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Surgery on contact 3-manifolds and Stein surfaces. (English) Zbl 1067.57024

Bolyai Society Mathematical Studies 13. Berlin: Springer. Budapest: János Bolyai Mathematical Society (ISBN 3-540-22944-2/hbk). 281 p. (2004).
The authors give a self-contained introduction to the recent exciting development in contact 3-topology. There are a couple of groundbreaking results in the near past. The first one is Giroux’s correspondence between contact structures and open book decompositions. Roughly speaking, an open book decomposition of a 3-manifold determines one and only one compatible contact structure (up to isotopic equivalence), and vice versa. Another one is the study of so called Lefschetz fibrations on Stein domains (which are bounded by contact 3-manifolds). On the other hand, we can also perform surgeries on contact 3-manifolds. All these are related. Let \(L\) denote a Legendrian link in the standard contact \(S^3\). Let \((Y,\xi)\) be the contact 3-manifold resulting from contact (\(\pm 1\))- surgery on \(L\). The surgery diagram for \((Y,\xi)\) gives rise to an (achiral) Lefschetz fibration on a 4-manifold \(X\) with boundary \(Y\), and hence an open book decomposition on \(\partial X=Y\). According to Giroux, there is a compatible contact structure associated to this open book decomposition, denoted by \(\xi_L\). The theorem says that \(\xi\) and \(\xi_L\) are isotopic on \(Y\). Along the way to prove this result, we need Giroux’s convex surface theory (analysis of the dividing set) applied to a Heegaard decomposition (Torisu’s result). The book also includes a chapter giving a synopsis of Heegaard-Floer theory developed by Ozsváth and Szabó.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57N10 Topology of general \(3\)-manifolds (MSC2010)
53D10 Contact manifolds (general theory)
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
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