Pseudo-rotations and Steenrod squares

TL;DR

This paper establishes that certain symplectic manifolds with specific homological properties cannot admit Hamiltonian pseudo-rotations without deforming the quantum Steenrod square, providing new restrictions on their existence.

Contribution

It proves that the quantum Steenrod square must be deformed in the presence of Hamiltonian pseudo-rotations on specific monotone symplectic manifolds, revealing new constraints.

Findings

Quantum Steenrod square deformation is necessary for pseudo-rotations.

Restrictions on pseudo-rotation existence based on homological conditions.

Method relies on equivariant pair-of-pants product-isomorphism techniques.

Abstract

In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.

Equations

$$N=N_M=[\mathbb{Z}:\mathrm{image}(c_1(TM):{\pi }_2(M)\to \mathbb{Z})].$$

$${\mathcal{Q}}_{-}=H_{\ast <2n}(M)\otimes \Lambda \subset QH(M,\Lambda ),$$

$$\mathcal{Q}\mathcal{S}([pt])=h^{2n}[pt].$$

$${⟨[pt],[pt],D⟩}_{\beta }$$

$$c([M],{\tilde{\varphi }}^{-1})=-c([pt],\tilde{\varphi }).$$

$$\overline{c}(a,\tilde{\varphi })=\underset{k\to \infty }{\lim }\frac{1}{k}c(a,{\tilde{\varphi }}^k).$$

$$\frac{1}{k}c([M],{\tilde{\varphi }}^k)>\overline{c}([M],\tilde{\varphi }).$$

$$\frac{1}{k}c([pt],{\tilde{\varphi }}^k)<\overline{c}([pt],\tilde{\varphi }).$$

$$c([pt],{\tilde{\psi }}^2)>2\cdot c([pt],\tilde{\psi }).$$

$$c([pt],{\tilde{\psi }}^2)\le 2\cdot c([pt],\tilde{\psi })$$

$${\mathcal{A}}_H:{\tilde{\mathcal{L}}}_{pt}^{\min }M\to \mathbb{R}$$

$${\mathcal{A}}_H(\overline{x})={\int }_0^{1}H(t,x(t))-{\int }_{\overline{x}}\omega ,$$

$${\Lambda }_{\mathbb{K}}={\Lambda }_{M,\text{min},\mathbb{K}}=\mathbb{K}[q^{-1},q]]$$

$$\nu (\sum a_j{q}^j)=j_0\cdot A_M,$$

$${\mathcal{A}}_H(\sum {\lambda }_j{\overline{x}}_j)=\max \{-\nu ({\lambda }_j)+{\mathcal{A}}_H({\overline{x}}_j)\},$$

$${\mathcal{E}}_{+}(F)={\int }_0^1\underset{M}{\max }(F_t)dt.$$

$$C{F}^{\ast }(H{)}^I=C{F}^{\ast }(H{)}^{<b}/C{F}^{\ast }(H{)}^{<a}.$$

$$c(\alpha ,(C,\mathcal{A}))=\inf \{a\in \mathbb{R}|\alpha \in \mathrm{im}(H(C{)}^{<a}\to H(C))\}\in \mathbb{R}\cup \{-\infty \}.$$

$$c({\alpha }_M,H;\mathcal{D})=c(PSS({\alpha }_M),(CF(H;\mathcal{D}),{\mathcal{A}}_{H;\mathcal{D}})),$$

$$c({\alpha }_M,H)\in \mathrm{Spec}(H).$$

$$|c({\alpha }_M,F)-c({\alpha }_M,G)|\le \mathcal{E}(F-G),$$

$$c({\alpha }_M\ast {\alpha }_M^{\prime },F\#G)\le c({\alpha }_M,F)+c({\alpha }_M^{\prime },G),$$

$${\mathcal{A}}_F(P+hQ)=\max \{{\mathcal{A}}_F(P),{\mathcal{A}}_F(Q)\}.$$

$$P+hQ=\sum ({\lambda }_j+h{\mu }_j){\overline{x}}_j.$$

$$\nu ({\lambda }_j+h{\mu }_j)=\min \{\nu ({\lambda }_j),\nu ({\mu }_j)\}$$

$${\lambda }_j+h{\mu }_j=\sum (a_l+h{b}_l)T^{c_l}.$$

$$\oplus H{F}_{\ast }^{\mathrm{loc}}(\tilde{\varphi },\overline{x})$$

$${\mathcal{A}}_H(\sum {\lambda }_j{y}_j)=\max \{-\nu ({\lambda }_j)+{\mathcal{A}}_H(y_j)\},$$

$${\mathcal{A}}_H(y_j)={\mathcal{A}}_H({\overline{x}}_{i(j)})$$

$$|{\mathcal{A}}_{H_1}({\overline{x}}_1)-{\mathcal{A}}_H(\overline{x})|<\delta \ll {ϵ}_0,$$

Full text

1. Introduction

This paper deduces obstructions, in terms of pseudo-holomorphic curves, to the existence of Hamiltonian pseudo-rotations. The latter are special Hamiltonian diffeomorphisms of symplectic manifolds, characterized by having the minimal possible number of periodic points, of all integer periods.

The classical notion of pseudo-rotations of the two-sphere, or the two-disk, has appeared in the work of Anosov and Katok [1] (see also Fayad and Katok [10]) and was further investigated extensively in the field of conservative dynamics. Indeed, the simplest example of a pseudo-rotation is an irrational point in the Hamiltonian $S^{1}={\mathbb{R}}/{\mathbb{Z}}$ action on $S^{2}$ with Hamiltonian (a constant multiple of) the standard height function. However, [1, 10] construct, by means of the conjugation method, examples of pseudo-rotations of $S^{2}$ with different dynamical properties than those of such irrational rotations. For example they admit precisely three ergodic measures: two fixed points, and the area measure.

Recent years saw a renewed interest in pseudo-rotations considered from the point of view of symplectic rigidity phenomena. For instance, Barney Bramham proved in [3, 4] that all Hamiltonian pseudo-rotations of the closed disk can be $C^{0}$ approximated by a sequence of conjugates of rational rotations, and that they are $C^{0}$ rigid in a suitable sense, provided that their rotation number is sufficiently Liouvillean. In the recent seminal paper [15] by Ginzburg and Gürel, the $C^{0}$-rigidity result of Bramham, as well as other results regarding the dynamics of Hamiltonian pseudo-rotations, were established for complex projective spaces of all dimensions.

This paper, as well as [8], takes a different point of view, considering pseudo-rotations to be strong counter-examples to the Conley conjecture. From this perspective, a conjecture of Chance and McDuff, arising from [18], asserts that the existence of such counter-examples, and hence that of pseudo-rotations, must imply the existence of non-trivial algebraic counts of pseudo-holomorphic spheres in the manifold. Here we provide an instance of such an implication, ruling out in particular the existence of pseudo-rotations on a closed monotone symplectic manifold of dimension $2n$ with minimal Chern number $n+2$ or greater. Further extensions of our results have very recently appeared in [7, 27].

