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 symmetric 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 page 8 of [this paper](http://www.maths.ed.ac.uk/~aar/papers/hirzrem.pdf)). 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?



Realization problem for Betti numbers
For $M^{4k}$, the intersection form on middle-dimensional homology $H\_{2k}$ is non-degenerate and symmetric, so the signature equals $b\_{2k} \pmod 2$. Therefore in your case the signature theorem implies $b\_6$ must be even if $b\_4=0$.


See also [Torsion in cohomology of smooth manifolds](https://mathoverflow.net/questions/116814/torsion-in-cohomology-of-smooth-manifolds)
for some related information concerning other restrictions.


