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. 



Subvarieties with different topology representing the same cycle
The answer to the first pair of questions is that there's no reason that such cycles should be homeomorphic. The question about algebraic subvarieties also has a negative answer, but presumably more restrictive questions might have more positive answers.


Let's stick to the case when the ambient space $X$ is an $n$-manifold, and let $d < n$ be the dimension for your homology class. Then you can simply take $Z$ to be the union of $Y$ with a $d$-dimensional sphere lying in a ball disjoint from $Y$. The number of components has changed so $Y$ and $Z$ are not homeomorphic; on the other hand the sphere is null-homologous so you haven't changed the homology class.


The other question depends on context. If you really mean subvariety, then you presumably allow some singularities; there are many examples where a non-singular curve can degenerate into a singular curve; this doesn't change the homology class. A simple example would be in $CP^2$, where a smooth curve of degree 2 (thus homeomorphic to a 2-sphere) can degenerate into a union of two lines (two 2-spheres meeting at a point). If you intended that your subvarieties be smooth, then this is more subtle and perhaps someone more knowledgeable about Chow rings can answer your question. 


