Hamiltonian circle actions and π_{1} of the space of Hamiltonians

Haydeé Herrera^{∗} and Rafael Herrera^{†}^{‡}

Abstract

We study the fundamental group of the space of Hamiltonians of certain compact symplectic 6-manifolds admitting Hamiltonian circle actions. In fact, we prove that the existence of a Hamiltonian S^{1} action provides and non-trivial element of the fundamental group of the space of Hamiltonians of such spaces.

1 Introduction

In the study of symplectic structures on manifolds, the topology of the Hamiltonian group plays a central role [4, 7, 2, 8] and, in particular, its fundamental group [10]. Let (M, Ω) be a closed symplectic manifold. Let Ham(M, Ω) denote the group of Hamiltonian symplectomorphims endowed with the C^{∞} topology, given as follows: a diffeomorphism ϕ belongs to Ham(M, Ω) if

1. ϕ^{∗}Ω = Ω,

2. there exists an isotopy ϕ_{t} : M → M, t ∈ [0, 1], with ϕ_{0} = id_{M} and ϕ_{1} = ϕ, and

∗Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA. E-mail: haydeeh@camden.rutgers.edu

†Mathematics Department, Princeton University, Fine Hall, Washington Road, NJ 08544 USA, and The City University of New York, Mathematics Department, College of Staten Island, 2800 Victory Blvd., Staten Island, NY 10314, USA. E-mail: rherrera@math.princeton.edu

‡Partially supported by a Guggenheim Fellowship, NSF grant DMS-0204002, and PSC-CUNY grant #60031-33-34.

3. there exists a time dependent Hamiltonian function H_{t} : M → ℝ such

that

Ω(·, \frac{∂ϕ_{t}}{∂t}) = d(H_{t} ∘ ϕ_{t}).

Since M is closed there is a unique Hamiltonian function F_{t} with the property that ∫_{M} F_{t}Ω^{n} = 0 for every t. Such a function will be called a normalized Hamiltonian.

The group π_{1}(Ham(M, Ω)) has been calculated only in very few cases, such as Riemann surfaces [3] or 4-dimensional ruled surfaces [4, 1, 2]. In the 6-dimensional case practically no information is available. For instance, Ham(ℝ^{2n}, dxdy) is contractible for n = 1, 2, while nothing is known for n = 3.

The Hofer norm [7] is defined as follows: for γ ∈ π_{1}(Ham(M, Ω)),

ν(γ) := inf_{F_{t}} ∫_{0}^{1}(max_{x} F_{t} − min_{x} F_{t})dt,

where the infimum is taken over all normalized Hamiltonian functions F_{t} that generate ϕ_{t}, and the maximum and minimum are taken over all the points x ∈ M. It is known that

ν(−γ) = ν(γ) and ν(γ + γ′) ≤ ν(γ) + ν(γ′),

where we have used additive notation, since π_{1}(Ham(M, Ω)) is an abelian group.

Let us define the positive and negative parts of ν separately,

ν_{+}(γ) = inf_{F_{t}} ∫_{0}^{1} max_{x} F_{t}(x)dt,
ν_{−}(γ) = inf_{F_{t}} ∫_{0}^{1} − min_{x} F_{t}(x)dt,

where the infimum is taken over all the normalized Hamiltonian functions that generate a loop representing γ. One can verify that

ν_{+}(γ) = ν_{−}(−γ) and ν(γ) ≥ ν_{+}(γ) + ν_{−}(γ).

Note that that equality holds in all the known examples. The calculation of ν_{+}(γ) is non-trivial even in the case M = S^{2} and γ is the class of the rotation with two fixed points (see [10]).

The main theorem of this note is the following.

Theorem 1. Let M = ℙ(E ⊕ℂ) where E is a complex line bundle and ℂ is the trivial line bundle over ℂℙ^{2}, with first Chern class c_{1}(E) = nρ, n = 0, 1, 2, 3, and ρ the generator of H^{2}(ℂℙ^{2}). Let M be endowed with the symplectic structure in the class of c_{1}(T M) + tρ, where t = 0 for n = 0, 1, 2 and 0 < t ≤ 1 for n = 3. Then

ν_{−}(kγ) ≥ k ≥ 0, k ∈ ℤ and n > 0,
ν_{−}(kγ) = 0,  if k is even and n = 0,
ν_{−}(kγ) = 1,  if k is odd and n = 0.

where γ is the loop of Hamiltonians given by a (global) S^{1}-action on M that rotates the fibers ℂℙ^{1} of M over ℂℙ^{2}. Then,

lim_{k→∞} \frac{ν_{−}(kγ)}{k} ≥ 1,

and γ is an element of infinite order in π_{1}(Ham(M, Ω)).