2. Setup and main results

In this paper, unless otherwise specified, we work with a closed monotone symplectic manifold $(M,\omega)$ of dimension $2n,$ and rescale the symplectic form so that $[\omega]=\kappa\cdot c_{1}(TM)$ on the image of the Hurewicz map $\pi_{2}(M)\to H_{2}(M;{\mathbb{Z}})$ for $\kappa=2.$ Recall that the minimal Chern number of $(M,\omega)$ is the index

[TABLE]

For a Hamiltonian diffeomorphism $\phi\in\operatorname{\mathrm{Ham}}(M,\omega),$ we denote by $\mathrm{Fix}(\phi)$ the set of contractible fixed points of $\phi,$ and by $x^{(k)}$ for $x\in\mathrm{Fix}(\phi)$ its image under the inclusion $\mathrm{Fix}(\phi)\subset\mathrm{Fix}(\phi^{k}).$ Contractible means that the homotopy class of the path $\alpha(x,\phi)=\{\phi^{t}_{H}(x)\}$ for a Hamiltonian $H\in C^{\infty}([0,1]\times M,{\mathbb{R}})$ generating $\phi$ as the time-one map $\phi^{1}_{H}=\phi$ of its Hamiltonian flow, is trivial. This class does not depend on the choice of Hamiltonian by a classical argument in Floer theory.

We say that a Hamiltonian diffeomorphism $\phi\in\operatorname{\mathrm{Ham}}(M,\omega)$ is a ${\mathbb{K}}={\mathbb{F}}_{2}$ Hamiltonian pseudo-rotation if:

1. (i)

It is perfect, that is for all iterations $k\geq 1$ of $\phi,$ $\mathrm{Fix}(\phi^{k})=\mathrm{Fix}(\phi)$ is finite. In other words, $\phi$ admits no simple periodic orbits of order $k>1.$ 2. (ii)

For each $x\in\mathrm{Fix}(\phi),$ the dimension of the local Floer homology of $\phi$ at $x$ satisfies $\dim_{{\mathbb{K}}}HF^{\operatorname{\mathrm{loc}}}(\phi,x)\geq 1,$ and furthermore, 3. (iii)

$N(\phi^{k},{\mathbb{K}})=\sum_{x\in\mathrm{Fix}(\phi)}\dim_{{\mathbb{K}}}HF^{\operatorname{\mathrm{loc}}}(\phi^{k},x^{(k)})=\dim_{{\mathbb{K}}}H_{*}(M)$ for all $k\geq 1.$

Remark 1*.*

We observe that a perfect Hamiltonian diffeomorphism necessarily has no symplectically degenerate maxima (see [16]). Furthermore, if all the points in $\mathrm{Fix}(\phi^{k})$ are non-degenerate, for all $k\geq 1,$ then condition (ii) is automatically satisfied, and all iterations are admissible, that is $\lambda^{k}\neq 1$ for all eigenvalues $\lambda\neq 1$ of $D(\phi)_{x}$. Furthermore, by the Smith inequality in local Floer homology [6, 28], conditions (i) and (iii) imply, for iterations of the form $k=2^{m},$ the stronger statement that for all $x\in\mathrm{Fix}(\phi),$ $\dim_{{\mathbb{K}}}HF^{\operatorname{\mathrm{loc}}}(\phi^{k},x^{(k)})=\dim_{{\mathbb{K}}}HF^{\operatorname{\mathrm{loc}}}(\phi,x).$ Moreover, [26, Theorem A] suggests that when a Hamiltonian diffeomorphism has a finite number of periodic points, then a condition like (iii) should be satisfied. Showing this would bridge the gap between the initial Chance-McDuff conjecture (see for example [16]) and the main result of this note, Theorem A. Finally, we include condition (ii) for compatibility with the literature: we do not use it below. We refer to [15] for further discussion of dynamics of Hamiltonian pseudo-rotations in higher dimensions.

We call a symplectic manifold strongly uniruled if there exists a non-trivial three point genus-zero Gromov-Witten invariant $\left<[pt],a,b\right>_{\beta},$ for $\beta\in H_{2}(M,{\mathbb{Z}})\setminus\{0\}.$ By [18, Lemma 2.1], $(M,\omega)$ is not strongly uniruled if and only if the $\Lambda$-linear subspace

[TABLE]

where $\Lambda$ is the minimal Novikov field of $(M,\omega),$ with quantum variable $q$ of degree $(-2N),$ is an ideal in the quantum homology ring $QH(M,\Lambda).$ Recall that the quantum product in $QH(M,\Lambda)$ is a deformation of the intersection product on homology given by three-point genus-zero Gromov-Witten invariants. Alternatively $[pt]\ast r=0$ for all $r\in{\mathcal{Q}}_{-}.$ Note that in this case, $[pt]\ast[pt]=0$ in particular.

A generally different stronger notion than $[pt]\ast[pt]=0,$ is that the quantum Steenrod square ${\mathcal{QS}}([pt]),$ defined in [30], of the point class satisfy

[TABLE]

This corresponds to there being no quantum corrections when passing from the classical Steenrod square of the point class on $M$ to its quantum version. In this case we say that $M$ is not ${{\mathbb{Z}}/(2)}$-Steenrod uniruled.

Remark 2*.*

Observe that when $(M,\omega)$ is ${{\mathbb{Z}}/(2)}$-Steenrod uniruled, then by a Gromov compactness argument there exists a $J$-holomorphic curve through each point of $M.$ Furthermore, recall from [31] that setting the quantum variable to be of cohomological degree $2N,$ $h$ to be of degree $1,$ and considering cohomological degree on the homology classes, ${\mathcal{QS}}([pt])$ must be of degree $2\deg([pt])=4n.$ Hence if $N>2n=\dim(M),$ then $(M,\omega)$ is automatically not ${{\mathbb{Z}}/(2)}$-Steenrod uniruled. By the same token, if $N=2n,$ then being ${{\mathbb{Z}}/(2)}$-Steenrod uniruled is equivalent to $[pt]\ast[pt]\neq 0.$ In fact, in this case $[pt]\ast[pt]=q[M].$ We note that while the above choice of degrees was convenient for this remark, throughout the paper we work with homology and use different conventions for degrees.

