Operad representations in Morse theory and Floer homology

Two equivariant operad representations are constructed using moduli spaces of graph and surface maps. The representations induce natural algebraic structures in the cohomology of smooth compact manifolds and monotone symplectic manifolds.; An operad, X, of finite labeled graphs is considered. The edge-labels are smooth vector fields on a manifold M. We construct a homotopy representation of the differential graded operad {dollar}Csb*(X){dollar} in the following way. Associated to every family of labelings, there are moduli spaces of maps from the graph to M. These moduli spaces are used to define endomorphisms of the Morse chain complex. The construction is equivariant under the action of each graph's automorphism group. Natural operations in the Morse cohomology of M are indexed by the homology of these groups.; The second example uses an operad of surface structures, {dollar}{lcub}cal J{rcub}.{dollar} Fix a monotone symplectic manifold M. Consider the space of conformal structures on a punctured surface {dollar}Sigma{dollar} together with perturbations of the standard Cauchy-Riemann equations. An operad structure is defined by joining surfaces at puncture points. Families of perturbed equations induce moduli spaces of maps from the surface to M. By counting points in these moduli spaces, we construct an equivalent homotopy representation of {dollar}Csb*({lcub}cal J{rcub}).{dollar} In this way, natural operations in the Floer cohomology of M are indexed by the homology of surface diffeomorphism groups.