Pseudo-rotations and Steenrod squares revisited
Egor Shelukhin












In this note we prove that if a closed monotone symplectic manifold M admits a Hamiltonian , which may be degenerate, then the quantum Steenrod square of the cohomology class Poincaré dual to the point must be deformed. This result gives restrictions on the existence of s, implying a form of uniruledness by pseudo-holomorphic spheres, and generalizes a recent result of the author. The new component in the proof consists in an elementary calculation with capped periodic orbits.





[2010]53D40, 37J10, 37J45

Pseudo-rotations and Steenrod squares revisited
    Egor Shelukhin
    
===============================================





§ SETUP










In this paper, (M,) denotes a closed monotone symplectic manifold  of dimension 2n, with the symplectic form rescaled so that [] = 2 c_1(TM) on the image of the Hurewicz map π_2(M) → H_2(M;). For a Hamiltonian diffeomorphism ϕ∈(M,), we denote by (ϕ) the set of its contractible fixed points, and by x^(k) for x ∈(ϕ) its image under the inclusion (ϕ) ⊂(ϕ^k). Contractible means that the homotopy class α(x,ϕ) of the path α(x,H) = {ϕ^t_H(x)} for a Hamiltonian H ∈H⊂[0,1] × M, generating ϕ as the time-one map ϕ^1_H = ϕ of its Hamiltonian flow, is trivial[This class does not depend on the choice of Hamiltonian by a classical argument in Floer theory.]. Here H = ∩_t ∈ [0,1](I_t), I_t(H) = ∫_M H(t,-) ^n. 

To an isolated fixed point x one associates (cf. <cit.>) a local cohomology group HF_(ϕ,x), which is naturally -graded. If we choose a capping x of α(x,H), we obtain a -graded version, HF_^∗(H,x) = HF_^∗(ϕ,x). It depends only on the class ϕ of {ϕ^t_H}_t ∈ [0,1] in the universal cover (M,), and the capped orbit x.

We say that ϕ∈(M,) is an _2 Hamiltonian  if:


  *  It is perfect, that is for all iterations k ≥ 1 of ϕ, (ϕ^k) = (ϕ). In other words, ϕ admits no simple periodic orbits of order k >1.




  *  for all iterations k≥ 1, 
    N(ϕ^k,_2) = ∑_x ∈(ϕ)__2 HF^*_(ϕ^k,x^(k)) = __2 H^*(M; _2).
 



Observe that a perfect Hamiltonian diffeomorphism necessarily has no symplectically degenerate maxima (see <cit.>). Furthermore, if ϕ is strongly non-degenerate, that is all the points in (ϕ^k) are non-degenerate, for all k ≥ 1, then HF_(ϕ,x) ≅, and all iterations are admissible, that is λ^k ≠ 1 for all eigenvalues λ≠ 1 of D(ϕ)_x. By the Smith inequality in local Floer homology <cit.>, conditions <ref> and <ref> imply for iterations k = 2^m, the stronger statement that for all x ∈(ϕ), __2 HF^(ϕ^k,x^(k)) = __2 HF^(ϕ, x). Moreover, <cit.> indicates that when a Hamiltonian diffeomorphism has a finite number of periodic points, then a condition like <ref> should be satisfied. Showing this in general would bridge the gap between the initial Chance-McDuff conjecture (see for example <cit.>) and the main result of this note, Theorem <ref>. Dynamics of Hamiltonian s in higher dimensions were recently studied by Ginzburg and Gürel <cit.>. We refer thereto for further discussion of this interesting notion, and survey results more closely related to the subject of this paper in Section <ref>.




