A surgery proof of Bing’s characterization of S
- M. Eisermann
- J. Knot Theory Ram
Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe’s n-splitting and Kobayashi’s k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S (resp. R). As a byproduct of the proof of the “if” part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M . We show that k-quasi-isotopy implies (k + 1)-cobordism of Cochran and Orr. It follows that each Cochran’s derived invariant β of PL links is invariant under sufficiently close C-approximation, and admits a (unique) extension, retaining this property, to all topological links. This strengthens a result of Milnor and overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who “became aware of impossibility to define” for wild links what for PL links is equivalent to the formal power series ∑ βz by a change of variable. Another corollary is that if z(c0 + c1z 2 + · · · + cnz) denotes the Conway polynomial of an m-component link, the residue class of ck modulo gcd(c0, . . . , ck−1) is invariant under k-quasi-isotopy.