Remark. In fact, the only property of the action needed in the proof is that the fixed point set be the two canonical sections of ℙ(E ⊕ ℂ).

This type of lower bound on the norm and the asymptotic growth of a loop are more general than those in [10] since the manifolds are not necessarily monotonic. Moreover, we see that the existence of a Hamiltonian circle action produces an element of infinite order in π_{1}(Ham(M, Ω)), which may also be true in more general situations.

Acknowledgements. The first named author wishes to thank Loring Tu for his encouragement and valuable discussions, and the Institute for Pure and Applied Mathematics (UCLA) for its hospitality during the Symplectic Geometry and Physics Program. The second named author wishes to thank the Max-Planck-Institut für Mathematik for its hospitality and support.

2 Preliminaries and Quantum Cohomology

The approach we shall use to study π_{1}(Ham(M, Ω)) is by means of symplectic fibrations over S^{2} (cf. [12, 8]) and quantum cohomology.

2.1 Symplectic fibrations

Given a loop ϕ : S^{1} × M → M of Hamiltonian diffeomorphisms, define

P_{ϕ} = M × D_{+}^{2} ∪_{ϕ} M × D_{−}^{2}

by

\begin{matrix}M × ∂D_{+}^{2} & ⟶ & M × ∂D_{−}^{2} \\ (x, t) & ↦ & (ϕ_{t}(x), −t), \end{matrix}

where we consider the two dimensional disc D_{+}^{2} with its positive orientation. Thus, P_{ϕ} fibers over S^{2} = D_{+}^{2}∪D_{−}^{2} with fiber equal to M, where the orientation in S^{2} is the one induced from D_{+}^{2}. We denote this fibration by p : P_{ϕ} → S^{2}. The assumption that ϕ is a Hamiltonian loop implies that p is a symplectic fibration, i.e. there exists a closed 2-form τ on P_{ϕ} that extends the symplectic form of the fibers. Homotopic loops produce isomorphic symplectic fibrations and, in particular, if [ϕ_{t}] = 0 in π_{1}(Ham(M, Ω)), then P_{ϕ} ≅ M ×S^{2}. For a loop ϕ_{t} representing the class γ, we write P_{γ} or P_{ϕ}. We denote by V ert the kernel of p_{∗}.

There are two canonical classes attached to P_{ϕ}. One is the first Chern class of the vertical bundle c_{1}(V ert) denoted by c_{ϕ}, which is the extension of c_{1}(T M) to the total space P_{ϕ} of the fibration. The other class is an extension of [Ω], the class of the symplectic form on the fiber. Such an extension is possible due to the Hamiltonian nature of the loop [9]. In particular, we consider the unique class with the following properties,

u_{ϕ} |_{F} = Ω and u_{ϕ}^{n+1} = 0,

where 2n =dimM. Such a class is called the coupling class of the fibration. In this case, we observe that Ω_{ϵ} = u_{ϕ} + ϵp^{∗}τ is a symplectic form on P_{ϕ} for ϵ big enough, where τ is a two form on S^{2} with ∫_{S^{2}} τ = 1.

2.2 Quantum Homology

In order to find the bound for ν_{−}, we need to use the homomorphism defined by Seidel [12]

Ψ : π_{1}(Ham(M, Ω)) ⟶ QH_{∗}^{×},

where QH_{∗}^{×} denotes the group of units in the even part of the small Quantum homology of M with coefficients in a real Novikov ring. Namely, let Λ be the Novikov ring of the group H := H_{2}^{S}(M, ℝ)/ ∼, where H_{2}^{S}(M, ℝ) is the image of π_{2}(M) in H_{2}(M, ℝ), and B ∼ B′ if Ω(B) = Ω(B′) and c_{1}(T M) · B = c_{1}(T M) · B′. Define the small quantum homology ring by

QH_{∗}(M, Λ) = H_{∗}(M, ℝ) ⊗ Λ,

so that for any a, b ∈ H_{∗}(M) their product is defined by

a ∗ b = ∑_{B∈H}(a ∗ b)_{B} ⊗ e^{−B},

where (a ∗ b)_{B} ∈ H_{∗}(M) is the class such that for any other c ∈ H_{∗}(M)

(a ∗ b)_{B} · c = Φ_{B}^{M}(a, b, c).

Here Φ_{B}^{M}(a, b, c) denotes the Gromov-Witten invariant that counts the number of pseudo-holomorphic curves representing the class B and intersecting generic representatives of a, b, c ∈ H_{∗}(M). We can also define a grading by

deg(a ⊗ e^{−B}) = deg(a) − 2c_{1}(T M)B.

Roughly speaking, Ψ(γ) counts the number of pseudo-holomorphic sections of p : P_{ϕ} → S^{2}. In general, there is no canonical section of p. Hence, we define a generalized section to be a homology class σ_{ϕ} ∈ H_{2}(P_{ϕ}, ℝ) such that p_{∗}(σ_{ϕ}) = [S^{2}] and c_{ϕ}(σ_{ϕ}) = 0 = u_{ϕ}(σ_{ϕ}) if Ω and c_{1}(T M) are linearly independent. If they are linearly dependent, then we just require c_{ϕ}(σ_{ϕ}) = 0. Thus,

Ψ(γ) = ∑_{B∈H} a_{B} ⊗ e^{B},

where σ_{ϕ} − B ∈ H_{2}(P_{ϕ}, ℤ), and a_{B} ∈ H_{∗}(M) is the class such that for c ∈ H_{∗}(M)

a_{B} ·_{M} c = Φ_{σ_{ϕ}−B}^{P_{ϕ}}([M], [M], c).  (1)

Notice that the left hand side in equation (1) is the intersection product in M, while the right hand side is the Gromov-Witten invariant in P_{ϕ}.

Next, consider the following valuation map on QH_{∗}(M),

val : QH_{∗}(M) ⟶ ℝ

val(∑_{B∈H} a_{B} ⊗ e^{B}) = sup{Ω(B) | a_{B} ≠ 0}.

The relevant property we will use is the following [13]. If γ ∈ π_{1}(Ham(M, Ω)), then

ν_{−}(γ) ≥ val(Ψ(γ)).

3 Proof of the main theorem

Consider the manifold M = ℙ(E ⊕ ℂ), where E → ℂℙ^{2} is a complex line bundle with first Chern class c_{1}(E) = nρ, n = 0, 1, 2, 3, and ρ is the generator of H^{2}(ℂℙ^{2}). Here, ℂ denotes a trivial line bundle.

The manifold M fibers over ℂℙ^{2} with fiber equal to the sphere S^{2}, π : M → ℂℙ^{2}. Clearly, there are two sections, Z_{+} = ℙ(0 ⊕ ℂ) and Z_{−} = ℙ(E ⊕ 0), with normal bundles E|_{Z_{+}} and E^{∗}|_{Z_{−}} respectively. Let ϕ_{t} : M → M be a loop of Hamiltonians whose fixed point set is Z_{+} ∪ Z_{−}, and such that that π ∘ ϕ_{t} = π, i.e. ϕ_{t} descends to the identity in ℂℙ^{2}. For instance, consider the loop of Hamiltonians that rotate the fiber 2πθ degrees, θ ∈ [0, 1].

In this case, P_{ϕ} = M × D_{+}^{2} ∪_{ϕ} M × D_{−}^{2}, fibers over ℂℙ^{2} × S^{2}, Π : M → ℂℙ^{2} × S^{2}, and the fibration p : P_{ϕ} → S^{2} is the composition of Π with the projection pr_{S^{2}} onto the second factor,

p : P_{ϕ} → ℂℙ^{2} × S^{2} → S^{2}.

In fact, it is not hard to see that

P_{ϕ} ≅ ℙ((pr_{S^{2}}^{∗}H^{−1} ⊗ pr_{ℂℙ^{2}}^{∗}E) ⊕ ℂ),

where H^{−1} is the complex line bundle over S^{2} of degree −1.

Next, we give a description of c_{ϕ}, the first Chern class of the vertical tangent bundle to P_{ϕ} with respect to p, and of the coupling class u_{ϕ}. Notice that T P_{ϕ} splits as V ert(p) ⊕ p^{∗}T S^{2} as well as V ert(Π) ⊕ Π^{∗}T (ℂℙ^{2} × S^{2}). Therefore c_{ϕ} = c_{1}(V ert(p)) = c_{1}(V ert(Π)) + c_{1}(Π^{∗}T ℂℙ^{2}).

We have the following form η = c_{1}(O_{M}(1)) on M. It generates the cohomology of the fiber, η^{2} = c_{1}(E)η = nρη, and c_{1}(T M) = 3ρ + (2η − c_{1}(E)) = 2η + (3 − n)ρ. This class can be represented by a symplectic form [9], which we shall denote by Ω, for n = 0, 1, 2. In the case of n = 3, we consider the class η + tρ, where 0 < t ≤ 1, and a representative shall also be denoted by Ω.