Remark 3*.*

The main result of [25] implies that if $N>n,$ and $[\omega]$ lies in the lattice $H^{2}(M,{\mathbb{Z}})/{{\mathrm{torsion}}}\subset H^{2}(M,{\mathbb{R}}),$ then $(M,\omega)$ being ${{\mathbb{Z}}/(2)}$-Steenrod uniruled implies that $(M,\omega)$ is strongly uniruled. In fact, in this case there exists a Gromov-Witten invariant

[TABLE]

that does not vanish modulo $2$ for a suitable divisor class $D\in H_{2n-2}(M;{\mathbb{Z}}).$ Note that when this holds, by a degree count, we obtain $N=n+1.$ Therefore Theorem A implies that there is no ${\mathbb{F}}_{2}$ Hamiltonian pseudo-rotation for $N>n+1.$ Under the additional assumption that $\left<[\omega]^{n},[M]\right>$ is odd, one may prove the above statement by a straightforward adaptation of the proof of [30, Lemma 6.1].

Finally, we say that $(M,\omega)$ satisfies the Poincaré duality property, if for all $\widetilde{\phi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega),$ the following Poincaré duality identity of Hamiltonian spectral invariants (defined in Section 3.2 below) holds:

[TABLE]

It is well-known (see [9]) that $(M,\omega)$ with $N>n$ satisfy this property. The main result of this note is the following.

Theorem A**.**

Let $(M,\omega)$ be a closed monotone symplectic manifold satisfying the Poincaré duality property, and admitting an ${\mathbb{F}}_{2}$ Hamiltonian pseudo-rotation $\phi.$ Then $(M,\omega)$ is ${{\mathbb{Z}}/(2)}$-Steenrod uniruled.

In view of Remarks 2 and 3 above we conclude the following result pertaining to the Chance-McDuff conjecture.

Corollary 4**.**

Let $(M,\omega)$ be a closed monotone symplectic manifold with $N>n.$ If $N>n+1$ then $(M,\omega)$ does not admit ${\mathbb{F}}_{2}$ Hamiltonian pseudo-rotations. If $N=n+1,$ and $(M,\omega)$ admits an ${\mathbb{F}}_{2}$ Hamiltonian pseudo-rotation, then $(M,\omega)$ satisfies $[pt]\ast[pt]\neq 0,$ and in particular it is strongly uniruled.

Remark 5*.*

The author was made aware that new relations between pseudo-rotations and holomorphic curves were also found in recent work of Çineli, Ginzburg, and Gürel [8].

Remark 6*.*

The symplectic manifold $({\mathbb{C}}P^{n},\omega_{st})$ has $N=n+1,$ and verifies the hypothesis and the conclusion of the theorem separately. Formally speaking, this result seems to be generally new even when $\phi$ comes as the irrational rotation with respect to a Hamiltonian $S^{1}$-action with isolated fixed points (however it does not seem to give new examples in that case: see [17, 21, 5]). We remark that by a result of McDuff [18] all Hamiltonian $S^{1}$-manifolds are uniruled, the latter being defined with $m$-point genus [math] Gromov-Witten invariants with arbitrary $m\geq 3.$

The strategy of the proof of the main result is the direct comparison between the following two results. First, the following Lusternik-Shnirelman111also transcribed from Russian as Lyusternik, Lusternick, Ljusternik, and respectively Schnirelmann, Shnirel’man. type result was shown in [16]. Define for $a\in QH(M),$ $\widetilde{\phi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ the asymptotic spectral invariant by

[TABLE]

Recall that a symplectic manifold is called rational if the period group of the symplectic form is a discrete subgroup of ${\mathbb{R}}.$

Theorem B**.**

Let $(M,\omega)$ be rational, and ${\mathbb{K}}$ be a ground field. Suppose $\phi$ has isolated periodic points of each period, none of which is a symplectically degenerate maximum. Then for each $k\geq 1,$ and lift $\widetilde{\phi}$ of $\phi$ to $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega),$

[TABLE]

We observe that in particular, when $\phi$ is perfect, by [12, Theorem 1.18], none of the fixed points of $\phi$ are symplectically degenerate maxima, and hence Theorem B applies. Hence, in the case when $(M,\omega)$ satisfies the Poincaré duality property, we obtain by applying Theorem B to $\widetilde{\phi}^{-1},$ for $\phi$ perfect, that

[TABLE]

In particular there exists $m\geq 1,$ such that for $\widetilde{\psi}=\widetilde{\phi}^{2^{m}},$

[TABLE]

Second, we prove below the following statement.

Theorem C**.**

Let $\psi$ be an ${\mathbb{F}}_{2}$ Hamiltonian pseudo-rotation on $(M,\omega)$ that is not ${{\mathbb{Z}}/(2)}$-Steenrod uniruled. Then

[TABLE]

for each $\widetilde{\psi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ covering $\psi.$

The proof of Theorem A is now the combination of (1) and (2).

Acknowledgements

I thank Viktor Ginzburg, Başak Gürel, and Nicholas Wilkins, for useful conversations. I thank Dusa McDuff for explaining to me her work on Hamiltonian $S^{1}$-manifolds. Finally, I thank the referees for useful comments and suggestions that lead to improvements in the exposition. At the University of Montréal I was supported by an NSERC Discovery Grant and by the Fonds de recherche du Québec - Nature et technologies.

3. Preliminaries

We present general preliminaries on Hamiltonian Floer homology and spectral invariants, to be applied in two settings below: usual and equivariant. To keep the paper short, we refer to [24, 30, 31, 28, 26] for preliminaries related to equivariant Floer homology, related natural operations and structures, and to [20] for further details on Hamiltonian Floer homology.

We denote by $\mathcal{H}\subset C^{\infty}([0,1]\times M,{\mathbb{R}})$ the space of time-dependent Hamiltonians on $M,$ that vanish near [math] and $1,$ where $H_{t}(-)=H(t,-)$ is normalized to have zero mean with respect to $\omega^{n}.$ We shall consider $H\in{\mathcal{H}}$ as a $1$-periodic function on ${\mathbb{R}}\times M$ in the ${\mathbb{R}}$-coordinate. The time-one maps of isotopies $\{\phi^{t}_{H}\}_{t\in[0,1]}$ generated by time-dependent vector fields $X^{t}_{H},$ such that $\iota_{X^{t}_{H}}\omega=-d(H_{t}),$ are called Hamiltonian diffeomorphisms and form the group $\operatorname{\mathrm{Ham}}(M,\omega).$ For $H\in\mathcal{H}$ we call $\overline{H}\in\mathcal{H}$ the Hamiltonian $\overline{H}(t,x)=-H(t,\phi^{t}_{H}x)$ generates the flow assigning $t\in[0,1]$ the map $\phi^{t}_{\overline{H}}=(\phi^{t}_{H})^{-1},$ and for $F,G\in{\mathcal{H}}$ the Hamiltonian $F\#G(t,x)=F(t,x)+G(t,(\phi^{t}_{F})^{-1}x)$ generates the path $\{\phi^{t}_{F}\phi^{t}_{G}\}_{t\in[0,1]}.$ Recall that homotopic Hamiltonian isotopies relative to endpoints give naturally isomorphic graded filtered Floer complexes. The universal cover $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ is constructed as the space of such homotopy classes, and carries a natural group structure. Further, for $H\in{\mathcal{H}}$ we set $H^{(k)}(t,x)=kH(kt,x)\in{\mathcal{H}}.$ If the flow of $H$ generates $\widetilde{\phi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega),$ then $H^{(k)}$ generates $\widetilde{\phi}^{k}.$

