manifold bounded by compact manifold with $b_1=0$

Question

Let $X$ be a non-compact manifold without boundary. Suppose that $b_1(X)=0$. Suppose $Y$ is a codimension zero compact submanifold with corner.

Q Can we find a compact submanifold $Z$ with smooth boundary such that $Y\subset Z\subset X$ and $b_1(Z)=0$.