Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in homology, then $Y$ and $Z$ have the same topology?
To be a bit more restrictive. Let $X$ be a smooth manifold. Is it always the case that two submanifolds $Y$ and $Z$ representing the same homology (nonzero) cycle must be homeomorphic? Is there some "natural" hypothesis to put on $X$ to make this true?
Let $X$ be a smooth projective variety over the complex numbers. Is it always the case that two algebraic subvarieties $Y$ and $Z$ representing the same (nonzero) cycle (in Chow, in some cohomology theory) must be homeomorphic?
I have the impression that my questions are a bit naive as formulated, so I would also be happy to hear if there is a good context/theory to formulate a nice question along these lines.