In Analysis Situs, Poincaré studies the following question: which sequences of integers $b_0,\ldots,b_n$ are the Betti numbers of an orientable compact manifold of dimension $n$?.

He knows that necessary conditions are $b_k=b_{n-k}$ and if $n=4k+2$, $b_{2k+1}$ is even. Then he computes the homology of a product of spheres, reducing the problem to finding a manifold of dimension $4k$ with $b_0=b_{2k}=b_{4k}=1$ and the other are 0. He proposes the symetric product of two spheres $S^{2k}$, missing that they are singular for $k>1$.

For $n=1$ and $2$, the projective plane and the quaternionic plane answer the question and I was very surprised to learn that the Hirzebruch signature theorem implies that a manifold of dimension 12 with $b_4=0$ has signature divisible by 62 (see http://www.maths.ed.ac.uk/~aar/papers/hirzrem.pdf p.8). Hence, the smallest odd value of $b_6$ of a 12-dimensional manifold with vanishing other Betti numbers is at least 63.

Here is the question: do we know the smallest odd value of $b_6$? More generally, do we know other obstructions for realizing an arbitrary sequence of Betti numbers?