We deal with some questions regarding concordance of knots in arbitrary closed $3$-manifolds. We first prove that, any non-trivial element in the fundamental group of a closed, oriented $3$-manifold gives rise to infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. In particular, we consider these distinct smooth almost-concordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PL-disk in the Mazur manifold. On the other hand, all the representatives we construct are topologically slice. We also prove that all knots in the free homotopy class of $S^1 \times pt$ in $S^1 \times S^2$ are smoothly concordant.
Read
