I am trying to have a mental picture of topological spaces like $S^3, T^3$ etc. I know about fact that the topological invariants are the essential tools to do this. But the topological invariants like homology, cohomology, etc. are not easily imaginable. I would rather identify the nature of a topological space by simple parameters like number of holes(I can imagine holes of different dimensions, say, a two dimensional hole is a disc removed from a 2d plane, a three dimensional hole is like a water bubble in a pond of water etc.)
I understand it would be difficult to uniquely identify a space using the number of holes alone, but I don't want a unique identification, but rather identifying from at least usual kind of spaces. Say I want to differentiate a $S^3$ from $S^2$ in a sense that whether $S^3$ has any hole or not, if it has, what is the dimensionality of the hole, etc.