Generating the Goeritz group of \(S^3\) (Q6601494)

Let \(M = A \cup_{T}B\) denote a Heegard splitting by \(T\) of the \(3\)-manifold \(M\). The Goeritz group \(G(M,T)\) of \(M\) is the group of isotopy classes of autodiffeomorphisms of the pair \((M,T)\) for which the induced diffeomorphism on \(M\) is isotopic to the identity. In this paper an element of this group is viewed as the final configuration of a loop of embeddings of \(T\) in \(M = S^3\). Thus the Goeritz group may be viewed here as the fundamental group of a space of ``configurations'' of a genus \(g\) Heegard surface in \(S^3\), a sort of higher dimensional braid group. \N\N\textit{L. Goeritz} [Abh. Math. Semin. Univ. Hamb. 9, 244--259 (1933; Zbl 0007.08102)], discovered a finite set of generators for the group in the case \(M = S^3\) and the genus of \(T\) is \(2\). A finite set (\(5\)) of generators for \(M = S^3\) and all genera was proposed by \textit{J. Powell} [Trans. Am. Math. Soc. 257, 193--216 (1980; Zbl 0445.57008)], and proved recently for genus \(3\) in [``Powell moves and the Goeritz group'', Preprint, \url{arXiv:1804.05909}] by \textit{M. Freedman} and \textit{M. Scharlemann}. The present paper completes the picture for \(S^3\) by showing elements of a certain type, first defined in the Freedman and Scharlemann paper (and called eyeglass twists), and \(4\) of Powell's generators (and their topological conjugates) generate the entire Goeritz group of \(S^3\) for all genera. \N\NThe result leans heavily on his earlier work with Freedmann, and to facilitate his explanation, the author introduces a good many (23) ad hoc definitions and an index to assist the reader. The exposition is also considerably aided by 40 or so figures, some of which have appeared before. The length of the paper (127 pages) and its complexity (the proof contains some 45 cases and subcases) are perhaps the result of three things going on at once: First there is a family of diffeomorphisms parameterized by the circle and arising from an element of the Goeritz group, second there is a sequence of pairs of ``reducing discs'' forming \(2\)-handles on modifications of \(T\), and third there is a Morse function with level sets \(2\)-spheres sweeping out \(S^3\). This state of affairs requires herding the changing (and disappearing) pieces of modifications of \(T\) and their complements (``compartments'') which are utilized to apply induction on genus. These compartments are sets and subsets of increasing complexity. The author has bravely confronted and managed these difficulties to obtain the result he desired.