Two smooth surfaces \(S\) and \(S'\) in a smooth \(4\)-manifold \(X\) are an exotic pair if \(S\) and \(S'\) are topologically, but not smoothly, isotopic. The main result in the paper under review is that there are infinitely many knots in \(\mathbb{S}^3\) each bounding countably many properly embedded, compact, orientable, smooth surfaces in the \(4\)-dimensional disc \(\mathbb{D}^4\) which are pairwise topologically isotopic but which cannot be sent one into the other via a diffeomorphism of \(\mathbb{D}^4\). In particular, one obtains infinitely many exotic pairs of properly embedded, smooth, orientable surfaces in \(\mathbb{D}^4\).
The construction of these surfaces is based on \(1\)-twist rim surgery. This technique was introduced by H. J. Kim [Geom. Topol. 10, 27–56 (2006; Zbl 1104.57018)] to produce compact oriented surfaces in \(\mathbb{CP}^2\) which are smoothly knotted, but topologically unknotted. Roughly speaking, twist rim surgery combines Zeeman twist-spinning construction of \(2\)-knots [E. C. Zeeman, Trans. Am. Math. Soc. 115, 471–495 (1965; Zbl 0134.42902)] and R. Fintushel and R. J. Stern’s rim surgery [Math. Res. Lett. 4, No. 6, 907–914 (1997; Zbl 0894.57014)]. Twist rim surgery allows the author to produce surfaces which are topologically isotopic, potentially not smoothly isotopic, without conditions on the fundamental group of their complement.
To obstruct two surfaces being sent one to the other via a diffeomorphism of \(\mathbb{D}^4\), the authors introduce a numerical invariant of properly embedded surfaces \[ \Omega(S)\in \mathbb{Z} \cup \{ -\infty \},\quad S\subset \mathbb{D}^4. \] This invariant is defined in terms of induced maps in Heegaard-Floer homology. The invariant \(\Omega\) behaves in a controlled way under twist rim surgery, and vanishes for quasi-positive surfaces pushed in the \(4\)-disk.
A key point in the proof of the main theorem is the non-vanishing of certain maps in Heegaard-Floer homology. To this end the authors prove that the (transverse version of the) LOSS invariant (see [P. Lisca et al., J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307–1363 (2009; Zbl 1232.57017)] and [J. A. Baldwin et al., Geom. Topol. 17, No. 2, 925–974 (2013; Zbl 1285.57005)]) is preserved under the maps induced by some link cobordisms which are ascending surfaces in Weinstein manifolds. This is an interesting result on its own, and fits into a family of similar results due to various authors, e.g. P. Ozsváth and Z. Szabó [Duke Math. J. 129, No. 1, 39–61 (2005; Zbl 1083.57042)], A. Juhász [Adv. Math. 299, 940–1038 (2016; Zbl 1358.57021)], J. A. Baldwin and S. Sivek [J. Symplectic Geom. 16, No. 4, 959–1000 (2018; Zbl 1411.57019); Geom. Topol. 25, No. 3, 1087–1164 (2021; Zbl 1479.53080)].