Handle attaching in symplectic homology and the chord conjecture
In his 1986 paper [Russ. Math. Surv. 41, No. 6, 1-21 (1986; Zbl 0649.58010)], \textit{V. I. Arnold} conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord, i. e. an orbit of the Reeb vector field intersecting the knot at least twice. The author of this paper formulates the same question in the following more general situation: Let \(N^{2n-1}\) be a closed oriented manifold of dimension \(2n-1\). A contact form on \(N\) is a \(1\)-form \(\lambda\) such that \(\lambda\wedge (d\lambda)^{n-1}\) is a positive volume form. The kernel \(\xi = \text{ker }\lambda\) of a contact form is called a (cooriented) contact structure, and the pair \((N,\xi)\) a contact manifold. A submanifold \(L\subset (N,\xi)\) is called isotropic if it is tangent to \(\xi\). An isotropic submanifold of the maximal possible dimension \(n-1\) is called Legendrian. A contact form determines a unique vector field \(Y_\lambda\), called the Reeb vector field, by the conditions \(i_{Y_\lambda}dX = 0\) and \(\lambda(Y_\lambda) = 1\). A characteristic chord for a contact form \(\lambda\) and a Legendrian submanifold is an orbit of the Reeb vector field intersecting \(L\) at least twice. Then the author says that the (generalised) chord conjecture holds for a triple \((N,\lambda,L)\) if \(L\) possesses a characteristic chord for \(\lambda\), and that it holds for a triple \((N,\xi,L)\) if it holds for every triple \((N,\lambda,L)\) with \(\xi = \text{ker} \lambda\). The author observes that since Legendrian isotopies extend to ambient contact isotopies, if the chord conjecture holds for \((N,\xi,L)\) then it holds for \((N,\xi,\widetilde L)\) with \(\widetilde L\) isotopic to \(L\) (through Legendrian embeddings). He also notices that the existence of a chord for a triple \((N,\lambda,L)\) can be checked directly from the Reeb flow of \(\lambda\) but the question becomes nontrivial if we fix the contact structure and allow the contact form and/or the Legendrian submanifolds to vary. Finally, in the preamble, the author mentions two (nontrivial) results concerning the chord conjecture. The first (Ginzburg and Givental) affirms that the chord conjecture holds for \((\mathbb{R} P^{2n-1},\xi_{st},L)\) where \(\xi_{st}\) is the standard structure on \(\mathbb{R} P^{2n-1}=(\mathbb{C}^n)^*/\mathbb{R}^*\) and \(L\) is isotopic to \(\mathbb{R} P^{n-1} = (\mathbb{R}^n)^*/\mathbb{R}^*\subset \mathbb{R} P^{2n-1}\) and the second (Abbas) says that the chord conjecture holds for any triple \((S^3,\lambda,L)\) where \(\lambda\) is a contact form induced by an embedding \(S^3\hookrightarrow R^4\) as a strictly convex hypersurface, and \(L\) is a Legendrian knot with the property that there exists an unknotted periodic Reeb orbit \(P\) with self-linking number \(-1\) such that \(L\) and \(P\) are not linked. For his main results in this paper, the author considers a lot of notions from \textit{A. Weinstein} [Hokkaido Math. J. 20, 241-251 (1991; Zbl 0737.57012)], \textit{Y. Eliashberg} [Symplectic geometry of plurisubharmonic functions. Lecture Notes by M. Abreu, NATO ASI Ser., Ser. C, Math. Phys. Sci. 488, 49-67 (1996; Zbl 0881.32010)] and \textit{Y. Eliashberg} and \textit{M. Gromov} [Proc. Symp. Pure Math. 52, Part 2, 135-162 (1991; Zbl 0742.53010). A Weinstein domain is a quadruple \((W^{2n},\omega, X,\phi)\) with: \((W,\omega)\) a compact symplectic manifold with boundary; \(X\) an expanding vector field, i. e. \(L_X\omega = \omega\); \(\phi\) a Morse function which has the boundary as a regular level set; the critical points of \(\phi\) are nondegenerate zeros of \(X\), and \(X.