Stable concordance of knots in $3$–manifolds

Abstract

Knots and links in $3$–manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato–Levine invariants and Milnor’s triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable $3$–manifolds, stable knot concordance in products of an orientable surface with the circle and stable link concordance for many links of null-homotopic knots in orientable $3$–manifolds.