TRIVIALITY OF SYMPLECTIC SU(2)-ACTIONS ON HOMOLOGY

YILDIRAY OZAN

Abstract. Lalonde and McDuff showed that the natural action of the rational homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold (M, ω) on the rational homology groups H_{∗}(M, ℚ) is trivial. In this note, given a symplectic action of SU(2), ϕ : SU(2) × M → M, we will construct a symplectic fiber bundle P_{ϕ} → ℂP^{2} with fiber (M, ω) and use it to construct the chains, which bound the images of the homology cycles under the trace map given by the SU(2)-action. It turns out that the natural chains bounded by the SU(2)-orbits in M are punctured ℂP^{2}’s, the counter parts of holomorphic discs bounding circles in case of Hamiltonian circle actions. We will also define some invariants of the action ϕ and do some explicit calculations.

1. Introduction

Let ϕ : G × M → M be a smooth action of a compact Lie group G on a smooth manifold M. The action induces a homomorphism on homology, called the trace homomorphism, ∂_{ϕ} : H_{k}(M, ℚ) → H_{k+d}(M, ℚ) defined as follows: If α ∈ H_{k}(M, ℚ) is a class represented by a cycle a : A → M, then ∂_{ϕ}(α) is the class in H_{k+d}(M, ℚ) represented by the cycle G × A → M, (g, x) ↦ ϕ(g, a(x)), where d is the dimension of G. In general this homomorphism is not trivial (just consider product spaces G × M). However, if M is a closed symplectic manifold and G is a compact Lie group acting on M in a Hamiltonian fashion, then McDuff and Lalonde showed that the homomorphism ∂_{ϕ} : H_{k}(M, ℚ) → H_{k+d}(M, ℚ) is trivial. Indeed, they have proved a much stronger result that the natural action of the homology of the group of Hamiltonian diffeomorphisms of the closed manifold (M, ω) on the homology groups H_{∗}(M, ℚ),

H_{k}(Ham(M, ω), ℚ) × H_{l}(M, ℚ) → H_{k+l}(M, ℚ)

is trivial ([LM, LMP]).

Below is the main result of this note, which determines the chains bounded by the images of the trace homomorphism in the case of G = SU(2).

Theorem 1.1. Let ϕ : SU(2)×M → M be a symplectic action on a closed symplectic manifold (M, ω). Then there is a closed symplectic manifold (P_{ϕ}, ω_{ϕ}), which fibers over ℂP^{2} with fiber M such that,

i) the rational homology of the fiber M injects into the rational homology of P_{ϕ}, ii) the symplectic form ω_{ϕ} restricts to ω at each fiber, and iii) if α ∈ H_{k}(M, ℚ) is a class represented by a cycle a : A → M, then in the

The work presented in this paper has been done during my stay at MSRI, Mathematical Sciences Research Institute, Berkeley, in the Spring of 2004. The author is grateful to the institute for the invitation and the kind hospitality.

manifold P_{ϕ}, the cycle

SU(2) × A → M ⊆ P_{ϕ}

representing the class ∂_{ϕ}(α), bounds a chain of the form ℂP_{0}^{2} × A → P_{ϕ}, where ℂP_{0}^{2} = ℂP^{2} − Int(D^{4}) is the punctured projective plane.

In particular, the induced homomorphism on homology ∂_{ϕ} : H_{k}(M, ℚ) → H_{k+3}(M, ℚ) is trivial.

Remark 1.2. Since SU(2) is simply connected any symplectic SU(2)-action on a symplectic manifold is Hamiltonian ([MS]).

Example 1.3. Let SU(2) act linearly on ℂP^{2} in the usual way (see the next section). Blowing up the isolated fixed point of the action we get an SU(2)-action on ℂP^{2}♯\overline{ℂP}^{2}. The action is Hamiltonian since it is algebraic. The orbit of a point with trivial stabilizer is a copy of SU(2) = S^{3}, which separates the two copies of the projective planes. So, the homology class represented by this orbit is trivial and it bounds a punctured ℂP^{2}, not a 4-ball.

The next section is devoted to the proof of Theorem 1.1. In the third section, we will construct some invariants of the action ϕ : SU(2) × M → M and compute them in some cases. Finally, we will mention some applications of these results to the study of the topology of real algebraic varieties.

2. Proof of Theorem 1.1

Let (M, ω) be a closed 2n-dimensional symplectic manifold and

ϕ : SU(2) × M → M

a symplectic action. The proof of Theorem 1.1 consists of three parts. In the first part, we will construct a smooth symplectic fiber bundle π_{S^{4}} : P_{ϕ}^{0} → S^{4} with fiber (M, ω) using the action ϕ : SU(2)×M → M as the clutching function. Moreover, the fibre bundle, both the total space and the base, will have an SU(2) and an S^{1}-action both preserving a closed two form ω_{S^{4}} on P_{ϕ}^{0}, which restricts to ω on each fiber. Moreover, the projection map will be equivariant with respect to both actions.

In the second part, using a natural SU(2) and S^{1}-equivariant degree one map ℂP^{2} → S^{4}, where the SU(2) and the S^{1}-action on the complex projective space are obtained from the natural actions of these groups on ℂ^{2}, we will pull back the bundle over the sphere to a bundle over ℂP^{2}, which we will denote π : P_{ϕ} → ℂP^{2}. Using the pull back of the two form ω_{S^{4}} on P_{ϕ}^{0} and the Fubuni-Study form on ℂP^{2} we will construct a symplectic form ω_{ϕ} on P_{ϕ}, which restricts to ω in each fiber. Moreover, the SU(2) and the S^{1}−actions on P_{ϕ}^{0} will induce Hamiltonian actions on P_{ϕ}.

In the last part, we will consider a symplectic reduction of the total space of the bundle π : P_{ϕ} → ℂP^{2} using the S^{1}-action. Finally, Kirwan’s Surjectivity Theorem on symplectic quotients ([K]) together with a topological observation will finish the proof.

2.1. Symplectic M-bundles over S^{4} with structure group SU(2). Given any smooth action ϕ : SU(2) × M → M we define the smooth manifold P_{ϕ}^{0} as the identification space

D_{+}^{4} × M ∪ D_{−}^{4} × M/(g, x) ∼ (g, ϕ(g, x)), for (g, x) ∈ ∂D_{+}^{4} × M

where we identify ∂D_{±}^{4} with S^{3} = SU(2). Note that we have a fiber bundle P_{ϕ}^{0} → S^{4} with fiber M, induced by the projection maps D_{±}^{4} × M → D_{±}^{4}.

Another description for this bundle, which is more suitable to define the SU(2)- action on, is as follows: Let ℍ denote the quaternion line and U(ℍ^{2}) the set of unit length vectors in ℍ^{2}. Also identify SU(2) with the set of unit quaternions ℍ. Let L → S^{4} denote the quaternion line bundle, whose unit disc bundle is the SU(2)-bundle U(ℍ^{2}) → S^{4}, (v_{1}, v_{2}) ↦ [v_{1} : v_{2}], for (v_{1}, v_{2}) ∈ U(ℍ^{2}). Note that the latter map is nothing but the orbit map of the SU(2)-action on U(ℍ^{2}) given by (v_{1}, v_{2}) ↦ (v_{1}g, v_{2}g), for g ∈ SU(2) and (v_{1}, v_{2}) ∈ U(ℍ^{2}).

Similarly, on U(ℍ^{2}) × M we have an SU(2)-action defined by

(g, (v_{1}, v_{2}), x) ↦ ((v_{1}g, v_{2}g), ϕ(g, x))

for g ∈ SU(2), x ∈ M and (v_{1}, v_{2}) ∈ U(ℍ^{2}). The projection map U(ℍ^{2})×M → U(ℍ^{2}) is equivariant and taking quotients by the respective actions on U(ℍ^{2})×M and U(ℍ^{2}) we recover the fiber bundle P_{ϕ}^{0} → S^{4}.

