  We investigate the disparity between smooth and topological almost
concordance of knots in general 3-manifolds Y. Almost concordance is defined by
considering knots in Y modulo concordance in Yx[0,1] and the action of the
concordance group of knots in the 3-sphere that ties in local knots. We prove
that the trivial free homotopy class in every 3-manifold other than the
3-sphere contains an infinite family of knots, all topologically concordant,
but not smoothly almost concordant to one another. Then, in every lens space
and for every free homotopy class, we find a pair of topologically concordant
but not smoothly almost concordant knots. Finally, as a topological
counterpoint to these results, we show that in every lens space every free
homotopy class contains infinitely many topological almost concordance classes.
