Journal article
Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers
Y Rieck, JH Rubinstein
Communications in Analysis and Geometry | INT PRESS BOSTON, INC | Published : 2009
Abstract
Let M be a 3-manifold admitting a strongly irreducible Heegaard surface Ε and f : M → M an involution.We construct an invariant Heegaard surface for M of genus at most 8g(Ε) - 7. As a consequence, given a (possibly branched) double cover π : M → N we obtain the following bound on the Heegaard genus of N: g(N) ≤ 4g(Ε) - 3. We also get a bound on the complexity of the branch set in terms of g(Ε). If we assume that M is non-Haken, by Casson and Gordon [3] we may replace g(Ε) by g(M) in all the statements above.
Grants
Awarded by JSPS
Funding Acknowledgements
Y.R. was supported in part by JSPS grant P00024 and The 21st Century COE Program "Constitution of wide-angle mathematical basis focused on knots" (Project LeaderAkio Kawauchi).We are very grateful to Tsuyoshi Kobayashi and Marc Lackenby for many helpful conversations and the anonymous referee for her/his comments. We thank Sean Bowman for the illustrations.Parts of Y.R.'s work was carried out when he was a JSPS post doctoral fellow of Tsuyoshi Kobayashi in Nara Women's University, and when he was visiting Akio Kawauchi in Osaka City University as a part of his 21st Century COE Program "Constitution of wide-angle mathematical basis focused on knots". He is grateful to both, and the math departments of Nara Women's University and Osaka City University for their warm hospitality.