Volume 10, issue 1 (2010)

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Stable concordance of knots in $3$–manifolds

Rob Schneiderman

Algebraic & Geometric Topology 10 (2010) 373–432
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 ${S}^{2}×{S}^{2}$. 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.

Keywords
$3$–manifold, Arf invariant, concordance, link invariant, stable concordance, stable embedding, Whitney disk, Whitney tower
Primary: 57M27
Secondary: 57M99