  This is the first of two papers devoted to showing how the rich algebraic
formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be
used to define higher algebraic structures on the symplectic cohomology of open
symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to
define a homotopy extension of the well-known Lie bracket on symplectic
cohomology. Apart from discussing applications to the existence of closed Reeb
orbits, we outline how the $L_{\infty}$-structure is conjecturally related via
mirror symmetry to the extended deformation theory of complex structures.
