## 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.
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
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