Possible homology of closed manifolds.

 2008-03-09 01:10:21 +0000 questions mathematics

In this fun paper,

it is pointed out that and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold?

It isn't very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify $(n-1)$-connected $2n$-manifolds).

Anyway, I was wondering: is this realization question solvable for homology with coefficients in $\Z/2\Z$ or $\Q$?