Monograph

#### Degree

Doctor of Philosophy

Mathematics

#### Supervisor

Pinsonnault, Martin

#### Abstract

In this thesis, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For Hamiltonian circle actions, we prove that the centralizers are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group $\Ham(M)$. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on $4$-manifolds, and on the Chen-Wilczy\'nski smooth classification of $\Z_n$-actions on Hirzebruch surfaces.

#### Summary for Lay Audience

The study of symplectic manifolds is motivated by classical mechanics. Consider a physical system such as a simple pendulum, or a spring with a mass attached. Associated to such a system is a space called the phase space which encapsulates every possible state that the system can attain. Such a space comes naturally equipped with a non-degenerate two form called a symplectic form and the time evolution of a particle corresponds to flowing along the symplectic gradient of the Hamiltonian of the system.

By Darboux's theorem, all symplectic manifolds are locally alike and hence there are no local invariants to distinguish symplectic manifolds. Global invariants of symplectic manifolds can be obtained by investigating the homotopy type of mapping spaces (such as symplectomorphism groups or symplectic embedding spaces) related to the symplectic structure. In his seminal paper, M. Gromov provided one such invariant called the Gromov width. This paper also shows that studying the topology of mapping spaces such as the space of symplectic embeddings of a ball into a symplectic manifold, or similarly, the group of self maps that preserve the symplectic structure, gives us key symplectic insights about the symplectic manifold.

In general, investigating symplectomorphism groups or embedding spaces are very hard problems. However, in dimension 4, due to certain special features of $J$-holomorphic curves we have more tools at our disposal to understand such questions of a global nature.

It is very natural for physical systems to have symmetries. These symmetries of a system correspond to Hamiltonian groups actions on the phase spaces. In this setting, we are interested in the time evolution of particles that preserve these symmetries (or group actions). The maps that preserve these symmetries are called equivariant symplectomorphims. These symmetries can be continuous symmetries like the action of circle $S^1$ or discrete symmetries like the n-th roots of unity.

In this thesis, we combine the theory of holomorphic curves together with moment map techniques to study the topology of spaces of all equivariant symplectomorphisms of certain 4-dimensional manifolds endowed with Hamiltonian actions of either the circle or a finite cyclic group.