let $X$ and $Y$ be compact, oriented manifolds and assume that $\partial X=Y$. Is it true that the normal bundle of $Y$ in $X$ is trivial? if it is the case, is there a simply explaination?
Yes, the normal bundle $\nu$ of $\partial X$ in $X$ is always trivial. To see why this is true, note that $\nu$ is a $1$-dimensional vector bundle and the outward pointing normal vector field to $\partial X$ is a nowhere vanishing smooth section of $\nu$. Since a $1$-dimensional vector bundle with a nowhere vanishing section is trivial, $\nu$ is trivial.