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Symplectic Tate homology. (English) Zbl 1337.57059

Authors’ 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 \(\underrightarrow {H}\underleftarrow {T}(W;\mathbb {Q})\) has the fixed point property and is therefore isomorphic to \(H(W;\mathbb {Q}) \otimes_{\mathbb{Q}} \mathbb {Q}[u,u^{-1}]\), 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.

MSC:

57R19 Algebraic topology on manifolds and differential topology
57T99 Homology and homotopy of topological groups and related structures
53D05 Symplectic manifolds (general theory)
57R17 Symplectic and contact topology in high or arbitrary dimension
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