  For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two
versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$
and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical
map $\kappa \colon \underrightarrow{H}\underleftarrow{T}(W) \to
\underleftarrow{H}\underrightarrow{T}(W)$. 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 $\underrightarrow{H}\underleftarrow{T}(W)$ has the
fixed point property and is therefore isomorphic to $H(W;\mathbb{Q}) \otimes
\mathrm{Q}[u,u^{-1}]$, where $\mathbb{Q}[u,u^{-1}]$ 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 $\kappa$ is not
surjective and examples where it is not injective.
