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

Symplectic snippets

surgery triangle from Lagrangian cobordism

  1. Let \(d\theta_{X,\CC}=\omega_{X,\CC}\) be primitives of each symplectic form. By assumption we have \((\theta_{X}\oplus\theta_\CC) |_{K}=df\) for some smooth function \(f:K\to \RR\). Then \(\theta_X|_{L_i}=d(\iota_{i}^*f)\), where \(\iota_{i}:L_i\into K\) includes \(L_i\) as a fiber over one of the ends. Indeed, for \(v\in TL_i\) we have \[ \theta_X (v)=(\theta_X\oplus\theta_{\CC})(v,0)=df(v,0)=(\iota_i^*(df))(v)=d(\iota_i^* f)(v) \] which proves that \(\theta_X|_{L_i}=d(\iota_i^*f)\).
  2. If \(p\in (L\times\gamma^-)\cap K\), then \(\pi_\CC(p)\in \pi_\CC(L\times\gamma^-)\cap \pi_\CC(K)=\set{z}\), so \((L\times\gamma^-)\cap K \cong L\cap L_2\) and these are precisely the generators of both vector spaces.
  3. By the choice of almost complex structure, the composition \(p:=\pi_\CC\circ u:\RR\times[0,1]\to\CC\) is holomorphic. First of all note that \(\text{Im}(p)\) must be bounded: the finite energy condition says that \(u\) extends to a map \(\tilde p:D\to \CC\) from the disc, which is compact and thus has compact --- in particular, bounded --- image. Now if \(\text{Im}(p)\not\subset\set{z}=\text{Im}(\gamma^-)\cap \RR_{<<0}\), then \(\text{Im}(p)\cap U\neq\emptyset\) for an open unbounded region \(U\subset \CC\setminus(\gamma^-\cup \pi_\CC(K))\). Furthermore:
    • \(\tilde p(D)\) is compact and thus closed, so \(\tilde p(D)\cap U=\text{Im}(p)\cap U\) is closed in \(U\).
    • \(p(\RR\times(0,1))\) is open by the open mapping theorem, thus \(p(\RR\times(0,1))\cap U=\text{Im}(p)\cap U\) is open in \(U\).
    We have a non-empty open and closed subset of \(U\); as \(U\) is connected, we conclude that \(\text{Im}(p)\cap U=U\), which implies that \(\text{Im}(p)\) contains the unbounded region \(U\). This is a contradiction, so \(\text{Im}(p)=\set{z}\). Now we know that all pseudoholomorphic strips with boundary on \(L\times\gamma^-\) and \(K\) --- which are precisely those contributing to the differential of \(CF^\bullet(L\times\gamma^-,K)\) --- are of the form \(u(s,t)=(v(s,t),z)\), where \(v\) is a pseudoholomorphic strip in \(X\) with boundary in \(L\) and \(L_2\) --- which are precisely those contributing to the differential of \(CF^\bullet(L,L_2)\). Thus we have a bijection betweeen generators of \(CF^\bullet(L\times \gamma^-,K)\) and \(CF^\bullet(L,L_2)\), as well as between their pseudoholomorhic strips.
  4. As a vector space, \(CF^\bullet(L\times\gamma^+,K)=CF^\bullet(L,L_1)\oplus CF^\bullet(L,L_2)\) by the same argument as in part 2. We now have three types of pseudoholomorphic strips:
    1. Those connecting two points in \(CF^\bullet(L,L_1)\), which are encoded in the differential \(\partial_{L,L_1}\).
    2. Those connecting two points in \(CF^\bullet(L,L_2)\), which are encoded in the differential \(\partial_{L,L_2}\).
    3. Those connecting points in \(CF^\bullet(L,L_1)\) with one in \(CF^\bullet(L,L_2)\), which make use of the bounded region between \(\gamma^+\) and \(\pi_\CC(K)\). Call this map \(f:CF^\bullet(L,L_1)\to CF^\bullet(L,L_2)\).
    Recalling the definition of a {\it mapping cone} of chain complexes, this says precisely that (up to grading considerations) \[ CF^\bullet(L\times\gamma^+,K)=cone(CF^\bullet(L,L_0)\xrightarrow{f} CF^\bullet(L,L_1)). \]
  5. By the previous part we have a long exact sequence \[ \dots HF^{i-1}(L\times\gamma^+,K)\to HF^i(L,L_0) \to HF^i(L,L_1) \to HF^i(L\times\gamma^-,K) \to HF^{i+1} (L,L_0)\to\dots \] The Hamiltonian isotopy between \(K\times\gamma^-\) and \(L\times\gamma^+\) together with the invariance of Lagrangian Floer homology under Hamiltonian isotopy gives \(HF^\bullet(L\times\gamma^+,K)\cong HF^\bullet(L\times\gamma^-,K)\), and by part 3 the latter is isomorphic to \(HF^\bullet(L_2,K)\). Putting all together we get a long exact sequence \[ \dots HF^{i-1}(L_2,K)\to HF^i(L,L_0) \to HF^i(L,L_1) \to HF^i(L_2,K) \to HF^{i+1} (L,L_0)\to\dots \] as required.