\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: lagrangian cobordisms

lagrangian cobordisms

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If \(M\) is a manifold, a surgery of \(M\) is the replacement of a \(D^{n-k}\times S^k\) with \(S^{n-k-1}\times D^{k+1}\). Surgery of manifolds is closely tied to cobordisms between manifolds. Let \(M, N\) be \(n\)-manifolds. A cobordism \(K: M\rightsquigarrow N\) is a manifold with boundary \(\partial K=M\sqcup N\). There is an analogous notion of cobordism exists for Lagrangian submanifolds.

definition 0.0.1 [Arn80]

Let \(\{L_0, \ldots, L_k\}\) be Lagrangian submanifolds of \(X\). A Lagrangian cobordism with positive ends \(L_0, \ldots L_k\) is a Lagrangian submanifold \(K\subset X\times T^*\RR\) for which there exists a compact subset \(D\subset \CC\) so that : \[K\setminus( \pi_\CC^{-1}(D))=\bigcup_{i=0}^k L_i\times \ell_i.\] Here, the \(\ell_i\) are rays of the form \(\ell_i(t)=i\cdot \jmath +t\), where \(t\in \RR_{\geq 0}\). We denote such a cobordism \(K:(L_0,L_1, \ldots L_k)\rightsquigarrow \emptyset\). One can also discuss Lagrangian cobordisms with both positive and negative ends.
figure 0.0.2:The projection of a Lagrangian cobordism \(K\subset X\times \CC\) to the \(\CC\) factor. This cobordism has ends \(K:(L_0, L_1, \ldots L_k)\rightsquigarrow \emptyset.\)

example 0.0.3 [ALP94]

Let \(\li_s: L\times \RR_t\to X\) be an exact Lagrangian homotopy generated by the function \(H_t: L\times \RR_t\to \RR\). The suspension of \(H_t\) is the Lagrangian cobordism \(K_{H_t}\) with topology \(L\times \RR\) parameterized by \begin{align*} L\times \RR\to & X\times \CC\\ (u, s)\mapsto& (\li_t(u), s+\jmath H_t(u))\in X\times \CC. \end{align*}

example 0.0.4

Let \(L_1, L_2\subset X\) be Lagrangian submanifolds which intersect transversely at a point \(q\), so we can define the Polterovich surgery \(L_1\#_q L_2\). Then there exists a Lagrangian cobordism \((L_1\#_U L_2)\rightsquigarrow (L_1, L_2)\). Lagrangian cobordisms give us new examples of exact sequences in the Fukaya category.

theorem 0.0.5 [BC14]

See also [Tan16]. Let \(L_0, \ldots L_k\in \Fuk(X)\) be Lagrangian submanifolds, and suppose there exists a monotone Lagrangian cobordism \(K: (L_0, \ldots, L_k)\rightsquigarrow \emptyset\). Then there exists an iterated cone decomposition in \(\text{mod}-\Fuk(X)\), where each triangle in the diagram an exact triangle, \(C_0=0\), and \(C_k=k\). In particular, if \(K: (L_0, L_1, L_2)\rightsquigarrow \emptyset\) is a Lagrangian cobordism, then we have an exact triangle \[L_2\to L_1\to L_0.\] In contrast to (surgery exact Triangle), such a Lagrangian cobordism does not identify which exact triangle one has --- it only identifies that you have an exact triangle. The iterated mapping cone given in theorem 0.0.5 is an example of a twisted complex.

References

[Arn80]Vladimir Igorevich Arnol'd. Lagrange and Legendre cobordisms. i. Funktsional'nyi Analiz i ego Prilozheniya, 14(3):1--13, 1980.
[ALP94]Michèle Audin, François Lalonde, and Leonid Polterovich. Symplectic rigidity: Lagrangian submanifolds. In Holomorphic curves in symplectic geometry, pages 271--321. Springer, 1994.
[BC14]Paul Biran and Octav Cornea. Lagrangian cobordism and fukaya categories. Geometric and functional analysis, 24(6):1731--1830, 2014.
[Tan16]Hiro Lee Tanaka. The fukaya category pairs with lagrangian cobordisms. arXiv preprint arXiv:1607.04976, 2016.