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Contact topology in dimension greater than three. (English) Zbl 1033.57013

Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10–14, 2000. Volume II. Basel: Birkhäuser (ISBN 3-7643-6418-1/hbk; 3-7643-6419-X/set). Prog. Math. 202, 535-545 (2001).
This paper gives an overview of known methods for constructing contact structures on manifolds \(M^{2n-1}\) with \(2n-1 >3\). It starts with the discussion of classical examples of contact structures coming from Liouville vector fields; contactizations of symplectic manifolds with an \(S^1\)-fiber due to Boothby and Wang; and contact structures on Brieskorn manifolds. The main part of the paper discusses various contact surgery techniques that started with the work of Ya. Eliashberg [Int. J. Math. 1, No.1, 29-46 (1990; Zbl 0699.58002)].
Contact surgeries are special surgeries in \((M^{2n-1}, \xi)\) done on spheres of dimension \(\leq (n-1)\) tangent to \(\xi\). Using contact surgeries the author was able to obtain an essentially complete solution to the question of existence of contact structures on \((n-1)\)-connected \(M^{2n-1}\), see H. Geiges [Topology 36, 1193–1220 (1997; Zbl 0912.57019)] and his earlier works.
Then the author discusses relations of contact surgery to spin bordisms outlined in his joint work with C. B. Thomas [Math. Ann. 320, 685–708 (2001; Zbl 0983.57017)]. In the last section the author briefly discusses the fibre connected sum, branched cover, contact reduction and heat flow constructions of contact structures.
For the entire collection see [Zbl 0972.00032].

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D35 Global theory of symplectic and contact manifolds
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