Any homology sphere is stably parallelizable and hence nullcobordant. Rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}({\Omega^{\text{SO}}_{5}}) \cong \mathbb Z/2\mathbb Z$. This leads to the following question.

Which classes in $\Omega^{\text{SO}}_{\ast}$ can be represented by rational homology spheres?

Of course, any such class is torsion, as all its composite Pontryagin numbers, as well as its signature, vanish.

Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}({\Omega^{\text{SO}}_{5}}) \cong \mathbb Z/2\mathbb Z$. This motivates the following question.

Which classes in $\Omega^{\text{SO}}_{\ast}$ can be represented by rational homology spheres?

Of course, any such class is torsion, as all its composite Pontryagin numbers, as well as its signature, vanish.