I've seen questions like this on a lot of old qualifying exams in algebraic topology. The type of question is usually of the form:
Let $X$, $Y$ be some topological spaces. Prove that $X$ can't be embedded in $Y$.
The problem I have is that I don't see how homology or homotopy or really anything I know helps. For example, in the general case taking e.g. $S^1$ we can embed it into the disk $D^2$. However, the latter is contractible, so all homotopy and homology groups are zero. Thus, knowing the homotopy/homology groups for the larger space does not give us any information about whether or not we can embed something into it. This would suggest that homotopy and homology doesn't provide any general methods as the spaces in this particular example are pretty much as nice as anything can get.
The question is a somewhat soft question, but I'm curious about which tools in algebraic topology are usually used to tackle questions like this and what are some of the standard tricks involved?