\phi > 0\) outside the critical points; \(X\) is outward pointing along the boundary. It follows that all critical points of \(\phi\) have index \(\leq n\). \((W^{2n},\omega,X,\phi)\) is called subcritical if all critical points of \(\phi\) have index \(<\lambda\). Note that the 1-form \(\lambda:= i_X\omega\) satisfies \(d\lambda= \omega\), and \(\lambda\mid \partial W\) is a contact form. A Weinstein homotopy is a smooth family \((W,\omega_t,X_t,\phi_t)_{t\in[0,1]}\) such that each \((W,\omega_t,X_t,\phi_t)\) has the property of a Weinstein domain except that critical points of \(\phi_t\) and zeros of \(X_t\) may be degenerate. It has been observed by A. Weinstein (op. cit) and \textit{Y. Eliashberg} [Int. J. Math. 1, 29-46 (1990; Zbl 0699.58002)] that the manifold \(W\cup_L H^{2n}_k\) obtained by attaching a \(2n\)-dimensional \(k\)-handle with a standard framing to a Weinstein domain \(W^{2n}\) along an isotropic sphere \(L^{k-1}\subset \partial W\) carries again a natural Weinstein structure. Call the handle subcritical if \(k < n\). The main result of the paper is the following: Theorem 1.3. Let \((N^{2n-1},\xi)\), \(n\geq 2\), be the boundary of a subcritical Weinstein domain \(W^{2n}\) and \(L\subset N\) a Legendrian sphere. Suppose that \(W\cup_L H^{2n}_k\cup\)\{subcritical handles\} is Weinstein homotopic to \(DT*Q\cup\) \{subcritical handles\} for some compact manifold \(Q^n\), where \(DT^*Q:=\{p\in T^*Q\mid \|p\|\leq 1\}\) is the unit disk cotangent bundle with respect to a Riemannian metric on \(Q^n\). Then the chord conjecture holds for \((N,\xi,L)\). This result is improved to yield chords in specific homotopy classes. Theorem 1.4. Let \((N,\xi)\), \(L,W, M\) be as above. Then for every \(\gamma\in\widetilde\pi_1(Q)\) with \(j_\#\gamma\) nontrivial there exists a chord \(x\) for \((N,\xi,L)\) whose class \([x]\in\widetilde\pi_1(W,L)\) satisfies \(i_\#[x] = j_\#\gamma\). Here, \(\widetilde\pi_1(Q)\) denotes the free homotopy classes of loops in \(Q\). \(\widetilde{\pi}_1(W,L)\) is the set of relative homotopy classes of maps \(x:[0,1]\to W\) with \(x(0),x(1)\in L\), and \(i\), \(j\) are the inclusions. These main results are continued by a lot of corollaries and examples. Some of these affirm that the chord conjecture holds for: -- any triple \((N^{2n-1},\xi,L),n\geq 2\), where \((N,\xi)\) is the boundary of a subcritical Weinstein domain, and \(L\) is a standard Legendrian unknot; -- \((S^{2n-1},\xi_{st}, L)\), where \(L\) is isotopic to a Legendrian great circle; -- triples \((S^3,\xi_{st},L)\) where \(L\) is a Legendrian unknot with Thurston-Bennequin invariant \(tb(L) =-1\) (Arnold conjecture); -- \(-(\#_k(S^1\times S^2),\xi_{st}, L)\), where \(L\) is a Legendrian unknot with Thurston-Bennequin invariant \(tb(L) =-1\) or a Legendrian knot running over the first \(2g\leq k\) handles with a special Kirby diagram. Some results refer to the existence of infinitely many chords. The proof relies on the behaviour of symplectic homology under handle attaching [the author, Math. Z. 228, 451-487 (1998; Zbl 0919.53015); the author, \textit{A. Floer} and \textit{H. Hofer}, Math. Z. 218, 103-122 (1995; Zbl 0869.58011) and \textit{A. Floer} and \textit{H. Hofer}, Math. Z. 215, 37-88 (1994; Zbl 0810.58013)].
scientific article; zbMATH DE number 1798901