For a Liouville domain W satisfying c1β(W)=0, we propose in this note two
versions of symplectic Tate homology HβTβ(W)
and HβTβ(W) which are related by a canonical
map ΞΊ:HβTβ(W)βHβTβ(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 S1-equivariant Tate homology. For rational coefficients the
symplectic Tate homology HβTβ(W) has the
fixed point property and is therefore isomorphic to H(W;Q)βQ[u,uβ1], where Q[u,uβ1] 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 ΞΊ is not
surjective and examples where it is not injective.Comment: 40 pages, 4 figures; v2: various improvement