Let $M$ be a closed smooth manifold of dimension $2n$ for some positive integer $n$. Let $\mathit{Diff}(M)$ be the group of diffeomorphisms of $M$. Let $L$ be a closed embedded $n$-dimensional submanifold of $M$.
Let $T(L)\subset \mathit{Diff}(M)$ be the subset of diffeomorphisms such that $f(L)$ is transverse to $L$. Let $I(L):T(L)\rightarrow \mathbb{Z}_{\geq 0}$ be the function defined as $I(L)(f)=\#(L\cap f(L))$.
What information about $L$ can one recover from $I(L)$? In particular, does there exist a compact closed subgroup $G\subset \mathit{Diff}(M)$ such that for any $L$ as above, the orbit of $L$ under $G$ can be recovered from $I(L)$?
I think that by looking at diffeomorphisms close to identity one should get an upper bound on the sum of Betti numbers (Morse theory).
P.S.: The question is inspired by integral geometry (e.g. I think that some variation of Funk transform should show that the answer is positive for even-dimensional spheres). But I got the impression that outside a certain very limited set of geometries (such as homogeneous spaces) integral geometry is useless. I wanted to clarify whether my impression was correct.