Similarly, in P_{ϕ} there is a form η̂ that generates the cohomology of the fiber (of Π), and such that, η̂^{2} = c_{1}(H^{−1} ⊗ E)η̂ = (nρ − τ)η̂, and

c_{1}(V ert(Π)) = 2η̂ − (nρ − τ).

Thus,

c_{ϕ} = 2η̂ − (nρ − τ) + 3ρ = 2η̂ + (3 − n)ρ + τ.

Notice that η̂ is an extension of η to P_{ϕ}. Hence, the coupling form u_{ϕ} is equal to 2η̂ + (n − 3)ρ + κτ, where κ is the constant for which u_{ϕ}^{4} = 0. That is

κ = \frac{− ∫_{P_{ϕ}}(2η̂ + δρ)^{4}}{∫_{P_{ϕ}} 4(2η̂ + δρ)^{3}τ} ,

where δ = 3 − n for n = 0, 1, 2 and δ = t for n = 3. Notice that for n = 1, 2, 3, κ < 0.

Let σ_{−} denote the section over S^{2} × {x} given by ℙ(0 ⊕ ℂ) ⊂ P_{ϕ}|_{S^{2}×{x}}, x ∈ ℂℙ^{2}. Observe that c_{ϕ}(σ_{−}) = −1, for n = 0, 1, 2, 3. For n = 0, 1, 2, c_{1}(T M) = [Ω] and we need to find a section σ_{ϕ} such that c_{ϕ}(σ_{ϕ}) = 0. The second homology group H_{2}(M, ℤ) is generated by the fiber class F, and the class L̂_{−} represented by a line in Z_{−}. The desired σ_{ϕ} is then given by σ_{−} + \frac{1}{2}F. For n = 3, c_{ϕ} = 2η̂ + τ, while u_{ϕ} = 2η̂ + tρ + κτ, thus the desired section is σ_{ϕ} = σ_{−} + \frac{1}{2}F + \frac{−κ+1}{t} L̂_{−}.

Now, Ψ(γ) = ∑_{B} a_{B} ⊗ e^{B}, where a_{b} · c = Φ_{σ_{γ}−B}^{P_{ϕ}}([M], [M], c). Instead, we shall calculate invariants of the form Φ_{σ_{−}−B}^{P_{ϕ}}([M], [M], c) since σ_{−} ∈ H_{2}(P_{ϕ}, ℤ). If this invariant is different from 0 then

dimM_{0,3}(σ_{−} − B, P_{ϕ}, J) + 6 + 6 + dim(c) = 3(8),

that is,

−3 ≤ c_{ϕ}(σ_{−} − B) ≤ 0.  (2)

Now σ_{γ} − B has to be of the form σ_{−} − αL̂_{−} − βF, for some α, β ∈ ℤ. The inequality (2) means that

−2 ≤ (n − 3) α − 2β ≤ 1  (3)

where α ≤ 0 to ensure that Π_{∗}(σ_{−}−B) is represented by a pseudo-holomorphic curve in S^{2} × ℂℙ^{2}. Furthermore, given the symplectic structures on P_{ϕ} and M we also need

c_{1}(V ert(Π))(σ_{−} − αL̂_{−} − βF) ≥ −c_{1}(H^{−1} ⊗ E)(Π_{∗}(σ_{−} − αL̂_{−} − βF)).

Equivalently,

\begin{cases}−1 − 2β ≥ 1, & if α < 0, \\ −1 − 2β ≥ −1, & if α = 0. \end{cases}  (4)

Therefore the only possibility is α = 0 = β, so that

dim(c) = −2c_{ϕ}(σ_{−}) = 2,

and the invariants we need are the following.

Lemma 1.

Φ_{σ_{−}}^{P_{ϕ}}([M], [M], L̂_{−}) = 0,
Φ_{σ_{−}}^{P_{ϕ}}([M], [M], F) = 1.

These invariants are computed as a direct application of Theorem 4.2 in [6]

Hence, for n = 0, 1, 2, Ψ(γ) = a_{\frac{1}{2} F} ⊗e^{\frac{1}{2} F}, where a_{\frac{1}{2} F} = Z_{+}, and val(Ψ(γ)) = Ω(\frac{1}{2}F) = 1. Thus

ν_{−}(γ) ≥ 1.

For n = 3, Ψ(γ) = a_{\frac{1}{2} F + \frac{1−κ}{t} L̂_{−}} ⊗ e^{\frac{1}{2} F + \frac{1−κ}{t} L̂_{−}}, where a_{\frac{1}{2} F + \frac{1−κ}{t} L̂_{−}} = Z_{+}, and

val(Ψ(γ)) = Ω(\frac{1}{2}F + \frac{1−κ}{t} L̂_{−}) = 1 + 1 − κ ≥ 1. Thus

ν_{−}(γ) ≥ 1,

which proves Theorem 1 for k = 1.

