Ph.D Thesis
Ph.D Student Sergei Lanzat Symplectic Quasi-Morphisms and Quasi-States for Non-Compact Symplectic Manifolds Department of Mathematics Full Professor Polyak Michael

Abstract

The main goal of the present thesis is to extend the Entov-Polterovich construction of (partial) quasi-morphisms (that is a certain remarkable ``almost homomorphisms") and quasi-states a (certain remarkable ``almost linear" functionals) to non-closed symplectic manifolds. For the construction we modify the quantum and Floer homology theories for a certain class of non-closed symplectic manifolds.

In the first part of the thesis we study the space of pseudo-holomorphic spheres in compact symplectic manifolds with non-empty convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This leads to a deformation of intersection products on the absolute and relative singular homologies. As a result, absolute and relative quantum homology algebras are defined analogously to the case of closed symplectic manifolds.

In the second part of the thesis we construct absolute and relative versions of Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products.  We establish the absolute and relative Piunikhin--Salamon--Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined.

In the last part of the thesis we prove that under appropriate algebraic conditions on the absolute and relative quantum homology algebras of a strongly semi-positive compact convex symplectic manifold (M, w), the universal cover of the group of its compactly supported Hamiltonian diffeomorphisms admits a real-valued homogeneous quasi-morphism. By a ``linearization" of the quasi-morphism we define a symplectic quasi-state on the space of smooth functions on M, which are constant near the boundary. Using the Biran-Cornea Lagrangian pearl homology, we construct examples of manifolds, for which the quasi-state is non-linear functional and hence, the quasi-morphism is not a homomorphism. Modifying the Seidel homomorphism construction, we prove that for our examples the quasi-morphism descends to the group of compactly supported Hamiltonian diffeomorphisms of (M, w). Next, we show that for a certain class of weakly exact compact convex symplectic manifolds the above construction leads to a non-trivial partial quasi-morphism and a partial quasi-state. Then we discuss (partial) quasi-morphisms and (partial) quasi-states for non-compact convex manifolds. Finally, we discuss several applications in the spirit of Entov-Polterovich to that class of symplectic manifolds.