This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that $\dim(M)+\dim(N)<\dim(X)$. Since there are enough dimensions to "move around," we can pull $M$ off of $N$ with a very small perturbation.
Is there a rigourous proof of this concept? Let me try to state my question more formally.
With $M$ and $N$ as above, does there always exist a deformation $\iota_t(M)$ of $M$ such that $\iota_1(M)\cap N=\emptyset$, but $|m-\iota_1(m)|\leq\epsilon$ for all $m\in M$ where $\epsilon>0$ is arbitrary.
(Please let me know if my question doesn't make sense as posed. Thanks.)
Yes, this follows from the transversality theorem, as long as you are working with smooth manifolds. This is exercise 5 of chapter 2, section 3 of Guillemin and Pollack. The basic idea is that we can do a small homotopy to make any two submanifolds of an ambient manifold intersect transversely, but if $\dim X+ \dim Y < \dim Z$, the only way for them to intersect transversely is for them to not intersect at all.