I was discussing with a friend about my very basic understanding of topology that it was "basically about holes" and she mentioned to me that the notion of holes was more complicated in higher dimensions. For $2$-dimensional surfaces in $3$-dimensional space we have the idea of genus that identifies a surface uniquely by the number of holes it has, but it occurred to me that in higher dimensions that this might not hold for higher dimensional surfaces and holes.
In four dimensions, for example, is there a counterexample where the Betti numbers are the same for two topological spaces but they are not homeomorphic?
I apologise if the question is not precise, but this is not my area of expertise.