In this talk we will describe two methods which shows that, under some rigidity assumptions on the involved Legendrian submanifolds, a Lagrangian cobordism is simply connected. The first one uses the functoriality of the fundamental class in Legendrian contact homology with twisted coefficients. The second uses an $L^2$-completion of the Floer complex associated to the cobordism. This is joint work with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko.