The equivariant pair-of-pants product in fixed point Floer cohomology
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 more of them than \(\phi\). In the same intuitive spirit (and with the localization theorem for equivariant cohomology in mind, one can think of tensoring with \(K((h))\) as throwing away the fixed points of \(\phi^2\) which are not fixed points of \(\phi\), leading to the isomorphism (1.2). Indeed, in a sense, the proofs ultimately reduce to such very basic considerations. Before one can get to that point, however, a map has to be defined which allows one to compare the two sides of isomorphism. We construct an equivariant refinement of the pair-of-pants product, which is a homomorphism of \(\mathbb Z/2\)-graded \(Kh\)-modules, \(H^*(\mathbb Z/2; {CF}^*(\phi) \otimes {CF}^*(\phi)) \longrightarrow {HF}^*_{eq}(\phi^2)\). Here \({CF}^*(\phi)\) is the chain complex underlying \({HF}^*(\phi)\). We take its tensor product with itself (as a chain complex), equip it with the involution that exchanges the two factors, and consider the associated group cohomology \(H^*(\mathbb Z/2; {CF}^*(\phi) \otimes {CF}^*(\phi))\). We will see, as part of the elementary formalism of group cohomology, that this depends only on \({HF}^*(\phi)\). Our main theorem is: The equivariant pair-of-pants product becomes an isomorphism after tensoring with \(K((h))\) on both sides. The isomorphism (1.2) of ungraded \(K((h))\)-modules is a purely algebraic consequence of this statement.''
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