Quasimorphisms on symplectic manifolds

Abstract

In this study, the main goal is to get an understanding of the concept of quasimorphisms. Quasimorphisms exist as the nontrivial elements of the second bounded cohomology group. The existence of quasimorphisms for certain groups of diffeomorphisms is an active research area among symplectic geometers. Main motivation of these studies focus on understanding the geometry of such groups. Basically, if there is a nontrivial quasimorphism on such diffeomorphism groups, then one concludes that the diameter of the group is infinite.The first part of the study is devoted to bounded cohomology. Roughly, bounded cohomology is the cohomology of chain complexes where cochains are bounded functions into reals, integers etc. Since the seminal paper of Gromov, various studies have been made on this subject. In this thesis, we recall the definitions from the usual cohomology of spaces, following Hatcher. Next, we examine the bounded cohomology of discrete groups in more detail. As an example, we analyze the theorem of Trauber proving that the bounded cohomology groups of amenable groups are trivial, folowing Bucher.The second bounded cohomology group is in the heart of this study. Namely, nontrivial quasimorphisms lie in the kernel of the comparison map between bounded and usual cohomology groups in the second degree. Roughly, quasimorphisms are homomorphisms up to an error. Since perfect groups do not admit nontrivial homomorphism into reals (or any abelian group), it is natural to seek nontrivial quasimorphisms defined for such groups. As an example, we include the result of Brooks in detail in which an infinite family of quasimorphisms are defined in the free group on two generators.In the final chapter, we recall basics of symplectic topology. The study of classical mechanical systems such as the planetary system is based on configuration spaces of particles satisfying certain equations (Hamilton equations). The configuration spaces are symplectic manifolds and there have been large number of studies on symplectic manifolds in the last three decades. In this thesis, we first review basics of symplectic topology and focus on the group of Hamiltonian diffeomorphisms of symplectic manifolds. A classical result due to Banyaga shows that this group is perfect and hence, quasimorphisms on this group become important. Recently, Polterovich and his students and Gambaudo and Ghys construct quasimorphisms for certain symplectic manifolds. We finish the thesis by stating these results. Their proofs include spectral invariants of Floer theoretic backgrounds and are beyond the scope of this study.