There is a second SU(2)-action on U(ℍ^{2})×M, which commutes with the first one:

(h, (v_{1}, v_{2}), x) ↦ ((h^{−1}v_{1}, v_{2}), x)

for h ∈ SU(2), x ∈ M and (v_{1}, v_{2}) ∈ U(ℍ^{2}). Clearly, this induces an action on S^{4} given as

(h, [v_{1} : v_{2}]) ↦ [h^{−1}v_{1} : v_{2}]

which makes the projection map equivariant. Since the two actions commute the second action induces actions on both the total space and the base of the fiber bundle P_{ϕ}^{0} → S^{4}, which makes the projection map equivariant. Note that the action on S^{4} is free outside the poles, namely [0 : 1] and [1 : 0], the only fixed points of the action.

There is also a left circle action on this space: Identify ℍ with ℂ^{2}, on which SU(2) acts by matrix multiplication. Also regard S^{1} as the set of matrices {e^{iθ}I_{2} | e^{iθ} ∈ S^{1}}, where I_{2} is the 2 × 2 identity matrix. Now let S^{1} act on U(ℍ^{2}) × M by

(e^{iθ}, (v_{1}, v_{2}), x) ↦ ((e^{−iθ}I_{2}v_{1}, v_{2}), x)

for e^{iθ} ∈ S^{1}, x ∈ M and (v_{1}, v_{2}) ∈ U(ℍ^{2}). Since e^{−iθ} I_{2} commutes with all elements in SU(2) we get an S^{1}-action on both the total space and the base of the fiber bundle P_{ϕ}^{0} → S^{4}, which makes the projection map equivariant. The circle action on S^{4} commutes with the SU(2)-action described in the above paragraph and it is a free action outside the poles.

The Wang sequence for cohomology associated to the M-bundle P_{ϕ}^{0} → S^{4} yields the isomorphism

0 = H^{−2}(M, ℚ) → H^{2}(P_{ϕ}^{0}, ℚ) →^{rest.} H^{2}(M, ℚ) →^{∂_{ϕ}^{∗}} H^{−1}(M, ℚ) = 0

given by the restriction map, where the last map is the dual of the trace homomorphism ∂_{ϕ} : H_{k}(M, ℚ) → H_{k+3}(M, ℚ) (k = −1 in this case). In particular, there is unique cohomology class on P_{ϕ}^{0}, which restricts to the cohomology class [ω] on each fiber. Indeed, we will construct a two form representing this cohomology class, which we will use to get a symplectic form on the M-fiber bundle over ℂP^{2}.

Lemma 2.1. There is an SU(2) and S^{1}-invariant closed two form ω_{S^{4}} on P_{ϕ}^{0}, which restricts to ω on each fiber and such that [ω_{S^{4}}]^{n+1} = 0 in cohomology, where the SU(2) and S^{1}-actions are the ones described in the above paragraphs. Moreover, the cohomology class of ω_{S^{4}} is uniquely determined by these conditions.

Before we prove this lemma we need some preliminaries. Let f : S^{3} ×M → S^{3} ×M be the smooth map given by f(g, x) = (g, ϕ(g, x)), (g, x) ∈ S^{3} × M. Consider the following diagram, where π_{i}, i = 1, 2, are the projections onto the second factors.

M M π_{1} ↑ ↑ π_{2}

S^{3} × M →^{f} S^{3} × M

Using the decomposition T_{∗}(S^{3} × M) = T_{∗}S^{3} × T_{∗}M we will write any tangent vector X on S^{3} × M as X = (X_{S}, X_{M}). Note that the differential of f has the form

f_{∗} = ( Id 0_{\frac{∂ϕ}{∂g} \frac{∂ϕ}{∂x}} ) .

Hence f_{∗}((X_{S}, 0)) = (X_{S}, X_{S}^{♯}) and f_{∗}((0, X_{M})) = (0, ϕ_{∗}(X_{M})), where X_{S}^{♯} is the vector field on M generated by the vector X_{S} via the action.

The SU(2) = S^{3}-action on M is Hamiltonian means that there is a smooth map μ : M → su(2)^{∗} such that for any vector X_{S} ∈ T_{∗}S^{3} we have

i_{X_{S}^{♯}}ω = d(μ(X_{S})).

Since SU(2) = S^{3} is parallelizable choosing a global frame de_{1}, de_{2}, de_{3} for the cotangent bundle for S^{3} we can regard μ as a one form on S^{3} × M, namely

μ(g, x) = A(x) de_{1} + B(x) de_{2} + C(x) de_{3}

for (g, x) ∈ S^{3} × M. One can easily check that d(μ(X_{S})) = −i_{X_{S}}dμ. Now we can state the next lemma.

