Unknotting nonorientable surfaces of genus 4 and 5 (Q6618715)

A closed, nonorientable, locally flatly embedded surfaces in the \(4\)-sphere is said to be topologically unknotted if it is topologically isotopic to a connected sum of standardly embedded projective planes. Every unknotted surface \(F\) in \(S^4\) has knot group \(\pi_1(S^4\setminus F)\cong\mathbb{Z}/2\). Surfaces with knot group \(\mathbb{Z}/2\) are called \(\mathbb{Z}/2\)-surfaces. The main question studied in this article is whether every \(\mathbb{Z}/2\)-surface is topologically unknotted.\N\NPrevious results on this question include the following. In 1984, [\textit{T. Lawson}, Math. Ann. 267, 439--448 (1984; Zbl 0525.57026)] proved that every projective plane with knot group \(\mathbb{Z}/2\) is unknotted. In 1988, [\textit{S. Finashin} et al., Lect. Notes Math. 1346, 157--198 (1988; Zbl 0639.00032)] proved that for fixed nonorientable genus \(h\) and normal Euler number \(e\), there are only finitely many topological isotopy types of smoothly embedded \(\mathbb{Z}/2\)-surfaces. They also produced infinitely many smoothly distinct, smoothly embedded \(\mathbb{Z}/2\)-surfaces with \((h,e)=(10,16)\). Later, [\textit{M. Kreck}, Lond. Math. Soc. Lect. Note Ser. 150, 63--72 (1990; Zbl 0837.57024)] showed that these are all topologically unknotted. Actually he showed that every smoothly embedded \(\mathbb{Z}/2\)-surface with \((h, e)\in\{(10, 16),(2,0)\}\) is topologically unknotted. [\textit{W. S. Massey}, Pac. J. Math. 31, 143--156 (1969; Zbl 0198.56701)] showed that for every locally flat embedding of a closed, nonorientable surface in \(S^4\) of nonorientable genus \(h\), the normal Euler number must lie in the range \(\{-2h,-2h+4,\ldots,2h-4,2h\}\). Recently, [\textit{A. Conway} et al., ``Unknotting nonorientable surfaces'', Preprint, \url{arXiv:2306.12305}] showed that every locally flat embedding with non-extremal normal Euler number, i.e.\ \(|e|\neq 2h\), is topologically unknotted. They also show the same for every locally flat embedding with \(h\leq 3\).\N\NIn this article, the author extends the latter bound to \(h\leq 5\), that is, he shows that every \(\mathbb{Z}/2\)-surface of genus at most \(5\) is topologically unknotted. He also considers \(\mathbb{Z}/2\)-surfaces in the \(4\)-ball with boundary a knot in the \(3\)-sphere. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. It is also shown that this methods fails when \(h=6,7\).