Let $M$ be an $n$-dimensional manifold and let $[M]$ denote the unoriented bordism class of $M$. Forming the usual commutative graded ring $\text{MO}_n$ we know that $\text{MO}_* \simeq \mathbb{Z}_2[u_n : n \neq 2^r -1, \deg u_n = n]$ as graded rings.
Writing out the first groups we have that $\text{MO}_0 = \mathbb{Z}_2$, $\text{MO}_1 = 0$, $\text{MO}_2 = \mathbb{Z}_2$ (generated by $[\mathbb{R}P^2]$), $\text{MO}_3 = 0$ and so on.
From this I can see that $\mathbb{R}P^2$ is not the boundary of some $3$-dimensional compact manifold. But is it true that any closed 3-dimensional manifold is the boundary of some 4-dimensional compact manifold? That is, does being cobordant to a manifold which bounds imply that the manifold itself bounds?