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**Lagrangian intersections and the Serre spectral sequence.**
*(English)*
Zbl 1141.53078

Consider a symplectic manifold \((M,\omega)\) which is convex at infinity. The dimension of \(M\) is fixed to be \(2n\) and it is assumed that \(\pi_1(L)=\pi_1(L')=0=c_1| _{\pi_2(M)}=\omega| _{\pi_2(M)}\) for a transversal pair of closed Lagrangian submanifolds \(L,L'\) and for a generic almost complex structure \(J\). The authors construct an invariant with a high homotopical content which consists in the pages of order \(\geq 2\) of a spectral sequence whose differentials provide an algebraic measure of the high-dimensional moduli spaces of pseudo-holomorphic strips of finite energy that join \(L\) and \(L'\). When \(L\) and \(L'\) are Hamiltonian isotopic, the authors show that the pages of the spectral sequence coincide (up to a horizontal translation) with the terms of the Serre spectral sequence of the path-loop fibration \(\Omega\rightarrow PL\rightarrow L\) and they deduce some applications. The machinery of A. Floer’s homology is applied [Commun. Pure Appl. Math. 42, 335–356 (1989; Zbl 0683.58017); J. Differ. Geom. 28, 513–547 (1988; Zbl 0674.57027); ibid. 30, 207–221 (1989; Zbl 0678.58012); Commun. Math. Phys. 120, 575–611 (1989; Zbl 0755.58022)], A. Floer and H. Hofer [Math. Z. 215, 37–88 (1994; Zbl 0810.58013)]. Also some results and techniques are used obtained by the second author in a series of papers on “Homotopical dynamics” [Ergodic Theory Dyn. Syst. 20, 379–391 (2000; Zbl 0984.37017); Ann. Sci. Éc. Norm. Supér. (4) 35, 549–573 (2002; Zbl 1021.37009); Duke Math. J. 109, 183–204 (2001; Zbl 1107.37300); Commun. Pure Appl. Math. 55, 1033–1088 (2002; Zbl 1024.37040)].

Fix a path-connected component \(\mathcal{P}_\eta(L,L')\) of the space \(\mathcal{P}(L,L')=\{\gamma\in C^\infty([0,1],M):\gamma(0)\in L\), \(\gamma(1)\in L'\}\). The construction of Floer homology depends on the choice of such a component. The corresponding Floer complex is denoted by \(CF_\ast(L,L';\eta)\) and the resulting homology by \(HF_\ast(L,L';\eta)\). In case \(L'=\varphi_1(L)\) with \(\varphi_1\) the time 1-map of a Hamiltonian isotopy \(\varphi:M\times [0,1]\rightarrow M\) (called a Hamiltonian diffeomorphism), by \(\mathcal{P}(L,L';\eta_0)\) is denoted the path-component of \(\mathcal{P}(L,L')\) such that \([\varphi^{-1}_t (\gamma(t)]=0\in\pi_1(M,L)\) for some (and thus all) \(\gamma\in\eta_0\). Given two spectral sequences \((E^r_{p,q},d^r)\) and \((G^r_{p,q},d^r)\) it is said that they are isomorphic up to translation if there exist an integer \(k\) and an isomorphism of chain complexes \((E^r_{\ast+k,\star},d^r)\approx (G^r_{\ast,\star},d^r)\) for all \(r\). Consider the path-loop fibration \(\Omega L\rightarrow PL\rightarrow L\) of base \(L\) which has as total space the space of paths based in \(L\) and as fibre the space of based loops. Given two points \(x,y\in L\cap L'\) by \(\mu(x,y)\) is denoted their relative Maslov index and by \(\mathcal{M}(x,y)\) the nonparametrized moduli space of Floer trajectories connecting \(x\) to \(y\). By \(\mathcal{M}\) is denoted the disjoint union of all the \(\mathcal{M}(x,y)\)’s. The space of all parametrized pseudo-holomorphic strips is denoted by \(\mathcal{M}'\). All homologies have \(\mathbb{Z}/2\)-coefficients.

The main result is the following Theorem 1.1: Under the assumptions above there exists a spectral sequence \(EF(L,L';\eta)= (EF^r_{p,q}(L,L';\eta),d^r_F)\), \(r\geq 1\), with the following properties:

a) If \(\varphi:M\times [0,1]\rightarrow M\) is a Hamiltonian isotopy, then \((EF^r_{p,q}(L,L';\eta),d^r)\) and \[ (EF^r_{p,q}(L,\varphi_1 L';\varphi_1 \eta),d^r) \] are isomorphic up to translation for \(r\geq 2\) (here \(\varphi_1\eta\) is the component represented by \(\varphi_t(\gamma(t))\) for \(\gamma\in \eta\)).

b) \(EF^1_{p,q}(L,L';\eta)\approx CF_p(L,L';\eta)\otimes H_q(\Omega L)\), \(EF^2_{p,q}(L,L';\eta)\approx HF_p(L,L';\eta)\otimes H_q(\Omega L)\).

c) If \(d^r_F\neq 0\), then there exist points \(x,y\in L\cap L'\) such that \(\mu(x,y)\leq r\) and \(\mathcal{M}(x,y)\neq\emptyset \).

d) If \(L'=\varphi' L\) with \(\varphi'\) a Hamiltonian diffeomorphism, then for \(r\geq 2\) the spectral sequence \((EF^r(L,L'),d^r_F)\) is isomorphic up to a translation to the \(\mathbb{Z}/2\)-Serre spectral sequence of the path loop fibration \(\Omega L\rightarrow PL\rightarrow L\).

