surprising when the converse holds: namely, the existence of a solution to the algebraic problem implies the existence of a solution of the partial differential relation. A theorem that proves this type of statement is referred to as an h-principle.
The language of jet bundles and holonomic sections will help make this more precise below.
The main results of Borman, Eliashberg, and Murphy that we are heading towards are an h-principle for contact and almost contact structures on higher dimensional closed manifolds which says that any almost contact structure is homotopic through almost contact structures to an actual (overtwisted) contact structure, and a parametric version of this h-principle which says that any family of almost contact structures connecting two genuine overtwisted contact structures can be homotoped to a family of genuine contact structures connecting the fixed overtwisted contact structures on the ends.
While contact structures on closed manifolds can have incredibly complicated classifications (because of the rigidity of tight contact structures), it is a result of Gromov that on open manifolds the geometric subtlety disappears and the classification of contact structures is reduced to algebraic topology by an h-principle. This post is based on a talk given by Kyler as part of the discussion of the proof of flexibility of overtwisted contact structures in higher dimensions, though the original source for the content is Gromov’s Partial Differential Relations book.
Define a (cooriented) almost contact structure on an odd dimensional manifold to be a cooriented hyperplane distribution, together with a non-degenerate 2-form on the distribution. In dimension 3, this is homotopy equivalent to the space of co-oriented 2-plane distributions. Gromov’s theorem is:
Let V be an open manifold. Then the inclusion of cooriented contact structures on V into cooriented almost contact structures on V is a homotopy equivalence.
The proof is based on two main ideas: the holonomic approximation theorem on neighborhoods of codimension one polyhedra, and the fact that all open smooth manifolds smoothly retract onto a neighborhood of a complex of codimension at least one. I’ll start with the former.
The 1-jet space of a fiber bundle , is a bundle where the fiber over consists of sections of X defined over a neighborhood of p up to an equivalence which equates sections that agree up to 1st order near p. (The r-jet bundle is defined similarly where you equate sections which agree up to rth order, but here we will only need the 1-jet bundle.) A section of chooses an equivalence class of sections over each point in : for each , where is a point in the fiber , and specifies the first partial derivatives of a function at that point. However, even though the section is smooth, need not specify the actual derivative of since is encoded as an independent direction in the fibers of . A holonomic section of a 1-jet space is one where this linear variation specified by agrees with the actual partial derivatives of the differentiable section of given by the 0th order information of the section. The holonomic approximation theorem aims to approximate an arbitrary section of the 1-jet bundle by a holonomic section as well as possible.
Here the blue curve represents a section of . The grey curve represents its projection to the 0th order information, and the 1st order information is encoded in the dimension coming out of the page. Representing the value the blue curve takes in this dimension by a green line of the appropriate slope centered at each point on the grey curve, we see that this is not a holonomic section because the 1st order information is not tangent to the curve.
The important relevant example for Gromov’s theorem is when , so sections of the bundle are 1-forms. Sections of the 1-jet space keep track of two coordinates: the pointwise values of the underlying 1-form and its formal linear variation. Locally, is a trivial bundle, and a section is just the graph of a function on . Modding out by the equivalence relation, we get that for a section , keeps track of the point , a point in and an n by n matrix at that point which specifies the formal first partial derivatives of a graph in that equivalence class (where n is the dimension of V). Symmetrizing this matrix (), gives the coefficients for a 2-form. When the section of is a holonomic section, this 2-form built from the 1st order information of the section, is the exterior derivative of the 1-form which gives the 0th order information of the section. Given any pair of a 1-form and a 2-form, there is a section of such that the 0th order information gives and after symmetrizing the 1st order information we get . For holonomic sections, this process gives a pair where is a 1-form.
There are certain limitations on the extent to which we can approximate an arbitrary section by a holonomic one. For example if we consider the 1-jet space of the bundle . A section of the 1-jet space is given by specifying the pointwise data and a formal 1st derivative. An example of a section has pointwise data given by the graph of , and formal derivative specified as 0 (horizontal lines at each point). To approximate this by a holonomic section, we would need to find a function g whose pointwise values only differ from those of by , and whose derivative only differs from zero by . Such a function would contradict the mean value theorem. So we cannot hope to approximate an arbitrary section by a holonomic one at every point. On the other hand, we can approximate the section in a small neighborhood of a point.
This motivates the idea to look at codimension 1 subspaces. Taking the previous example and just taking the product with a trivial extra dimension with coordinate y, we run into the same problem: that if we can only move up with a tiny slope in the x-direction, we cannot get up far enough by just moving along a path that has slope 1 in the x direction and does not move in the y-direction. However if we are allowed to perturb the path to lengthen it in the extra y-dimension that we have by adding many zig-zags, then we can do this approximation.
Moving along the black curve, there is no holonomic approximation which stays close to the horizontal planes. However, moving along the perturbed red curve, we can find a closer approximation which is holonomic.
This leads us to the precise theorem:
Holonomic approximation theorem: Let be a polyhedrong of codimension at least 1 and suppose we have a section of the jet bundle defined over a neighborhood of A. Then for any , there is a small isotopy of A (measured in the topology), and a holonomic section defined in a smaller neighborhood of which is close to the chosen section.
Suppose we have an almost contact structure on the open manifold V of dimension n. In order to use this theorem to prove Gromov’s theorem, we must identify a good codimension 1 subset of our open manifold V, where we can use holonomic approximation to find a genuine contact structure on a neighborhood of this subset which is close to the almost contact structure we are considering. Choose a triangulation of V, and for each top dimensional simplex, choose a path from the barycenter of that simplex out to infinity which avoids the barycenters of other simplices. The parts of the 2-skeleton which do not intersect these paths form a codimension 1 subcomplex S. The entire manifold smoothly deformation retracts onto arbitrarily small neighborhoods of S.
Now apply the holonomic approximation theorem to the pair (which corresponds to a section of ) along S. Then on a tiny perturbation of S, there is an actual holonomic section corresponding to which is very close to . By choosing our sufficiently small so that and are sufficiently close, we can ensure that the straight line homotopy between them stays in the space of almost contact structures (since the almost contact condition is an open condition (). Therefore the holonomic approximation theorem implies we can homotope our almost contact structure to be contact on a neighborhood of the perturbed S.
Observe that if is the end of the deformation retraction which sends V into the neighborhood of S where the almost contact structure is now genuinely contact, then pulls back the almost contact structure on V to a genuine contact structure on V. The deformation retraction provides a homotopy between the almost contact structure which is contact on the neighborhood of S to the genuine contact structure coming from this pullback. Therefore concatenating the homotopy provided by the holonomic approximation theorem with the homotopy provided by the deformation retract, gives a homotopy from our original almost contact structure to an actual contact structure on V.