A note on Lagrangian intersections and Legendrian cobordism (Q2139697)

``The notion of Legendrian cobordism between two compact Legendrian submanifolds was introduced by \textit{V. I. Arnol'd} in [Funkts. Anal. Prilozh. 14, No. 3, 1--13 (1980; Zbl 0448.57017) and ibid. 14, No. 4, 8--17 (1980; Zbl 0472.55002)]. Arnol'd defined Legendrian cobordisms between Legendrians in jet spaces, the space of contact elements and the space of co-oriented contact elements, and computed the immersed Legendrian cobordism group (oriented and nonoriented) when the immersed Legendrian has dimension 1. The cases of higher dimensions were computed by \textit{M. Audin} [Ann. Inst. Fourier 35, No. 3, 159--194 (1985; Zbl 0542.57026)] and \textit{J. Eliashberg} [in: Géométrie symplectique et de contact, Journ. lyonnaises Soc. math. France 1983. Sémin. sud-rhodanien Géom. I, 17--31 (1984; Zbl 0542.57024)] independently. Then \textit{E. Ferrand} [Transl., Ser. 2, Am. Math. Soc. 190, 23--35 (1999; Zbl 1067.57501)] extended Arnold's definition of Legendrian cobordism to arbitrary contact manifolds.'' In general, an embedded Legendrian cobordism does not preserve pseudoholomorphic curves type invariants. However, under certain restrictions, the author of the paper under review shows that Legendrian cobordisms do preserve the Floer homology group defined by Eliashberg-Hofer-Salamon [\textit{Y. Eliashberg} et al., Geom. Funct. Anal. 5, No. 2, 244--269 (1995; Zbl 0844.58038)]. More precisely, the author proves the following result: Theorem. Let \(Y\) be a closed hypertight contact manifold and \(\Lambda, \Lambda'\subset Y\) a pair of connected hypertight Legendrian submanifolds related by a connected hypertight Legendrian cobordism \((W; \Lambda, \Lambda') \subset \mathbb C\times Y\). If \(P\subset Y\) is a closed pre-Lagrangian submanifold which is either weakly exact or monotone with minimal Maslov number strictly greater than two, then \(HF(P,\Lambda)\simeq HF(P,\Lambda')\). This result admits the following corollary: Corollary. Under the assumptions of the theorem, consider a pair \((P, (W; \Lambda, \Lambda'))\), where \(P\) is weakly exact and \(\Lambda \subset P\). Assume moreover that the boundary homomorphism \(\pi_2(Y ,P)\to \pi_1(P)\) is trivial and that \(\{\phi_t\}_{0\leq t\leq 1}\) is a contact isotopy such that \(P\) transversely intersects \(\phi_1(\Lambda')\). Then \[\# \phi_1(\Lambda')\cap P \geq rank (H_{\ast}(\Lambda,\mathbb Z_2)).\]