Lemma 2.2. π_{1}^{∗}(ω) − (π_{2} ∘ f)^{∗}(ω) is an exact two form on S^{3} × M, which vanishes on T_{∗}M identically.

Proof. Let X = (X_{S}, X_{M}) and Y = (Y_{S}, Y_{M}) be tangent vectors at any point of S^{3} × M. Then

I = (π_{1}^{∗}(ω) − (π_{2} ∘ f)^{∗}(ω)) ((X_{S}, X_{M}), (Y_{S}, Y_{M}))
= ω(X_{M}, Y_{M}) − ω(X_{S}^{♯} + ϕ_{∗}(X_{M}), Y_{S}^{♯} + ϕ_{∗}(Y_{M}))
= ω(X_{M}, Y_{M}) − ω(ϕ_{∗}(X_{M}), ϕ_{∗}(Y_{M}))
− ω(X_{S}^{♯}, Y_{S}^{♯} + ϕ_{∗}(Y_{M})) − ω(ϕ_{∗}(X_{M}), Y_{S}^{♯})
= −ω(X_{S}^{♯}, Y_{S}^{♯} + ϕ_{∗}(Y_{M})) − ω(ϕ_{∗}(X_{M}), Y_{S}^{♯}),

because ω(ϕ_{∗}(X_{M}), ϕ_{∗}(Y_{M})) = ϕ^{∗}(ω)(X_{M}, Y_{M}) = ω(X_{M}, Y_{M}). Note that this calculation already shows that π_{1}^{∗}(ω) − (π_{2} ∘ f)^{∗}(ω) is identically zero on T_{∗}M.

Now using i_{X_{S}^{♯}}ω = d(μ(X_{S})) we can write

I = −d(μ(X_{S}))(Y_{S}^{♯} + ϕ_{∗}(Y_{M})) + d(μ(Y_{S}))(ϕ_{∗}(X_{M})). Since

d(μ(X_{S})) = −i_{X_{S}}dμ

we get I = dμ(X_{S}, Y_{S}^{♯} + ϕ_{∗}(Y_{M})) − dμ(Y_{S}, ϕ_{∗}(X_{M}))

= dμ(X_{S}, Y_{S}^{♯}) + dμ(X_{S}, ϕ_{∗}(Y_{M})) + dμ(ϕ_{∗}(X_{M}), Y_{S}).

On the other hand, similar calculations yield dμ(f_{∗}(X), f_{∗}(Y)) = dμ(X_{S}, Y_{S}^{♯}) + dμ(X_{S}^{♯}, Y_{S})

+ dμ(X_{S}, ϕ_{∗}(Y_{M})) + dμ(ϕ_{∗}(X_{M}), Y_{S}).

Comparing the two equations we deduce that I = dμ(f_{∗}(X), f_{∗}(Y)) − dμ(X_{S}^{♯}, Y_{S}). For the last term we can write dμ(X_{S}^{♯}, Y_{S}) = −dμ(Y_{S}, X_{S}^{♯}) = −(i_{Y_{S}}dμ)(X_{S}^{♯}) = d(μ(Y_{S}))(X_{S}^{♯})

= (i_{Y_{S}^{♯}}ω)(X_{S}^{♯}) = ω(Y_{S}^{♯}, X_{S}^{♯}). So, we have obtained

I = dμ(f_{∗}(X), f_{∗}(Y)) + ω(X_{S}^{♯}, Y_{S}^{♯}).

Writing ω(X_{S}^{♯}, Y_{S}^{♯}) = I − (f^{∗}(dμ))(X, Y) we see that the map

(X, Y) ↦ ω(X_{S}^{♯}, Y_{S}^{♯})

is a closed two form on S^{3} × M. Moreover, it vanishes if X_{S} or Y_{S} is zero and hence it is identically zero on the T_{∗}M component of the tangent space. So, the de Rham cohomology class represented by this closed two form evaluates zero on any two dimensional homology class of the product S^{3} × M provided that the homology class is represented by a cycle lying in some {pt}×M. However, since S^{3} has no first and second homology, by the Künneth formula, this de Rham class must be trivial. Hence, there is a one form u on S^{3} × M such that I = (f^{∗}(dμ))(X, Y) + du(X, Y). This finishes the proof of the lemma. □

Proof of Lemma 2.1. By the isomorphism obtained from the Wang sequence, the cohomology class of ω_{S^{4}} is uniquely determined by that of ω (see the paragraph above Lemma 2.1).

