Topology of (small) Lagrangian cobordisms

Abstract

We study the following quantitative phenomenon in symplectic topology: In many situations, if a Lagrangian cobordism is sufficiently small (in a sense specified below) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms which, by Biran-Cornea's Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev's theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.Corollary 7. Let L, L ′ ∈ L 1 and suppose at least one of them is spin. If there exists a small Lagrangian cobordism V :for every field F. Of course, if neither L nor L ′ is spin then the same conclusion holds for F = Z 2 .Corollary 8. No oriented L ∈ L with χ(L) = 0 admits an oriented Lagrangian null-cobordism. Similarly, no L ∈ L admits a small Lagrangian null-cobordism.Of course the only case where the first conclusion in Corollary 8 is a symplectic phenomenon is when both n and χ(L) are even. 1 The corollary in particular implies 1 Recall that every closed odd dimensional manifold N satisfies χ(N ) = 0. Similarly it is wellknown that the boundary of a compact manifold has even Euler characteristic, so for χ(L) odd the conclusion of the corollary follows from classical topology.

Invariants of Lagrangian cobordisms via spectral numbers

“…On the other hand there are many explicit examples of non-monotone Lagrangian cobordisms which do not arise as Lagrangian suspensions [3], [6], [12]. In [5] we study how properties similar to (2) of such cobordisms are intimately linked to their topology.…”

“…Note that the existence of such a Lagrangian in (M, ω) implies that M is symplectically aspherical in the sense that ω| π 2 (M ) ≡ 0 and c 1 (T M)| π 2 (M ) ≡ 0. We denote by L 0 the space of all closed Lagrangians L ⊂ (M, ω) satisfying (5). Note that for any L ∈ L 0 , QH * (L) reduces to H * (L) = H * (L; Z 2 ).…”