Surjections between homology groups

Question

When does there exist a continuous surjection between two cell complexes $X$ and $Y$ such that $H_*(X)$ is not isomorphic to $H_*(Y)$. What properties must be satisfied?

Answer 1

It is actually somewhat rare for there to not exist a continuous surjection between two arbitrary CW-complexes, and the only real obstructions are connectedness and compactness, not with homology as you seem to think. In particular, for instance, if $Y$ is any finite connected CW-complex, then there exists a continuous surjection $[0,1]\to Y$ by the Hahn-Mazurkiewicz theorem. It follows that if $X$ is any CW-complex of positive dimension, there exists a continuous surjection $X\to Y$ (since we can compose with a continuous surjection $X\to [0,1]$).

It follows easily, for instance, that if $X$ and $Y$ are both finite CW-complexes, there exists a continuous surjection $X\to Y$ iff $X$ has at least as many connected components as $Y$ and $X$ has at least as many non-singleton connected components as $Y$.