Real open books and real contact structures

Details Real open books and real contact structures Real open books and real contact structures

2012-02-27

arxiv.org/abs/1202.5928v1

Mathematics

Geometric Topology

23 pages, 10 figures

Scientific paper

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. We prove that every real 3-manifold admits a real open book -an open book compatible with the real structure. The proof also shows that every real Heegaard decomposition of a real 3-manifold comes from a real open book. This is not true in general in the non-real setting. We observe that there are infinitely many closed oriented real 3-manifolds which cannot have a maximal real Heegaard decomposition. Furthermore, we show that there is a Giroux correspondence in the existence of a real structure; namely we prove that there is a one to one correspondence between the real contact structures on a 3-manifold up to equivariant contact isotopy and the real open books up to positive real stabilization. Finally, we construct a real 3-manifold with a real open book which is the canonical boundary of a Lefschetz fibration but which cannot be filled by any real Lefschetz fibration.

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