This dissertation is comprised of two papers studying the topology of certain spaces of symplectic embeddings. The first paper shows how given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result which claims that given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, one can prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible.The second paper proves how given two 2n-dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, a certain map from the n-torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map at the level of homology with mod 2 coefficients