We remind the reader that the minimal Chern number of (M,) is the index 
    N = N_M = [:image(c_1(TM): π_2(M) →)].
 The notion of mean-index, introduced in symplectic topology in <cit.> is described as follows. For a Hamiltonian H ∈H generating ϕ∈(M,) and capped periodic orbit x of H, we set 
    Δ(H,x) = Δ(ϕ_H,x) = lim_k →∞1/k CZ(ϕ^k, x^(k)),
 where x^(k) is x iterated k times, which is indeed a capped periodic orbit of a Hamiltonian generating ϕ^k. The limit exists, since the Conley-Zehnder index comes from a quasi-morphism Sp(2n,) → (see <cit.>). The mean-index satisfies the following properties that we use below:


	
  * homogeneity: Δ(ϕ^k,x^(k)) = k ·Δ(ϕ,x), for all k ∈_>0.
	
  * recapping: Δ(ϕ,x# A) = Δ(ϕ,x) - 2<c_1(TM),A>, A ∈ H_2^S(M;).
	
  * distance to index: CZ(ϕ,x) ∈ [Δ(ϕ,x) - n, Δ(ϕ,x) + n], 
	
	
	
	
  * support of local Floer cohomology: HF_^r(ϕ,x) = 0, unless r ∈ [Δ(ϕ,x) - n, Δ(ϕ,x) + n].
	
	
  * symplectically degenerate maxima: by definition, x is not a SDM of ϕ, if HF_^r(ϕ,x) = 0, unless r ∈ [Δ(ϕ,x) - n, Δ(ϕ,x) + n). 



	Observe that, by the symmetry of the Conley-Zehnder index and a simple duality argument, if the reversal x^(-1) of x is not an SDM of ϕ^-1, then HF_^r(ϕ,x) = 0, unless r ∈ (Δ(ϕ,x) - n, Δ(ϕ,x) + n].




Finally, for a quantum cohomology class μ∈ QH^*(M;Λ__2) ∖{0} and a Hamiltonian H ∈ with isolated contractible fixed points (ϕ^1_H), we recall that the Hamiltonian spectral invariant c(μ,H) of μ is carried by a capped 1-periodic orbit x of H, if in a suitable sense x is a lowest action term in a highest minimal action representative of the image PSS_H(μ) of μ under the PSS isomorphism <cit.> from the quantum cohomology QH^*(M,Λ__2) to the filtered Floer cohomology of the Hamiltonian H. Keeping in mind the duality between Floer homology and Floer cohomology <cit.>, we refer the reader to <cit.> for a detailed description of this notion, recording only the following two facts:


	
  * spectrality: for (M,) rational, in particular monotone, for each non-zero class μ∈ QH^*(M;Λ__2), and H ∈ with #(ϕ^1_H) < ∞, c(μ,H) is carried by at least one capped 1-periodic orbit x of H.
	
	
  * contribution to local Floer cohomology: if μ is homogeneous of degree k, and x carries c(μ, H), then HF^k_(H,x) ≠ 0. 









§ RESULTS


We call a symplectic manifold strongly uniruled if there exists a non-trivial 3-point genus-0 Gromov-Witten invariant < [pt], a, b>_β, for β∈ H_2(M,) ∖{0}. By <cit.>, if (M,) is not strongly uniruled, then the quantum square μ∗μ = 0 for the degree 2n cohomology class μ Poincaré dual to the point class. A generally different stronger notion than μ∗μ = 0, is that the quantum Steenrod square  QS(μ), defined in <cit.>, of the volume class μ satisfy 
    QS(μ) = h^2nμ.
 Note that QS(μ) is equal to the classical Steenrod square Sq(μ) = h^2nμ plus quantum corrections coming from certain pseudo-holomorphic curves. When (<ref>) does not hold, we say that M is -Steenrod uniruled. The main result of this note is the following.




































Let (M,) be a closed monotone symplectic manifold admitting an _2 Hamiltonian pseudo-rotation ϕ. Then (M,) is -Steenrod uniruled. 