Finally, let $\mathcal{J}(M,\omega)$ be the space of $\omega$-compatible almost complex structures on $M.$

3.1. Hamiltonian Floer homology.

Consider $H\in\mathcal{H}.$ Let $\mathcal{L}_{pt}M$ be the space of contractible loops in $M.$ Let $c_{M}:\pi_{1}(\mathcal{L}_{pt}M)\cong\pi_{2}(M)\to 2N_{M}\cdot{\mathbb{Z}},$ be the surjection given by $c_{M}(A)=2\left<c_{1}(M,\omega),A\right>.$ Let $\widetilde{\mathcal{L}}^{\min}_{pt}M=\widetilde{\mathcal{L}_{pt}}\times_{c_{M}}(2N_{M}\cdot{\mathbb{Z}})$ be the cover of $\mathcal{L}_{pt}M$ associated to $c_{M}.$ The elements of $\widetilde{\mathcal{L}}^{\min}_{pt}M$ can be considered to be equivalence classes of pairs $(x,\overline{x})$ of $x\in{\mathcal{L}}_{pt}M$ and its capping $\overline{x}:{\mathbb{D}}\to M,$ $\overline{x}|_{\partial{\mathbb{D}}}=x.$ Of course $x$ is determined by $\overline{x}.$ The symplectic action functional

[TABLE]

is given by

[TABLE]

that is well-defined by monotonicity: $[\omega]=\kappa\cdot c_{M}.$ Assuming that $H$ is non-degenerate, that is the graph ${\mathrm{graph}}(\phi^{1}_{H})=\{(\phi^{1}_{H}(x),x)\,|\,x\in M\}$ intersects the diagonal $\Delta_{M}\subset M\times M$ transversely, the generators over the base field ${\mathbb{K}}$ of the Floer complex $CF(H;J)$ are the lifts $\widetilde{\mathcal{O}}(H)$ to $\widetilde{\mathcal{L}}^{\min}_{pt}M$ of $1$-periodic orbits $\mathcal{O}(H)$ of the Hamiltonian flow $\{\phi^{t}_{H}\}_{t\in[0,1]}.$ These are the critical points of $\mathcal{A}_{H},$ and we denote by $\operatorname{\mathrm{Spec}}(H)=\mathcal{A}_{H}(\widetilde{\mathcal{O}}(H))$ the set of its critical values. Choosing a generic time-dependent $\omega$-compatible almost complex structure $\{J_{t}\in\mathcal{J}(M,\omega)\}_{t\in[0,1]},$ and writing the asymptotic boundary value problem on maps $u:{\mathbb{R}}\times S^{1}\to M$ defined by the negative formal gradient on $\mathcal{L}_{pt}M$ of $\mathcal{A}_{H},$ the count of isolated solutions with signs determined by a suitable orienation scheme, modulo ${\mathbb{R}}$-translation, gives a differential $d_{H;J}$ on the complex $CF(H;J),$ $d^{2}_{H;J}=0.$ This complex is graded by the Conley-Zehnder index $CZ(H,\bar{x})$ [22, 23]. The Conley-Zehnder index has the property that the action of the generator $A=2N_{M}$ of $2N_{M}\cdot{\mathbb{Z}}$ has the effect $CZ(H,\bar{x}\#A)=CZ(H,\bar{x})-2N_{M},$ and it is normalized to be equal to $n$ at a maximum of a small autonomous Morse Hamiltonian. Its homology $HF_{*}(H)$ does not depend on the generic choice of $J.$ Moreover, considering generic families interpolating between different Hamiltonians $H,H^{\prime},$ and writing the Floer continuation map, where the negative gradient depends on the ${\mathbb{R}}$-coordinate we obtain that $HF_{*}(H)$ in fact does not depend on $H$ either. While $CF_{*}(H,J)$ is finite-dimensional in each degree, it is worthwhile to consider its completion in the direction of decreasing action. In this case it becomes a free graded module of finite rank over the Novikov field

[TABLE]

with $q$ being a variable of degree $(-2N_{M}).$ This field carries a non-Archimedean valuation $\nu:\Lambda_{{\mathbb{K}}}\to{\mathbb{R}}\cup\{+\infty\}$ given by $\nu(0)=\infty,$ and

[TABLE]

where $A_{M}=\kappa\cdot N_{M},$ and $j_{0}=\min\{j\;|\;a_{j}\neq 0\}.$ It satisfies the properties:

1. (1)

$\nu(x)=+\infty$ if and only if $x=0,$ 2. (2)

$\nu(xy)=\nu(x)+\nu(y)$ for all $x,y\in\Lambda_{{\mathbb{K}}},$ 3. (3)

$\nu(x+y)\geq\min\{\nu(x),\nu(y)\},$ for all $x,y\in\Lambda_{{\mathbb{K}}}.$

Moreover, we extend ${\mathcal{A}}_{H}$ to a non-Archimedean filtration on $CF(H,J)$ by

[TABLE]

for a $\Lambda_{{\mathbb{K}}}$-basis of $CF(H,J)$ consisting of capped orbits $\overline{x}_{j}\in\widetilde{{\mathcal{O}}}(H).$

Recall, following [29], that for a field $\Lambda$ with non-Archimedean valuation $\nu,$ a function $l:C\to{\mathbb{R}}\cup\{-\infty\}$ on a finite-dimensional $\Lambda$-module $C,$ is called a non-Archimedean filtration (function), if it satisfies the following properties:

1. (1)

$l(x)=-\infty$ if and only if $x=0,$ 2. (2)

$l(\lambda x)=l(x)-\nu(\lambda)$ for all $\lambda\in\Lambda,x\in C,$ 3. (3)

$l(x+y)\leq\max\{l(x),l(y)\},$ for all $x,y\in C.$

We call a complex $(C,d)$ with $C$ a finite-dimensional $\Lambda$-module with a non-Archimedean filtration $l$ filtered if $l(dy)\leq l(y)$ for all chains $y$ and strict if $l(dy)<l(y)$ for all chains $y\neq 0.$ It is straightforward to see that $CF(H;J)$ with ${\mathcal{A}}_{H}$ is a strict filtered complex over $\Lambda_{{\mathbb{K}}}.$

Furthermore, for $a\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H)$ the subspace $CF(H,J)^{<a}$ spanned by all generators $\bar{x}$ with $\mathcal{A}_{H}(\bar{x})<a$ forms a subcomplex with respect to $d_{H;J},$ and its homology $HF(H)^{<a}$ does not depend on $J.$ Arguing up to $\epsilon,$ one can show that a suitable continuation map sends $HF(H)^{<a}$ to $HF(H^{\prime})^{<a+\mathcal{E}_{+}(H-H^{\prime})},$ for

