On homological rigidity and flexibility of exact Lagrangian endocobordisms

Abstract

We show that an exact Lagrangian cobordism $L\subset \mathbb R \times P \times \mathbb R$ from a Legendrian submanifold $\Lambda\subset P\times \mathbb R$ to itself satisfies $H_i(L;\mathbb F)=H_i(\Lambda;\mathbb F)$ for any field $\mathbb F$ in the case when $\Lambda$ admits a spin exact Lagrangian filling and the concatenation of any spin exact Lagrangian filling of $\Lambda$ and $L$ is also spin. The main tool used is Seidel's isomorphism in wrapped Floer homology. In contrast to that, for loose Legendrian submanifolds of $\mathbb{C}^n \times \mathbb R$, we construct examples of such cobordisms whose homology groups have arbitrary high ranks. In addition, we prove that the front $S^m$-spinning construction preserves looseness, which implies certain forgetfulness properties of it.