Fredholm theory and transversality for the parametrized and for the $S_1$-invariant symplectic action

Abstract

We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the _L_2-gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic _S_1-invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define _S_1-equivariant Floer homology. As an intermediate result of independent interest, we generalize Aronszajn’s unique continuation theorem to a class of elliptic integro-differential inequalities of order two.