[TABLE]

It shall also be useful to define $\mathcal{E}_{-}(F)=\mathcal{E}_{+}(-F),\;\mathcal{E}(F)=\mathcal{E}_{+}(F)+\mathcal{E}_{-}(F).$ Moreover, for an admissible action window, that is an interval $I=(a,b),$ $a<b,$ $a,b\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H),$ we define the Floer homology of $H$ in this window $HF^{*}(H)^{I}$ as the homology of the quotient complex

[TABLE]

Finally, one can show that for each $a\in{\mathbb{R}},$ $HF(H)^{<a}$ as well as $HF(H)^{I}$ for an admissible action window, depends only on the class $\widetilde{\phi}_{H}$ of the path $\{\phi^{t}_{H}\}_{t\in[0,1]}$ in the universal cover $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ of the Hamiltonian group of $M.$ When we wish to emphasize the coefficient field in the definition of Floer homology, we write $HF(H,\Lambda_{{\mathbb{K}}})^{<a}.$ We shall mostly work with ${\mathbb{K}}={\mathbb{F}}_{2}$ and ${\mathbb{K}}=\mathcal{K}={\mathbb{F}}_{2}[h^{-1},h]],$ for a formal variable $h,$ and write $\Lambda=\Lambda_{{\mathbb{K}}}$ if the choice of ${\mathbb{K}}$ is clear.

In the case when $H$ is degenerate, we consider a perturbation $\mathcal{D}=(K^{H},J^{H}),$ with $K^{H}\in\mathcal{H},$ such that $H^{\mathcal{D}}=H\#K^{H}$ is non-degenerate, and $J^{H}$ is generic with respect to $H^{\mathcal{D}},$ and define the complex $CF(H;\mathcal{D})=CF(H^{\mathcal{D}};J^{H})$ generated by $\widetilde{\mathcal{O}}(H;\mathcal{D})=\widetilde{\mathcal{O}}(H^{\mathcal{D}}),$ and filtered by the action functional $\mathcal{A}_{H;\mathcal{D}}=\mathcal{A}_{H^{\mathcal{D}}}.$ An admissible action window $I=(a,b)$ for $H,$ remains admissible for all $K^{H}$ sufficiently $C^{2}$-small, and the associated homology groups $HF(H;\mathcal{D})^{I}$ are canonically isomorphic for all $K^{H}$ sufficiently $C^{2}$-small. Hence $HF(H)^{I}$ is defined as the colimit of the associated indiscrete groupoid.

3.2. Spectral invariants.

Given a filtered complex $(C,\mathcal{A}),$ to each homology class $\alpha\in H(C)$, denoting by $H(C)^{<a}=H(C^{<a}),$ $C^{<a}=\mathcal{A}^{-1}(-\infty,a),$ we define a spectral invariant by

[TABLE]

For $(C,\mathcal{A})=(CF(H;\mathcal{D}),\mathcal{A}_{H;\mathcal{D}})$ we denote $c(\alpha,H;\mathcal{D})=c(\alpha,(C,\mathcal{A})).$ Furthermore, one can obtain classes $\alpha$ in the Hamiltonian Floer homology by the PSS isomorphism. This lets us define spectral invariants by:

[TABLE]

for $\alpha_{M}\in QH(M).$ From the definition it is clear that the spectral invariants do not depend on the almost complex structure term in $\mathcal{D}.$ Moreover, if $H$ is non-degenerate, we may choose the Hamiltonian term in $\mathcal{D}$ to vanish identically, and denote the resulting invariants by $c(-,H).$ Moreover, by [2, Section 5.4] spectral invariants remain the same under extension of coefficients, hence below we do not have to specify the Novikov field $\Lambda$ that we work over. Spectral invariants enjoy numerous useful properties that hold for rational symplectic manifolds, the relevant ones of which we summarize below:

1. (1)

spectrality: for each $\alpha_{M}\in QH(M)\setminus\{0\},$ and $H\in\mathcal{H},$

[TABLE] 2. (2)

non-Archimedean property: $c(-,H;\mathcal{D})$ is a non-Archimedean filtration function on $QH(M),$ as a module over the Novikov field $\Lambda$ with its natural valuation. 3. (3)

continuity: for each $\alpha_{M}\in QH(M)\setminus\{0\},$ and $F,G\in\mathcal{H},$

[TABLE] 4. (4)

triangle inquequality: for each $\alpha_{M},\alpha^{\prime}_{M}\in QH(M),$ and $F,G\in\mathcal{H},$

[TABLE] 5. (5)

invariance: $c(\alpha_{M},H)$ depends only on the element $\widetilde{\phi}=[\{\phi^{t}_{H}\}_{t\in[0,1]}]$ in the universal cover $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ generated by $H.$

We remark that by the continuity property, the spectral invariants are indeed defined for all $H\in\mathcal{H}$ and all the properties above apply in this generality. Finally, by the invariance property, we shall consider the spectral invariants as functions on $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega),$ and shall sometimes denote for brevity by ${\mathcal{A}}_{\widetilde{\phi}},$ the action functional ${\mathcal{A}}_{H}$ for a certain Hamiltonian generating $\widetilde{\phi}.$

Below, we will sometimes be using the ground field $\mathcal{K}={\mathbb{F}}_{2}[h^{-1},h]],$ for a formal variable $h,$ for the Novikov field. This is the field of fractions of the ring ${\mathcal{L}}={\mathbb{F}}_{2}[[h]].$ In this case, we require the following observation.

Lemma 7**.**

Let $F\in{\mathcal{H}}$ be a non-degenerate Hamiltonian. Consider elements $P\in CF(F,J;\Lambda_{{\mathbb{F}}_{2}})\subset CF(F,J;\Lambda_{\mathcal{K}}),$ and $Q\in CF(F,J;\Lambda_{{\mathcal{L}}})\subset CF(F,J;\Lambda_{\mathcal{K}}).$ Then

[TABLE]

The proof of this lemma is essentially immediate, since writing $P=\sum\lambda_{j}\overline{x}_{j}$ with $\lambda_{j}\in\Lambda_{{\mathbb{F}}_{2}}$ in the basis $\{\overline{x}_{j}\}$ and $hQ=\sum h\mu_{j}\overline{x}_{j}$ with $\mu_{j}\in\Lambda_{{\mathcal{L}}},$ we have

[TABLE]

The lemma now follows from (3) and

[TABLE]

for each $j.$ We prove the latter statement for a given fixed $j.$ Writing $\lambda_{j}=\sum a_{l}T^{\sigma_{l}},$ $a_{l}\in{\mathbb{F}}_{2},$ $\mu_{j}=\sum b_{l}T^{\sigma_{l}},$ $b_{l}\in{\mathcal{L}}$ with the sets $\{\sigma_{l}>c\,|\,a_{l}\neq 0\},$ $\{\sigma_{l}>c\,|\,b_{l}\neq 0\},$ finite for all $c\in{\mathbb{R}},$ we get

[TABLE]

As $a_{l}+hb_{l}=0$ if and only if $a_{l}=0$ and $b_{l}=0,$ because $a_{l}\in{\mathbb{F}}_{2},$ $hb_{l}\in h{\mathcal{L}},$ the conclusion now follows.

