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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.

##### MSC:
 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)
##### Keywords:
Heegaard splitting; reducing sphere; curve complex