The authors give a geometric interpretation of this spectral sequence and discuss the role of the Serre spectral sequence. Finally some applications are given: algebraic consequences, existence of pseudo-holomorphic strips, and nonsqueezing and Hofer’s energy.

Fix a path-connected component \(\mathcal{P}_\eta(L,L')\) of the space \(\mathcal{P}(L,L')=\{\gamma\in C^\infty([0,1],M):\gamma(0)\in L\), \(\gamma(1)\in L'\}\). The construction of Floer homology depends on the choice of such a component. The corresponding Floer complex is denoted by \(CF_\ast(L,L';\eta)\) and the resulting homology by \(HF_\ast(L,L';\eta)\). In case \(L'=\varphi_1(L)\) with \(\varphi_1\) the time 1-map of a Hamiltonian isotopy \(\varphi:M\times [0,1]\rightarrow M\) (called a Hamiltonian diffeomorphism), by \(\mathcal{P}(L,L';\eta_0)\) is denoted the path-component of \(\mathcal{P}(L,L')\) such that \([\varphi^{-1}_t (\gamma(t)]=0\in\pi_1(M,L)\) for some (and thus all) \(\gamma\in\eta_0\). Given two spectral sequences \((E^r_{p,q},d^r)\) and \((G^r_{p,q},d^r)\) it is said that they are isomorphic up to translation if there exist an integer \(k\) and an isomorphism of chain complexes \((E^r_{\ast+k,\star},d^r)\approx (G^r_{\ast,\star},d^r)\) for all \(r\). Consider the path-loop fibration \(\Omega L\rightarrow PL\rightarrow L\) of base \(L\) which has as total space the space of paths based in \(L\) and as fibre the space of based loops. Given two points \(x,y\in L\cap L'\) by \(\mu(x,y)\) is denoted their relative Maslov index and by \(\mathcal{M}(x,y)\) the nonparametrized moduli space of Floer trajectories connecting \(x\) to \(y\). By \(\mathcal{M}\) is denoted the disjoint union of all the \(\mathcal{M}(x,y)\)’s. The space of all parametrized pseudo-holomorphic strips is denoted by \(\mathcal{M}'\). All homologies have \(\mathbb{Z}/2\)-coefficients.

The main result is the following Theorem 1.1: Under the assumptions above there exists a spectral sequence \(EF(L,L';\eta)= (EF^r_{p,q}(L,L';\eta),d^r_F)\), \(r\geq 1\), with the following properties:

a) If \(\varphi:M\times [0,1]\rightarrow M\) is a Hamiltonian isotopy, then \((EF^r_{p,q}(L,L';\eta),d^r)\) and \[ (EF^r_{p,q}(L,\varphi_1 L';\varphi_1 \eta),d^r) \] are isomorphic up to translation for \(r\geq 2\) (here \(\varphi_1\eta\) is the component represented by \(\varphi_t(\gamma(t))\) for \(\gamma\in \eta\)).

b) \(EF^1_{p,q}(L,L';\eta)\approx CF_p(L,L';\eta)\otimes H_q(\Omega L)\), \(EF^2_{p,q}(L,L';\eta)\approx HF_p(L,L';\eta)\otimes H_q(\Omega L)\).

c) If \(d^r_F\neq 0\), then there exist points \(x,y\in L\cap L'\) such that \(\mu(x,y)\leq r\) and \(\mathcal{M}(x,y)\neq\emptyset \).

d) If \(L'=\varphi' L\) with \(\varphi'\) a Hamiltonian diffeomorphism, then for \(r\geq 2\) the spectral sequence \((EF^r(L,L'),d^r_F)\) is isomorphic up to a translation to the \(\mathbb{Z}/2\)-Serre spectral sequence of the path loop fibration \(\Omega L\rightarrow PL\rightarrow L\).

The authors give a geometric interpretation of this spectral sequence and discuss the role of the Serre spectral sequence. Finally some applications are given: algebraic consequences, existence of pseudo-holomorphic strips, and nonsqueezing and Hofer’s energy.

Reviewer: Ioan Pop (Iaşi)

### MSC:

53D12 | Lagrangian submanifolds; Maslov index |

53D40 | Symplectic aspects of Floer homology and cohomology |

55R20 | Spectral sequences and homology of fiber spaces in algebraic topology |

55T10 | Serre spectral sequences |