Automorphisms of the 3-sphere preserving a genus two Heegaard splitting. (English) Zbl 1095.57017

Goeritz found a finite set of generators for the group \(\mathcal H\) of isotopy classes of orientation preserving automorphisms of \(S^3\) that leave a genus two Heegaard decomposition invariant [see L. Goeritz, Abh. Math Sem. Univ. Hamburg 9, 244–259 (1933; Zbl 0007.08102)]. This result was generalized to Heegard splittings of any genus by J. Powell [Trans. Am. Math. Soc. 257, No. 1, 193–216 (1980; Zbl 0445.57008)].
The author explains that Powell’s proof contains a gap. He provides a modern proof of Goeritz’ result, in the hope that it may be helpful for studying the case of higher genus Heegaard splittings. The proof is based on considering a natural \(2\)-complex on which \(\mathcal H\) acts transitively.


57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
57N10 Topology of general \(3\)-manifolds (MSC2010)