Generating the genus \(g+1\) Goeritz group of a genus \(g\) handlebody. (English) Zbl 1288.57014

Scharlemann, Martin

Hodgson, Craig D. (ed.) et al., Geometry and topology down under. A conference in honour of Hyam Rubinstein, Melbourne, Australia, July 11–22, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8480-5/pbk). Contemporary Mathematics 597, 347-369 (2013).

For the \(3\)-sphere, the genus \(g\) Goeritz group consists of the isotopy classes of orientation-preserving homeomorphisms of the \(3\)-sphere that leave the genus \(g\) Heegaard splitting invariant. When \(g=2\), its finite presentation is known by E. Akbas [Pac. J. Math. 236, No. 2, 201–222 (2008; Zbl 1157.57002)]. For higher genus cases, J. Powell [Trans. Am. Math. Soc. 257, 193–216 (1980; Zbl 0445.57008)] gave a set of generators, but his proof contained a gap, pointed out by M. Scharlemann [Bol. Soc. Mat. Mex., III. Ser. 10, 503–514 (2004; Zbl 1095.57017)].
The purpose of the paper under review is to give a finite set of generators of the genus \(g+1\) Goeritz group \(G(H,\Sigma)\) of a genus \(g\;(\geq 1)\) handlebody \(H\). Here, \(\Sigma\) is a genus \(g+1\) Heegaard surface of \(H\). Indeed, a concrete set of \(4g+1\) generators is described. If \(g\geq 2\), then \(G(H,\Sigma)\) is shown to be isomorphic to the fundamental group of the space \(\mathrm{Unk}(I,H)\) of unknotted arcs in \(H\). When \(g=1\), there is a surjection from \(\mathrm{Unk}(I,H)\) to \(G(H,\Sigma)\). The author exhibits two proofs, using classical techniques and thin position.
For the entire collection see [Zbl 1272.57002].

Reviewer: Masakazu Teragaito (Hiroshima)

MSC:

57M40	Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Goeritz group; handlebody; thin position

Citations:

Zbl 1157.57002; Zbl 0445.57008; Zbl 1095.57017