Elements of the Lagrangian Whitney trick (Q1434502)

Let \(D\) be a symplectic \(2\)-manifold diffeomorphic to an open disc \(\widetilde{D}\) in \(\mathbb{R}^2\), \(l_0,l_1\) open arcs closed in \(D\) and intersecting transversely at exactly two points \(x_0,y_0\), and \(B_r^{n-1}\) the Euclidean open \((n-1)\)-ball of radius \(r\) centered at the origin. The authors define the area (resp. capacity) condition for the triple \((D,l_0,l_1)\) [resp. \((D\times B_a^{n-1}\times B_b^{n-1},l_0,l_1)\)]. Let \(f:M^n\to P^{2n}\) be a self-transverse Lagrangian immersion into a symplectic manifold and assume that \(\psi:D\times B_a^{n-1}\times B_b^{n-1}\to P^{2n}\) is a symplectic Whitney neighborhood associated to \(f\) [i.e. \(\psi\) is a symplectic embedding such that \(\psi^{-1}\circ f(M^n)=l_0 \times B_a^{n-1}\times\{0\}\cup l_1\times\{0\}\times B_b^{n-1}\)] for a pair of self-intersections \(p:=f(x_0,0,0)\), \(q:=f(y_0,0,0)\). If the area and capacity conditions are satisfied by the domain of \(\psi\), the authors prove the existence of a homotopy \(f_u:M^n\to P^{2n}\), \(u\in [0,1]\), with compact support, through Lagrangian immersions such that \(f_0=f\) and the set of self-intersections of \(f_1\) is that of \(f\) minus \(\{p,q\}\). The proof is based on the existence of a ``Lagrange-Whitney trick'' in \(\widetilde{D}\times\mathbb{R}^{n-1}\times \mathbb{R}^{n-1}\) (for the Whitney trick in the differential category, see [\textit{J. Milnor}, Lectures on the \(h\)-cobordism theorem, Princeton Mathematical Notes (1965; Zbl 0161.20302)] and on a result of \textit{B. Dacorogna} and \textit{J. Moser} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 1, 1--26 (1990; Zbl 0707.35041)]). Given \(f,D,l_0,l_1\), where \(f\) is as above and \(D\subset P^{2n}\) a symplectic open disc such that there are embedded open arcs \(l_0,l_1\) closed in \(D\), \(l_0\cup l_1=D\cap f(M)\) and \(l_0,l_1\) meet at exactly two points, they show that there is a symplectic Whitney neighborhood associated to \(f\) if a Maslov-Viterbo index closely related to \(D\) is \(1\). An example of a Lagrangian immersion of the Klein bottle into \(\mathbb{C}^2\) that cannot be homotopic to a Lagrangian embedding through Lagrangian immersions by a result of \textit{Y. G. Oh} [Int. Math. Res. Not. 1996, No. 7, 305--346 (1996; Zbl 0858.58017)] is considered.