Symplectic Tate homology

For a Liouville domainW satisfying c(1)(W) = 0, we propose in this note two versions of symplectic Tate homology, <(H)under right arrow>(T) under left arrow-(W) and (H) under left arrow(T) under left arrow (W), which are related by a canonical map kappa: <(H)under right arrow>(T) under left arrow -> -(H) under left arrow(T) under left arrow (W). Our geometric approach to Tate homology uses the moduli space of finite energy gradient flow lines of the Rabinowitz action functional for a circle in the complex plane as a classifying space for S-1-equivariant Tate homology. For rational coefficients the symplectic Tate homology <(H)under right arrow>(T) under left arrow-(W; Q) has the fixed point property and is therefore isomorphic to H(W; Q)circle times(Q) Q[u, u(-1)], where Q[u, u(-1)] is the ring of Laurent polynomials over the rationals. Using a deep theorem of Goodwillie, we construct examples of Liouville domains where the canonical map. is not surjective and examples where it is not injective.