Do the Betti numbers determine the topology?

Question

Do the Betti numbers determine the topology? I.e., if we are given a list of the Betti numbers of a topological space, then are we able to find the topology? What about in the smooth manifold case? Or in the compact case?

Answer 1

In general topological spaces this does not work, consider $$X_n = \mathbb R^n \setminus \mathbb R^{n-2}.$$ Then the homotopy type of all the $X_n$ is identical, so they have the same Betti numbers (but are not homeomorphic). So a better question might be whether they determin the homotopy type. The answer is still no: consider the space $$\mathbb R \mathbb P^3 = S^3/\pm1.$$ This space falls in all of your categories, has the same Betti numbers as $S^3$, but is not homeomorphic to $S^3$ (it has nontrivial but finite fundamental group).