Generalised cohomology and relatively exact Lagrangian submanifolds

Abstract

In this thesis, we study the topology of relatively exact Lagrangian submanifolds. One of our main goals is to study their generalised cohomology, extending known results about their singular cohomology. We do this using different (and simpler) technical set-ups to that of Cohen, Jones and Segal [18, 16]. In Chapter 2, we prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ1 of a symplectic manifold (M,ω) fixing a relatively exact Lagrangian L setwise must act trivially on R∗(L), where R∗is some multiplicative generalised homology theory. We use a strategy inspired by that of Hu, Lalonde and Leclercq [44], who proved an analogous result over Z/2and over Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, ψ1|L is homotopic to the identity. We also prove (under similar conditions) that ψ1|L acts trivially on R∗(LL), where LL is the free loop space of L. From this we deduce that when L is a surface or a K(π,1), ψ1|L is homotopic to the identity. We also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over Z/2. This uses the methods of Lalonde and McDuff from [53]. In Chapter 3 (which is joint work with Amanda Hirschi), we find lower bounds on the number of intersection points between a relatively exact Lagrangian submanifold L and its image L′under a Hamiltonian diffeomorphism, coming from the cup length of L in various multiplicative generalised cohomology theories R∗, under similar orientation assumptions to those in Chapter 2. As an intermediate result which may be of independent interest, we show that the space of holomorphic discs with certain boundary conditions admits a natural evaluation map to L which is injective on R-cohomology groups.This extends work of Hofer [42], who showed both of these things in singular cohomology with coefficients in Z/2. We follow his strategy, adapting his argument where necessary.