Suppose you have a closed oriented smooth manifold $X$. And two orientable submanifolds $S_{1},S_{2}$ with $\dim(S_{1}) + \dim(S_{2}) = \dim(X)$ intersecting transversally and with $[S_{1}] \cdot [S_{2}] = 0$. Suppose that we are interested in finding a cycle $C \in C_{\dim(S_{1})}(X)$, such that $[C]=[S_{1}] \in H_{\dim(S_{1})}(X,\mathbb{Z})$ and $C$ is disjoint from $S_{2}$.
I have heard cautionary tale that it is not always possible to find such a cycle. On the other hand, I have a vague recollection of an argument in geometric topology when we "bring together" (probably by some kind of homotopy of $S_{1}$) the +1 and -1 intersection points and they annihilate each other or something like that, although that was probably in some specific situation or is perhaps not applicable here.
I would be interested to know a counter-example, or some reference that addresses this in a reasonably general way. I wonder if this is a phenomina "linearises" in the sense algebraic topology for high dimensions i.e. is it possible to find such a $C$ if we assume that $\dim(X)>>0$ or $\dim(X)>> \dim(S_{1})$?