3.3. Homotopy-canonical filtered complexes and local Floer homology

The theme of this section is that the situation of filtered Hamiltonian Floer homology of $H\in{\mathcal{H}}$ with $\mathrm{Fix}(\phi^{1}_{H})$ finite is very similar to that of the non-degenerate case, once we allow a finite-dimensional graded vector space of generators to be supported at each $x\in\mathrm{Fix}(\phi^{1}_{H}).$ We first prove the following result, which is a chain-level enhancement of [15, Lemma 2.2]. Its proof below relies on homological perturbation techniques and constitutes a Novikov-field version of the canonical $\Lambda^{0}$-complexes from [26]. While we formulate it for monotone symplectic manifolds to simplify notation, it extends almost verbatim to the case of rational symplectic manifolds.

Theorem D**.**

Let $(M,\omega)$ be a closed monotone symplectic manifold. Consider the class $\widetilde{\phi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ of the Hamiltonian flow $\{\phi^{t}_{H}\}_{t\in[0,1]}$ of $H\in{\mathcal{H}},$ with $\mathrm{Fix}(\phi^{1}_{H})$ finite. For a ground field ${\mathbb{K}},$ there is a homotopy-canonical complex $C(H)$ over the Novikov field $\Lambda_{{\mathbb{K}}}$ on the action-completion of

[TABLE]

the sum running over all capped one-periodic orbits $\overline{x}\in\widetilde{\mathcal{O}}(H),$ that is free and graded over $\Lambda_{{\mathbb{K}}},$ and is strict, that is ${\mathcal{A}}_{H}(d_{H}y)<{\mathcal{A}}(y)$ for all $y\in C(H),$ with respect to the non-Archimedean action-filtration ${\mathcal{A}}_{H}$ on $C(H)$ defined as follows:

[TABLE]

[TABLE]

for a $\Lambda$-basis $\{y_{j}\}$ of $C(H)$ given by a basis $\{y_{i}\;|\;i(j)=i\}$ of $HF^{\operatorname{\mathrm{loc}}}_{*}(\widetilde{\phi},\overline{x}_{i(j)}),$ for a choice of lifts $\{\overline{x}_{i}\}$ of $\mathrm{Fix}(\phi)=\{x_{i}\}$ to capped orbits $\widetilde{{\mathcal{O}}}(H).$ Furthermore the filtered homology $HF(H)^{<a}$ is given by $HF(C(H)^{<a}),$ $C(H)^{<a}=({\mathcal{A}}_{H})^{-1}\,(-\infty,a),$ for all $a\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H).$ In particular $HF(H)=H(C(H),d_{H})\cong QH(M;\Lambda_{{\mathbb{K}}}).$

Moreover all the Floer-theoretical operations that we consider extend naturally to this chain-level setting. We shall only need one instance of this naturality formulated in Proposition 9 below. We now outline a proof of Theorem D.

Proof of Theorem D.

We let $H_{1}$ be a sufficiently $C^{2}$-small perturbation of a Hamiltonian $H.$ It generates a Hamiltonian diffeomorphism $\phi_{1},$ whose contractible fixed points separate into clusters $\mathrm{Fix}(\phi_{1},x)\subset\mathrm{Fix}(\phi_{1})$ of fixed points of $\phi_{1}$ near $x\in\mathrm{Fix}(\phi).$ Furthermore the corresponding capped periodic orbits $\widetilde{\mathcal{O}}(H_{1})$ split into clusters $\widetilde{\mathcal{O}}(H_{1},\overline{x})$ of orbits near $\overline{x}\in\widetilde{\mathcal{O}}(H)$ in $\widetilde{{\mathcal{L}}}^{\min}_{pt}M,$ in such a way that the evaluation of a periodic orbit at [math] is an isomorphism of sets $\widetilde{\mathcal{O}}(H_{1},\overline{x})\to\mathrm{Fix}(\phi_{1},x)$ for each capping $\overline{x}$ of $\alpha(x,H),$ and for each $A\in(2N_{M}){\mathbb{Z}},$ $\widetilde{\mathcal{O}}(H_{1},\overline{x}\#A)=\widetilde{\mathcal{O}}(H_{1},\overline{x})\#A.$ Following [23, 13] we observe that the elements of $\widetilde{\mathcal{O}}(H_{1},\overline{x})$ and the Floer trajectories between them form a complex $CF_{*}^{\operatorname{\mathrm{loc}}}(H_{1},\overline{x}),$ with differential $d^{\operatorname{\mathrm{loc}},\overline{x}}$ whose homology $HF_{*}^{\operatorname{\mathrm{loc}}}(\widetilde{\phi},\overline{x})$ depends only on the class $\widetilde{\phi}\in\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ of the path $\{\phi^{t}_{H}\}_{t\in[0,1]}$ (in particular it does not depend on the choice of $H_{1}$ given that it is sufficiently close to $H$). Furthermore, by [14, 16, 19], as well as [11, 28], there is a crossing energy $2\epsilon_{0}>0$ depending only on $H$ and $x\in\mathrm{Fix}(\phi),$ such that each $H_{1}$ Floer trajectory $u(s,t)$ asymptotic to $\alpha(x_{1},H_{1})$ as $s\to{-\infty}$ where $x_{1}\in\mathrm{Fix}(\phi_{1},x)$ is either contained in a small isolating neighborhood $U_{x}$ of $\alpha(x,H),$ in $S^{1}\times M,$ and hence connects two elements of $\widetilde{\mathcal{O}}(H_{1},\overline{x})$ for each capping $\overline{x}$ of $\alpha(x,H),$ or has energy $E(u)=E_{H_{1}}(u)\geq 2\epsilon_{0}.$ Finally, the actions ${\mathcal{A}}_{H_{1}}(\overline{x}_{1})$ and indices $CZ(H_{1},\overline{x}_{1})$ for $\overline{x}_{1}\in\widetilde{\mathcal{O}}(H_{1},\overline{x})$ satisfy

[TABLE]

[TABLE]

This implies that the Floer differential of the Floer complex $CF(H_{1};{\mathcal{D}})$ splits as

[TABLE]

with $d^{\operatorname{\mathrm{loc}}}=\oplus_{\overline{x}\in{\mathcal{O}}(H)}\,d^{\operatorname{\mathrm{loc}},\overline{x}}$ suitably completed, and ${\mathcal{A}}_{H_{1}}(dy)\leq{\mathcal{A}}_{H_{1}}(y)-\epsilon_{0}$ for all chains $y\in CF(H_{1};{\mathcal{D}}).$ Observe that $d^{\operatorname{\mathrm{loc}}}$ is a differential of $CF(H_{1};{\mathcal{D}})$ as a $\Lambda$-module.

