Symplectic Tate homology

TL;DR

This paper introduces two versions of symplectic Tate homology for Liouville domains, explores their properties using geometric methods, and provides examples illustrating differences in their canonical maps.

Contribution

It proposes two new symplectic Tate homology theories, establishes their relation via a canonical map, and constructs examples showing the map's non-injectivity and non-surjectivity.

Findings

For rational coefficients, symplectic Tate homology has the fixed point property.

The homology is isomorphic to the tensor product of the domain's homology with Laurent polynomials.

Examples demonstrate the canonical map can fail to be injective or surjective.

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