Steenrod's problem asks wheter a simplicial homology class of a topological space $x$, $$ x\in H_n(X, \mathbb{Z})$$ can be represented by a triangulation of an $n$-dimensional, closed and oriented manifold.
It is known that if $n\leq 6$, then all classes can be realized by smooth manifolds, and that this does not hold in higher degrees.
Moreover, in dimensions $2$ and $1$, the circle and the orientable surfaces catch all homology classes.
Is there out there a list of orientable smooth 3,4, 5 and 6- Manifolds which generate the homology groups in the relevant degrees? What about rational coefficients?
