Topology on the space of foliations

Question

Let $(M^3,g)$ be a closed Riemannian manifold. Is there a “natural” topology on the space $\operatorname{Fol}(M)$ of smooth codimension $1$ foliations on $M$? Is there any other relevant structure on this set?

Answer 1

A smooth codimension 1 foliation (in any dimension) is the same thing as an integrable codimension 1 distribution (tangent spaces to the leaves of the foliation). The space of smooth distributions is the space of smooth sections of the projectivization of the cotangent bundle of the manifold. If you have any smooth bundle $E\to M$ over any manifold, then the space of sections $\Gamma(M,E)$, has a family of topologies depending on how much smoothness you want to keep track of. In your case, for instance, if your foliation is of class $C^1$, the distribution will be of class $C^0$, hence, the natural topology to use on the space of continuous sections of $P(T^*M)$ is the compact-open topology (or topology of uniform convergence since $M$ is compact); then restrict this topology to the subspace topology on the subspace of integrable distributions. The Riemannian metric is mostly irrelevant here, but you can use it to identify $TM$ and $T^*M$ if you like.

Answer 2

As mentioned in the answer by Moishe, a possibility to give a topology to the set of regular (the dimension of the leaves is constant) $C^r$-foliations on a manifold $M$ is to consider them as $C^r$-continuous sections of the grassmannian bundle $\mathrm{Gr}(M)$, and then set one of the weak or strong $C^r$-topology on the space of sections $\Gamma^r(M, \mathrm{Gr}(M))$.

But this approach does not capture the dynamics of the foliation around a fixed leaf, i.e. the holonomy, due to the fact that you are loosing one degree of differentiability.

David B. Epstein in A topology for the space of foliations, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), pp. 132–150 Lecture Notes in Math., Vol. 597, introduced an alternative topology for foliations on a smooth manifold. (see section 3 in On $C^0$-stability of compact leaves with amenable fundamental group).

Fix $\mathcal{F}$ a $C^r$-foliation on an n-dimensional manifold $M$, and consider $\{U_i,\phi_i,K_i\}_{i\in I}$, where $\{U\}_{i\in I}$ is an open cover of $M$, homeomorphisms $\phi_i\colon U_i\to V_i\subset\mathbb{R}^n$ such that the connected components of $\mathcal{F}$ in $U_i$ correspond to the preimages under $\phi_i$ of $\mathbb{R}^k\times \{y\}$ and the composition $\phi_i\circ \phi_j^{-1}$ is $C^r$-continuous (i.e. a $C^r$-chart), and $K_i\subset U_i$ is compact with $\phi_i(K_i)$ an $n$-cube in $\mathbb{R}^n$ whose sides are parallel to the axes and $\{K_i\}_{i\in I}$ is a locally finite cover of $M$.

A $\delta$-neighborhood of $\mathcal{F}$ consist of the foliations $\mathcal{F}'$ of the same codimension as $\mathcal{F}$ such that for each $i\in I$, there exists a local chart $\phi_i'\colon U_i'\to V_i'\subset \mathbb{R}^n$ of $\mathcal{F}'$ for which there exists $K'_i\subset U_i'$ compact with $\phi_i'(K_i')$ homeomorphic to an $n$-cube with sides parallel to the axes, $K_i\subset K_i'$, and $\phi_i'\circ\phi_i^{-1}$ is $\delta$-close to $\mathrm{Id}_{\mathbb{R}^n}$ over $\phi_i(K_i)$ in the $C^r$-topology.

With this definition the tangent distributions are getting close (w.r.t. the weak topology) as sections of the grassmannian, but also the integrable submanifolds of the distributions (i.e. the leaves) should be getting close to each other in local charts.