Non-diffeomorphic but homeomorphic knottings of surfaces in the 4-sphere

Knotted surfaces in 4-manifolds and stabilizations

“…[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.…”

“…When double branched covers along exotically knotted Σ i in X result in an exotic pair of 4-manifolds X i (as it is the way to argue that they are smoothly knotted in many examples in the literature, e.g. [9,10]) there is a direct connection. In this case, our stabilization of the embedded surfaces amounts to taking relative connect sum of (X, Σ i ) with (S 4 , T 2 ), the standard unknotted embedding of T 2 in S 4 , so the double cover along the stabilized surface then gives a connected sum of X i with S 2 × S 2 , (which is the double branched cover of S 4 along the unknotted T 2 ).…”

“…1 We will examine the analogous question for surface stabilizations: how many are needed to make exotic knottings smoothly isotopic? We analyze in detail various constructions of exotically knotted and exotically embedded surfaces 2 , which produced non-orientable surfaces in S 4 in the pioneering works of Finashin, Kreck and Viro [10] and Finashin [9], and orientable surfaces (often topologically isotopic to symplectic/complex curves) in many other 4-manifolds in the works of Fintushel and Stern [14,15], Kim and Ruberman [21,22,23,24], Finashin [7,8], Mark [26], Hoffman and Sunukjian [17].…”

Smooth structures on complex surfaces with fundamental group $\mathbf {Z}_2$

“…It is interesting to examine when the quotients are also diffeomorphic to each other. Compare with Finashin-Kreck-Viro [6], where they find non-diffeomorphic knottings from different anti-holomorphic involutions on (different) Dolgachev surfaces although the quotient manifolds are all diffeomorphic to S 4 .…”

Scharlemann’s 4-manifolds and smooth 2-knots in 𝑆²×𝑆²

“…Proposition 2 implies that if a Scharlemann's manifold Xa is not diffeomorphic to S3 x SX\]S2 x S2, then the 2-knot S in Proposition 2 will give an exotic knotting of the trivial 2-knot in S2 x S2 . Any example of exotic knottings of S2 into a 4-manifold has not been known yet, but some examples of exotic knottings of 2-disc or nonorientable surfaces into a 4-manifold are known [5,26].…”