Symplectic Tate homology

Abstract

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 $\kappa :\underrightarrow{H}\underleftarrow{T}(W)\to \underleftarrow{H}\underrightarrow{T}(W)$. 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.

40 pages, 4 figures; v2: various improvements

References in corpus (1)

• Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture

Cited by in corpus (2)

• Symplectic homology and the Eilenberg-Steenrod axioms
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures