Heegaard splittings and open books

Abstract

We show that if the monodromy of an open book decomposition has sufficiently high displacement distance, acting on the loop and arc complex for a page, then it is the unique minimal Euler characteristic open book for the manifold. In particular, we show that such an open book induces the unique (up to isotopy) minimal genus Heegaard surface for the manifold, and that this Heegaard surface has cyclic mapping class group.

25 pages, 4 figures

References in corpus (3)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Calculating isotopy classes of Heegaard splittings
• An upper bound on common stabilizations of Heegaard splittings

Cited by in corpus (7)

• One-sided and two-sided Heegaard splittings
• Mapping class groups of Heegaard splittings of surface bundles
• Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds
• The arc complex and contact geometry: non-destabilizable planar open book decompositions of the tight contact 3-sphere
• A Distance for Circular Heegaard Splittings
• Thick isotopy property and the mapping class groups of Heegaard splittings
• Twisted book decompositions and the Goeritz groups