# Steenrod Problem and realization of rational homology classes by manifolds

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?

• This would be a long list, since for any closed oriented $n$-manifold $X$ the list would have to contain a closed oriented $n$-manifold $M$ with a degree $1$ map $f : M \to X$, and then $f_* : H_*(M) \to H_*(X)$ would have to be split surjective. Feb 13, 2017 at 15:39