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

Symplectic snippets

Lagrangian cobordisms give exact sequences

theorem 0.0.1 [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 \(\text{mod}-\Fuk(X)\), exact sequences from cobordismswhere each triangle in the diagram an exact triangle, \(C_0=0\), and \(C_k=k\).

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References

[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.