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Rizell, Georgios Dimitroglou; Golovko, Roman

On homological rigidity and flexibility of exact Lagrangian endocobordisms. (English) Zbl 1315.53093

Int. J. Math. 25, No. 10, Article ID 1450098, 24 p. (2014).
Summary: 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 \(\mathbb F\), given that \(\Lambda\) admits a spin exact Lagrangian filling and that 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 arbitrarily high ranks. In addition, we prove that the front \(S^m\)-spinning construction preserves looseness, which implies certain forgetfulness properties of it.

Cited in 5 Documents

MSC:

53D12 Lagrangian submanifolds; Maslov index
53D42 Symplectic field theory; contact homology

Keywords:

exact Lagrangian cobordism; Legendrian contact homology; wrapped Floer homology; rigidity; flexibility
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\textit{G. D. Rizell} and \textit{R. Golovko}, Int. J. Math. 25, No. 10, Article ID 1450098, 24 p. (2014; Zbl 1315.53093)
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