Let, $X$ be a compact submanifold of $Y$ and let $X$ intersect another submanifold $Z$. Assume that dim$X$ + dim$Z$ $<$ dim$Y$. Given $\epsilon> 0$ show that there exists a deformation $X_{t}=i_{t}(X)$ such that $X_{1}$ does not intersect Z and that $\mid x - i_{1}(x)\mid < \epsilon $ for all $x \in X $.
This is a problem from Guillemin and Pollack's Differential Topology book. Can anyone give me any suggestions on how to approach this problem?
This is an immediate application of the Transversality Theorem. Because of the dimension hypothesis, the only way $i_t$ can be transverse to $Z$ is for the image to be disjoint from $Z$. Do this argument first when $Y=S=\Bbb R^N$ and let $i_s(x)=x+s$ for $s\in B(0,\epsilon)\subset\Bbb R^N$. Once you've written that out, go back to the case $Y\subset\Bbb R^N$ and use the $\epsilon$-neighborhood theorem as with a lot of the proofs in that section of the book.