This is a difficult topic. I shall use $X$ to denote the whole space.
For example let $X=\mathbb{R}^{3}$ and $A\cong\mathbb{S}^{1}$, then the homology of $X-A$ is equal to $\mathbb{Z}$ if and only if $A$ is an unknot. But the proof is not easy. In general you require an isotopy that can move the $l$-th dimensional submanifold to a general position such that shrinking it to a point would not interfere the hole created by $A$. I think you need a strong embedding theorem to achieve that.
This is not the same as transversality where two manifolds have intersections in the general position. For a counter-example, let $X=\mathbb{R}^{3}$ and $A=\mathbb{R}^{2}\times {0}$. Then any submanifold lying in one of the half spaces must be contractible. But we have $\dim A+\dim D^{3}=5$ in this case, where $D^{3}$ is in the upper-half space.