Separating two submanifolds [closed]

Question

Let $M, N, X$ are compact manifolds. Let $f_1:M \rightarrow X$ and $g_1: N \rightarrow X$ be any two embeddings. Is it always possible to find embeddings $f_2 $ homotopic to $f_1$ and $g_2$ homotopic to $g_1$ such that images of $f_2$ and $g_2$ are disjoint? I think not, but can't find a counter example. Is there sufficient conditions for this to be possible?

Answer 1

Homology might be a helpful word. Point x circle, circle x point, in the 2-torus.

Answer 2

No, but if the codimension of the two submanifolds is high enough, then it is possible. Generically the intersection $A$ will be a submanifold of dimension $\dim(A)=\dim M+\dim N-\dim X$. Hence if the dimension of $X$ is big enough the intersection will be empty.