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.

22 pages, 2 figures. Serious revisions for publication, the main rigidity result remains the same; accepted for publication in the International Journal of Mathematics

References in corpus (2)

• A note on Lagrangian cobordisms between Legendrian submanifolds of R^{2n+1}
• A note on the front spinning construction

Cited by in corpus (2)

• Symplectic homology and the Eilenberg-Steenrod axioms
• On Lagrangian embeddings of closed non-orientable 3-manifolds