Now choose subspaces $X_{\ast}^{\operatorname{\mathrm{loc}}}(\overline{x})\subset\ker(d^{\operatorname{\mathrm{loc}},\overline{x}})\subset CF^{\operatorname{\mathrm{loc}}}_{\ast}(H_{1},\overline{x})$ such that

[TABLE]

for all $k\in{\mathbb{Z}},$ and the inclusion

[TABLE]

is a quasi-isomorphism. Choose projections

[TABLE]

similarly compatible with the Novikov action, such that

[TABLE]

[TABLE]

for homotopies

[TABLE]

again compatible with the Novikov action, and satisfying ${\Theta_{\overline{x}}}^{2}=0$ and ${\mathcal{A}}(\Theta_{\overline{x}}y)<{\mathcal{A}}(y)+\delta$ for all chains $y.$ We refer to [26, Section 6] for a detailed construction of such $\Theta_{\overline{x}},\iota_{\overline{x}},\pi_{\overline{x}}$ in a similar setting, as the two local settings can be identified by fixing one capping $\overline{x}_{0},$ making the choices for it, and then extending to all other cappings $\overline{x}$ by compatibility with the Novikov action.

The homological perturbation formulae now yield a differential $d_{H}$ on the $\Lambda$-module $C_{*}(H)$ given by completing $\oplus X^{\operatorname{\mathrm{loc}}}_{\ast}$ with respect to the filtration ${\mathcal{A}}_{H_{1}},$ as well as injection $\overline{\iota}:C_{*}(H)\to CF_{*}(H_{1};{\mathcal{D}}),$ projection $\overline{\pi}:CF_{*}(H_{1};{\mathcal{D}})\to C_{*}(H),$ and homotopy $\overline{\Theta}:CF_{*}(H_{1};{\mathcal{D}})\to CF_{*+1}(H_{1};{\mathcal{D}})$ satisfying

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

for all chains $y.$ Furthermore

[TABLE]

for all chains $y\in C(H).$ In particular for each $y,z\in X^{\operatorname{\mathrm{loc}}}_{\ast}(\overline{x})$ the coefficient $\left<d_{H}(y),z\right>$ vanishes. Now define the filtration $\mathcal{A}_{H}$ on $C(H)$ by setting

[TABLE]

for all $y\in X^{\operatorname{\mathrm{loc}}}_{\ast}(\overline{x}),$ and extending it naturally to the completion. This definition is easily seen to coincide with the description in Theorem D. Then

[TABLE]

for all chains $y\in C(H)\setminus\{0\}.$ In particular $(C(H),d_{H})$ is strict with respect to ${\mathcal{A}}_{H}.$ Finally, it is easy to see by a filtration argument that the complexes $C(H)$ obtained from different sufficiently small perturbations $H_{1}$ of $H$ are all filtered-isomorphic: indeed the continuation maps between them yield chain maps that induce isomorphisms on the local homology groups, and split similarly to $d$ in (5). Hence the homological perturbation formulae produce maps on the complexes $C(H)$ that are of the form $\gamma^{\operatorname{\mathrm{loc}}}+\Gamma,$ where ${\mathcal{A}}_{H}(\Gamma(y))\leq{\mathcal{A}}_{H}(y)-\epsilon_{0}$ for all chains $y,$ and $\gamma^{\operatorname{\mathrm{loc}}}$ is a linear isomorphism preserving the filtration. The conclusion follows.

Now we use monotonicity, or in fact rationality, to prove that the filtered complex $(C(H),{\mathcal{A}}_{H})$ calculates the filtered Floer homology of $H.$ By rationality, and $\phi$ having isolated contractible fixed points, we obtain that there is $\epsilon_{1}>0$ such that

[TABLE]

for each two capped orbits $\overline{x},\overline{y}$ of $H$ with distinct actions. Now for $a\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(H),$ by the independence of $C(H)$ up to filtered isomorphism on $H_{1},$ we choose $H_{1}$ such that $d(a,\operatorname{\mathrm{Spec}}(H))>\delta$ and hence ${\mathcal{A}}_{H_{1}}({\mathcal{O}}(H_{1},\overline{x}))$ is contained either in $(-\infty,a)$ or in $(a,\infty)$ for all $\overline{x}\in{\mathcal{O}}(H).$ Therefore $HF(H)^{<a}=HF(H_{1})^{<a}$ by definition, and $HF(H_{1})^{<a}=H(CF(H_{1})^{<a})=H(CF(H)^{<a})$ by construction.

Finally, it is not hard to see that the construction of $C(H)$ does not depend on the choices made in the proof up to filtration-preserving chain-homotopies. ∎

Moreover, the proof of Lemma 7 adapts tautologically to prove the following.

Lemma 8**.**

Let $F\in{\mathcal{H}}$ be a Hamiltonian with $\mathrm{Fix}(\phi^{1}_{F})$ finite. Let $P$ be in $C(F;\Lambda_{{\mathbb{F}}_{2}})\subset CF(F;\Lambda_{\mathcal{K}}).$ Let $Q$ be in $C(F;\Lambda_{{\mathcal{L}}})\subset C(F;\Lambda_{\mathcal{K}}).$ Then

[TABLE]

4. Proof of Theorem A

Consider the filtered Floer homology $HF(\widetilde{\phi},\Lambda)^{<c},$ $\Lambda=\Lambda_{{\mathbb{F}}_{2}},$ where $c\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(\widetilde{\phi}),$ and the filtered ${{\mathbb{Z}}/(2)}$-equivariant Tate Floer homology $\widehat{HF}(\widetilde{\phi}^{2})^{<c}$ for level $c\in{\mathbb{R}}\setminus\operatorname{\mathrm{Spec}}(\widetilde{\phi}^{2}).$ The spectral invariant $c(a,\widetilde{\phi}),$ for $a\in QH(M)\setminus\{0\},$ is defined as

[TABLE]

For $a\in QH(M,\Lambda_{\mathcal{K}})$ let the ${{\mathbb{Z}}/(2)}$-equivariant Tate spectral invariant $\widehat{c}(a,\widetilde{\phi}^{2})$ be

[TABLE]

where $PSS_{{{\mathbb{Z}}/(2)}}$ is the equivariant PSS isomorphism introduced in [31].

