Contact round surgery and symplectic round handlebodies (Q2874725)

Let \(M\) be an \(n\)-dimensional manifold with nonempty boundary \(\partial M\). In [Ann. Math. (2) 102, 41--54 (1975; Zbl 0316.57020)], \textit{D.~Asimov} introduced the notion of a round handle to study non-singular Morse-Smale flows. A round handle of dimension \(n\) and index \(k\) attached to \(M\) is a pair \(R_k=(D^k\times D^{n-k-1}\times S^1,f)\) consisting of a product of an \((n-1)\)-dimensional disk \(D^k\times D^{n-k-1}\) with corner, a circle \(S^1\), and an attaching embedding \(f:\partial D^k\times D^{n-k-1}\times S^1\to \partial M\). A closed, nondegenerate 2-form \(\omega\) on an even-dimensional manifold is called a symplectic structure. A completely nonintegrable hyperplane field on an odd-dimensional manifold is a contact structure. On a symplectic manifold \((W,\omega)\), a vector field \(X\) is called a Liouville vector field if it satisfies \({\mathcal L}_X\omega=\omega\). In [Hokkaido Math. J. 20, No. 2, 241--251 (1991; Zbl 0737.57012)], \textit{A.~Weinstein} showed that the manifold obtained from a contact manifold by a certain handle surgery carries a contact structure. In this paper, the author introduces a new method of round surgery for contact manifolds by the attachment of the standard symplectic round handle. It is shown that if \(M\) is a convex contact type subset of the boundary of a \(2n\)-dimensional symplectic manifold \((W,\omega)\) with respect to a Liouville vector field \(X\) defined near \(M\subset W\), and \(\widetilde S^k=S^{k-1}\times S^1\subset M\) is an isotropic product of a sphere and a circle with a trivialization of the conformal symplectic normal bundle \(\text{CSN}(\widetilde S^k,M)\) with respect to the contact structure induced on \(M\) from \(X\) and \(\omega\), then the Liouville vector field \(X\) and the symplectic structure \(\omega\) extend to the manifold obtained from \(W\) by attaching a round handle of index \(k\) along \(\widetilde S^k\) so that the modified boundary is still convex. Also, it is proven that if \((M,\xi)\) is a contact \(3\)-manifold, then the contact \(3\)-manifold \((\widetilde M, \widetilde \xi)\) obtained from \((M,\xi)\) by a Legendrian contact round surgery along a Legendrian link \(L\subset (M,\xi)\) with two components is contactomorphic to the contact \(3\)-manifold \((\widetilde M', \widetilde \xi')\) obtained from \((M,\xi)\) by the contact round surgery along \(L\).