	We observe following <cit.> that when (M,) is -Steenrod uniruled, then there exists a J-holomorphic curve through each point of M for each -compatible almost complex structure J. Hence, Theorem <ref> provides a geometric obstruction to the existence of pseudo-rotations. Other arguments described in <cit.> provide obstructions to the existence of pseudo-rotations for monotone manifolds with N>n, and rule it out completely for N>n+1. We remark that the existence of pseudo-rotations was ruled out in <cit.> (and references therein), in a strong way, by proving the Conley conjecture, for manifolds such that (A) · c_1(A) ≤ 0 for all spherical homology classes A ∈ H_2^S(M;). 
 


Theorem <ref> was proven by the author in <cit.> under the additional assumption that (M,) satisfies the Poincaré duality property for Hamiltonian spectral invariants (which holds in particular whenever the minimal Chern number of (M,) satisfies N > n). Under the assumption that ϕ is strongly non-degenerate, Theorem <ref> was also proved by Çineli, Ginzburg, and Gürel in <cit.>, using different additional arguments extending <cit.>. A related, but quite different, result relating the existence of pseudo-rotations to pseudo-holomorphic spheres was shown in <cit.>. 





















§ PROOF


The proof of the main result relies on the following observations. First, as mentioned in Remark <ref>, no fixed point of ϕ or ϕ^-1 is a symplectically degenerate maximum (see <cit.>). In particular, if the capping x of a contractible fixed point x ∈(ϕ) carries a cohomology class μ of Conley-Zehnder index n in HF^n(ϕ) ≅ QH^2n(M,Λ__2) for a lift ϕ of ϕ to the universal cover (M,) of (M,), then its mean-index Δ = Δ(ϕ,x) satisfies n< Δ + n. Hence 
    Δ(ϕ,x) > 0.














Second, the following result was proven in <cit.> specifically in the setting of a pseudo-rotation assuming that (<ref>) holds. Here μ∈ QH^2n(M,Λ__2) denotes the cohomology class Poincaré dual to the point.





Let ψ be an _2  of (M,) that is not -Steenrod uniruled. Then 
    c(μ,ψ^2) ≥ 2 · c(μ, ψ)
 for each ψ∈(M,) covering ψ.




















We proceed to the proof of the main result, which follows a calculation from <cit.>.

 

Choose H ∈ H, such that the path {ϕ^t_H }_t ∈ [0,1] represents the class ϕ lifting ϕ. By the pigeonhole principle, there exists a fixed point x ∈(ϕ), and an increasing sequence k_i such that c(μ, H^(r_i)) for r_i =  2^k_i is carried by a capping y_i of the 1-periodic orbit of the r_i-iterated Hamiltonian H^(r_i) corresponding to x^(r_i). By taking a power of ϕ, we can assume that r_1 = 1, and set y= y_1. Write y_i as a recapped iteration of y, that is 
    y_i = y^(r_i)# A_i.
 We claim that for r_i large, (A_i) ≤ 0, and c_1(A_i) > 0 contradicting  monotonicity. Indeed, write _i for the action functional of H^(r_i), and :=_1. Then by (<ref>) and Theorem <ref>, 
    r_i (y) - (A_i) = _i(y_i) = c(μ, H^(r_i)) ≥ r_i c(μ, H) = r_i (y).
Hence 
    (A_i) ≤ 0.
 However, we know that y_i carries c(μ, H^(r_i)), hence Δ(H^(r_i),y_i) ∈ (0,2n) and also Δ(H,y) ∈ (0,2n). Hence r_i Δ(H,y) > 2n for r_i large enough, and 
    2n > Δ(H^(r_i), y_i) = r_i Δ(H, y) - 2c_1(A_i).
 Therefore 
    c_1(A_i) > 0.
 
	





§ ACKNOWLEDGEMENTS

I thank Paul Seidel for a very useful discussion, and Erman Çineli, Viktor Ginzburg, and Başak Gürel for sharing their preprint <cit.> with me. At the University of Montréal I was supported by an NSERC Discovery Grant, by the Fonds de recherche du Québec - Nature et technologies, and by the Fondation Courtois.































































































abbrv