By the construction of Seidel [24] and Wilkins [31], combined with the action estimates as in [28] for example, the equivariant pair-of-pants product precomposed with Kaledin’s quasi-Frobenius map yields an injective map, for $\mathcal{K}={\mathbb{F}}_{2}[h^{-1},h]],$ and $c\in{\mathbb{R}},$

[TABLE]

Furthermore, by [31], this map commutes with vertical maps

[TABLE]

[TABLE]

where $PSS,$ $PSS_{{{\mathbb{Z}}/(2)}}$ may also be replaced by the inverse $PSS$ maps, and the horizontal map:

[TABLE]

given by the quantum Steenrod square from [30]. From this commutativity, we immediately obtain the inequality

[TABLE]

We shall require one feature of the equivariant PSS map $PSS_{{{\mathbb{Z}}/(2)}}$: for each chain $z\in CM(f,\Lambda_{\mathcal{K}})$ in the Morse complex of a Morse function $f$ on $M,$ we have the identity

[TABLE]

where $hR(z)=\sum_{j\geq 1}h^{j}PSS_{{{\mathbb{Z}}/(2)},j}(z)$ is a collection of terms of higher order in $h.$ We require the following analogue of this statement in the case of isolated, but possibly degenerate, contractible fixed points. It is deduced from its non-degerate version by adapting the proof of Theorem D above to the equivariant Tate complex, as in [26, Section 6] for the $\Lambda^{0}$ case, and inducing maps on the homotopy-canonical complexes from the PSS and equivariant PSS maps respectively.

Proposition 9**.**

Consider $H\in{\mathcal{H}}$ with $\mathrm{Fix}(\phi^{1}_{H^{(2)}})$ finite. Then the PSS isomorphism construction induces quasi-isomorphisms $PSS:C(M;\mathcal{D},\Lambda_{{\mathbb{K}}})\to C(H^{(2)})$ and $PSS_{{{\mathbb{Z}}/(2)}}:C(M;\mathcal{D},\Lambda_{\mathcal{K}})\to\widehat{C}(H^{(2)})=C(H^{(2)})\otimes_{\Lambda_{{\mathbb{K}}}}\Lambda_{\mathcal{K}}.$ Furthermore, for compatible choices of auxiliary data we have the relation:

[TABLE]

for $hR=\sum_{j\geq 1}h^{j}PSS_{{{\mathbb{Z}}/(2)},j}$ is a collection of terms of higher order in $h.$ In fact $PSS_{{{\mathbb{Z}}/(2)},j}$ is defined over $\Lambda_{{\mathbb{K}}}.$

Furthermore, we have the following key technical result specific to pseudo-rotations.

Proposition 10**.**

For each class $z\in QH(M,\Lambda_{\mathcal{K}}),$ and $H\in{\mathcal{H}},$ with $\phi^{1}_{H}$ a pseudo-rotation, the chain representatives of $PSS(z)$ in $C(H)$ and $PSS_{{{\mathbb{Z}}/(2)}}(z)$ in $\widehat{C}_{{{\mathbb{Z}}/(2)}}(H^{(2)})$ are unique and coincide and their action levels coincide with their spectral invariants: $c(z,\widetilde{\phi})$ and $\widehat{c}(z,\widetilde{\phi}^{2}).$

Remark 11*.*

We shall use Proposition 10 to relate $c([pt],H^{(2)})$ and $\widehat{c}([pt],H^{(2)}).$

Proof of Proposition 10.

First of all, by comparing the dimensions of $C(H)$ and $QH(M,\Lambda_{{\mathbb{F}}_{2}})$ over $\Lambda_{{\mathbb{F}}_{2}}$ condition (iii) of the pseudo-rotation $\phi^{1}_{H}$ implies that the differential $d_{H}$ on $C(H)$ vanishes. Hence each homology class $a\in C(H)\setminus\{0\}$ has a unique representative. This is in particular true for $a=PSS(z)$ for $z\in QH(M,\Lambda_{{\mathbb{F}}_{2}}),$ and the compatibility of $C(H)$ with the action filtration shows that $\mathcal{A}_{H}(PSS(z))=c(z,H).$ The same holds for coefficients in the field extension $\Lambda_{\mathcal{K}}$ of $\Lambda_{{\mathbb{F}}_{2}}.$

A similar, yet slightly more complicated, argument applies to the equivariant Tate case. Indeed, by condition (i) of a pseudo-rotation, the homotopy-canonical equivariant Tate complex $\widehat{C}(H^{(2)})$ is given as the action-completion of

[TABLE]

where each such capped $\overline{x}$ is a recapping of an iterated capped orbit $\overline{y}^{(2)}$ of $H,$ and its differential, which is $\Lambda_{\mathcal{K}}$-linear, is given by

[TABLE]

where ${\mathcal{A}}_{H^{(2)}}(\widehat{D}_{H^{(2)}}(y))\leq{\mathcal{A}}_{H^{(2)}}(y)-\epsilon_{0}$ for all chains $y.$ We claim that $\widehat{d}_{H^{(2)}}=0.$ Indeed, by the equivariant PSS isomorphism, $(\widehat{C}(H^{(2)}),\widehat{d}_{H^{(2)}})$ is quasi-isomorphic as a $\Lambda_{\mathcal{K}}$-module to $QH(M,\Lambda_{\mathcal{K}}).$ However, the dimension of $\widehat{C}(H^{(2)})$ over $\Lambda_{\mathcal{K}}$ is given by $N(\phi^{2},{\mathbb{F}}_{2})=\dim_{\mathcal{K}}QH(M,\Lambda_{\mathcal{K}})$ by condition (iii) of a pseudo-rotation. This finishes the proof. ∎

Remark 12*.*

In fact, for pseudo-rotations (6) becomes an equality, that is for $y\in QH(M,\Lambda_{\mathcal{K}}),$ we have the identity of spectral invariants:

[TABLE]

Indeed, consider $y\in QH(M),$ and let $c=c(y,\widetilde{\phi})$ be its critical level. Since the equivariant pair of pants product is an isomorphism on local Floer homology (see [28, 26]), it gives an isomorphism of homologies in action windows

[TABLE]

for all $\epsilon$ sufficiently small. This shows that $2c$ is the critical level of ${\mathcal{QS}}(y)$ for $\widetilde{\phi^{2}}.$ Indeed, otherwise the non-zero image of the chain representative of $y$ in the leftmost homology would go to zero in the rightmost homology, for $\epsilon$ sufficiently small.

Proof of Theorem C.

From estimate (6) we obtain the bound

[TABLE]

However,

[TABLE]

Identifying between the class $[pt]$ and its chain level representative, by choosing a Morse function on $M$ with unique minimum, which represents the point class, by Proposition 10 the following identities hold

[TABLE]

[TABLE]

Furthermore, by Proposition 9 combined with Lemma 8,

[TABLE]

This finishes the proof. ∎