Smooth maps transverse to a foliation

2012-12-13T17:36:25Z

Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$ ($\dim S < \dim N$). Let $\mathfrak S$ be a foliation of $S$. We say that a map between $M$ and $N$ is transverse to $\mathfrak S$ if it is transverse to every leaf of $\mathfrak S$.

Now, suppose $f : M \rightarrow N$ is a smooth map transverse to a foliation $\mathfrak S$ of $S$ at a point $x \in M$ ($f(x) \in S$ and $f$ is transverse to the leaf of $\mathfrak S$ passing through $f(x)$ at the point $x \in M$). Is it possible to find a map $g : M \rightarrow N$, arbitrarily close to $f$ in the strong (Whitney) topology, with $g(x) = f(x)$ and such that $g$ is transverse to $\mathfrak S$ at all points of $M$?

Answer by Ryan Budney for Smooth maps transverse to a foliation

2012-12-13T18:42:19Z

I originally misunderstood your question. Here is a fairly simple example.

Let $N = \mathbb R^2$, $M=S^2$ and $S$ be the unit circle in $\mathbb R^2$ considered as foliated by its points -- the leaves are $0$-dimensional.

The map $M \to N$ will be a linear projection map. Let's project $S^2$ onto $\mathbb R^2$ in a way so that the image of $S^2$ is a disc in the plane, which half covers the unit circle.

So this map is transverse to the foliation at a point. But no nearby map is transverse to the foliation on all $M$. This is because the image of $M$ is a compact subset of the plane which does not contain the unit circle. So for a sufficiently small perturbation, you similarly do not cover the unit circle. Wherever the boundary of the image intersects the circle, you will not be transverse to the foliation of the circle.

Smooth maps transverse to a foliation - MathOverflow

Smooth maps transverse to a foliation

Answer by Ryan Budney for Smooth maps transverse to a foliation