Stable isotopy in four dimensions

Abstract

We construct infinite families of topologically isotopic, but smoothly distinct knotted spheres, in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with S 2 ×S 2 , and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all of whose corresponding proper sublinks are smoothly isotopic, that become smoothly isotopic after stabilizing.

Knotted surfaces in 4-manifolds and stabilizations

“…In [3], Auckly, Kim, Melvin and Ruberman construct exotic surfaces through a method that does not follow the pattern above, i.e. their knotted surfaces do not arise via a 3-dimensional surgery crossed with S 1 .…”

“…[3,10,7,8,9,14,15,17,21,22,23,24,26]. Notably, the phenomenon of infinite exotic knottings is unique to dimension 4.…”

“…For any exotic pair of almost completely decomposable X i , inclusions of CP 1 ⊂ CP 2 into the standard manifold X = X i #CP 2 make up a pair of exotically knotted 2-spheres Σ i in X. Recently in [3], Auckly, Kim, Melvin and Ruberman explored this recipe (also see Akbulut's note [1]) to produce exotically knotted 2-spheres Σ i which become smoothly isotopic in the 4-manifold…”

Positive scalar curvature and higher‐dimensional families of Seiberg–Witten equations

“…One therefore needs another technique to attack Problem . Although it is, of course, a difficult problem to show some vanishing/non‐vanishing result for homotopy groups of in general, the author expects that some combination of the invariant defined in this paper and ideas given in Auckly–Kim–Melvin–Ruberman provides a way to approach it.…”

“…Proof. For two families ϕ • and ϕ • , we can take a generic map 1] since π n (Π) is trivial. Here F n−1 0,j and F n−1 1,j are the facets of [0, 1] n obtained by putting k = n − 1 in (1) and (2).…”