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Contact surgery and symplectic handlebodies. (English) Zbl 0737.57012

C. Meckert [Ann. Inst. Fourier 32, No. 3, 251-260 (1982; Zbl 0471.58001)] has shown that the connected sum of two contact manifolds has a contact structure. The author of the paper extends Meckert’s result as follows. Let \(Y\) be any isotropic sphere in a contact manifold \(X\), with a trivialization of the conformal symplectic normal bundle of \(Y\) (the conformal symplectic normal bundle is a direct summand of the ordinary normal bundle of \(Y\) in \(X\)). Then the manifold \(X'\) obtained from \(X\) by elementary surgery along \(Y\) (using the framing induced by the trivialization) has a contact structure. Moreover, \(X\) and \(X'\) both form the boundary of an elementary cobordism \(P\) having a symplectic structure together with a conformally symplectic vector field which is transverse to the boundary and which actually produces the original contact structure on \(X\) and the contact structure on \(X'\).

MSC:

57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R25 Vector fields, frame fields in differential topology
57R65 Surgery and handlebodies
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