Let $M,N\subset P$ be two manifolds such that dim($M$) + dim($N$) < dim($P$), suppose that $M$ is compact and $N$ is closed, is it true that there exists an isotopy $F$ of $M$ such that $F(M,1)\cap N =\emptyset$?
I already proved that we can suppose $M$ and $N$ transversal but I don't know what to do next, any help would be appreciated.
Yes, the Transversality Theorem says that you can take a (small) homotopy $F_t$ [with $F_0=\iota$, the inclusion map] of $M$ so that for most $t$ we have $F_t(M)$ transverse to $N$. By your dimension hypothesis, the only way we can have transversality is for the two submanifolds to be disjoint. If $M$ is compact, the set of embeddings is open in the $C^0$ topology, so our small homotopy will, in fact, be an isotopy.
