The equivariant pair-of-pants product in fixed point Floer cohomology
type: Item
instance of: scholarly article
title: The equivariant pair-of-pants product in fixed point Floer cohomology
zbMATH Open document ID: 1331.53119
author: Paul Seidel
published in: Geometric and Functional Analysis. GAFA
publication date: 2015-07-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.2639
full work available at URL: http://hdl.handle.net/1721.1/106971
review text: From the authors' introduction: ``This paper concerns the Floer cohomology of symplectic automorphisms, and its behaviour under iterations: more specifically, when passing to the square of a given automorphism (one expects parallel results for odd prime powers, but they are beyond our scope here). The concrete situation is as follows. Let \(\phi\) be an exact symplectic automorphism of a Liouville domain \(M\) (there are some additional conditions on \(\phi\).) The Floer cohomology \({HF}^*(\phi)\) is a \(\mathbb Z/2\)-graded \(K\)-vector space. Here and throughout the paper, \(K = F_2\) is the field with two elements. The Floer cohomology of \(\phi^2\) carries additional structure, namely an action of \(\mathbb Z/2\). Denote the invariant part by \({HF}^*(\phi^2)^{\mathbb Z/2}\). From the viewpoint of applications, our most significant result is the following Smith-type inequality of total dimensions: \(\dim {HF}^*(\phi^2)^{\mathbb Z/2} \geq \dim {HF}^*(\phi)\). This is not entirely new: under additional topological restrictions it has been previously proved by Hendricks. The proof involves an equivariant form of Floer cohomology, written as \({HF}^*_{eq}(\phi^2)\). This is a finitely generated \(\mathbb Z/2\)-graded module over \(Kh\), the ring of formal power series in one variable \(h\) (the variable has degree \(1\)). The information encoded in this equivariant theory can be viewed as a refinement of the previously mentioned \(\mathbb Z/2\)-action. What we obtain is a description of equivariant Floer cohomology after inverting \(h\), which means after tensoring with the ring \(K((h))\) of Laurent series. Namely, there is an isomorphism of ungraded \(K((h))\)-modules: \({HF}^*(\phi)((h)) = {HF}^*(\phi) \otimes K((h)) \cong {HF}^*_{eq}(\phi^2) \otimes_{Kh} K((h))\) (1.2). The first result follows from this by purely algebraic arguments. Naively it may not be surprising: if one thinks of Floer cohomology as a measure of fixed points, \(\phi^2\) clearly has…
reviewed by: Alexander Felshtyn
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6466330
MaRDI profile type: Publication
OpenAlex ID: W1867211284
arXiv ID: 1411.2639
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DOI: 10.1007/S00039-015-0331-X
description: scientific article