Recall that c_{1}(M) = 2η + (3 − n)ρ. For n = 0, 1, 2, c_{1}(M) has a representative given by a symplectic form Ω. For n = 3, we take Ω = 2η + tρ, where 0 < t ≤ 1.

3.1 Case n = 0

In this case Z_{+} = Z_{−} = [pt × ℂP^{2}]. Now, Z ∗ Z = ∑ a_{B} ⊗ e^{−B}, so that for any other c ∈ H_{2}(M, ℤ)

a_{B} · c = Φ_{B}^{M}(Z, Z, c),

which is different form 0 only if

2c_{1}(M) · B + dim c = 4.

The only possibilities are B = 0, and B = F,

Φ_{0}^{M}(Z, Z, c) = Z · Z · c = 0,
Φ_{F}^{M}(Z, Z, pt) = 1.

Thus,

a_{F} = [M] and Z ∗ Z = [M] ⊗ e^{−F} ,

so that

Z ⊗ e^{F/2} ∗ Z ⊗ e^{F/2} = [M],

and

val(Ψ(2γ)) = 0.

3.2 Case n = 1

The (small) quantum cohomology of M is given by (cf. [11])

QH^{∗}(M, Λ) = ℝ{η, ρ, e^{F} , eL̂_{−}}/R,

where the set of relations is

R = {ρ^{3} = (η − ρ)e^{−L̂_{−}}, η^{2} = ηρ + e^{−F} }.

By Poincaré duality and the correspondence between the intersection product in homology and the exterior product in cohomology, we can use the above description to find information about the quantum homology of M. In particular, Z_{+} = PD(η), so that,

(Z_{+} ⊗ e^{F/2})^{∗2} = L̂_{+} ⊗ e^{F} + [M],
(Z_{+} ⊗ e^{F/2})^{∗3} = [pt] ⊗ e^{3F/2} + (Z_{+} + π^{−1}(L)) ⊗ e^{F/2}.

By using an induction argument, it is not hard to see that val(Z_{+}^{∗k}) = 0 and so,

val(Ψ(kγ)) = Ω(kF/2) = k.

3.3 Case n = 2

The (small) quantum cohomology of M is given by (cf. [5, 14])

QH^{∗}(M, Λ) = ℝ{η, ρ, e^{F} , eL̂_{−}}/R,

where the set of relations is R = {ρ^{3} = (ρ^{2} − 2ηρ) ⊗ e^{−L̂_{−}} + (8ρ − 4η) ⊗ e^{−2L̂_{−}} + e^{−L̂_{−}−F} , η^{2} = 2ηρ + e^{−F} }. This time we have, for example,

(Z_{+} ⊗ e^{F/2})^{∗2} = 2L̂_{+} ⊗ e^{F} + [M],

(Z_{+} ⊗ e^{F/2})^{∗3} = 4[pt]e^{3F/2} + (Z_{+} + 2π^{−1}(L)) ⊗ e^{F/2} − 4[M] ⊗ e^{−L̂_{−}+F/2}. Once again, it is not hard to see by induction that val(Z_{+}^{∗k}) = 0 and so

val(Ψ(kγ)) = Ω(kF/2) = k.

3.4 Case n = 3

The (small) quantum cohomology of M is of the form

QH^{∗}(M, Λ) = ℝ{η, ρ, e^{F} , eL̂_{−}}/R,

where the set of relations is

R = {ρ^{3} = ∑_{β≥1} a_{F +βL̂_{−}} ⊗ e^{−F −βL̂_{−}}, η^{2} = 3ηρ + e^{−F} + ∑_{β≥1} b_{F +βL̂_{−}}e^{−F −βL̂_{−}}},

a_{F +βL̂_{−}} ∈ H^{2}(M, ℤ) and b_{F +βL̂_{−}} ∈ H^{0}(M, ℤ). For example,

(Z_{+} ⊗ e^{F/2+(1−κ)L̂_{−}/t})^{∗2} = 3L̂_{+} ⊗ e^{F +2(1−κ)L̂_{−}/t} + [M]e^{2(1−κ)L̂_{−}/t}
+ ∑_{β≥1} a_{F +βL̂_{−}} ⊗ e^{−βL̂_{−}+2(1−κ)L̂_{−}/t}.

By induction, val(Z_{+}^{∗k}) = 0 and

val(Ψ(kγ)) = Ω(kF/2 + k(1 − κ)L̂_{−}/t) = k(2 − κ) ≥ k. This concludes the proof of Theorem 1.

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