We will regard the total space P_{ϕ}^{0} as the identification space

ℝ_{+}^{4} × M ∪ ℝ_{−}^{4} × M/(t, g, x) ∼ F (t, g, x) ≐ (t^{−1}, g, ϕ(g, x)),

for (t, g, x) ∈ (ℝ_{+}^{4} − {0}) × M, where we identify ℝ^{4} − {0} with (0, ∞) × S^{3} in the obvious way. Let π_{1} and π_{2} denote the projections onto the M factors of the products ℝ_{+}^{4} × M and ℝ_{−}^{4} × M, respectively. Also, we will denote the projection of (0, ∞) × S^{3} × M onto the S^{3} × M component by π_{SM}.

Let ω_{i} = π_{i}^{∗}(ω), i = 1, 2. Then by Lemma 2.2

ω_{1} − F^{∗}(ω_{2}) = π_{SM}^{∗}(dv_{1})

for some one form v_{1} on S^{3} × M. Let v_{2} = (f^{−1})^{∗}(v_{1}), where f is as in Lemma 2.2. So, ω_{1} = F^{∗}(ω_{2} + dv_{2}) on (ℝ_{+}^{4} − {0}) × M. Let ω̃_{2} = ω_{2} + d(ρ(t)v_{2}), where ρ is a smooth function on ℝ, which vanishes on (−∞, 0.5] and equals one on [0.75, ∞).

Now we have

F^{∗}(ω̃_{2}) = F^{∗}(ω_{2}) + d(F^{∗}(ρ(t)v_{2})) = ω_{1} − dv_{1} + d(ρ(1/t)v_{1}).

So, letting ω̃_{1} = ω_{1} + d((ρ(1/t) − 1)v_{1}) we obtain F^{∗}(ω̃_{2}) = ω̃_{1}. By the choice of the function ρ the forms ω̃_{i} are defined on all of ℝ^{4} × M, and indeed, they are equal to ω_{i} on D_{1/2} × M, where D_{1/2} denotes the disc of radius 1/2 in ℝ^{4}. Moreover, since both dv_{i} and dt vanishes on T_{∗}M, each ω̃_{i} restricts to ω on each fiber {pt} × M. Hence, together they define a global closed two form on P_{ϕ}^{0}, say ω_{S^{4}}, which restricts to ω in each fiber.

To see that [ω_{S^{4}}]^{n+1} is zero just open up ω̃_{i}^{n+1} and note that ω_{i}^{n+1} = 0.

The form ω_{S^{4}} may not be SU(2)-invariant. However, we can average it over the SU(2)-orbits to get an SU(2)-invariant form with the desired properties. Namely, let dH denote the Haar measure on SU(2) with total volume one and define the average of ω_{S^{4}} as the form

(X, Y) ↦ ∫_{SU(2)}(ψ^{∗}(h, p) (ω_{S^{4}})) (X, Y) dH

where ψ : SU(2) × P_{ϕ}^{0} → P_{ϕ}^{0} is the action map and the integration is over h ∈ SU(2) for fixed vectors X, Y ∈ T_{p}(P_{ϕ}^{0}). Since on U(ℍ^{2}) × M the action is given by (h, ((v_{1}, v_{2}), x)) ↦ ((h^{−1}v_{1}, v_{2}), x) and the restriction of ω_{S^{4}} to each fiber, which is a copy of M, is just ω, so will be the restriction of the average of ω_{S^{4}}. Once, the form is SU(2)-invariant then we can average it to make also S^{1}-invariant in the same way. Since the two actions commute, averaging over the S^{1}-orbits will not spoil the SU(2)−invariance of the form. Also averaging commutes with exterior derivative and therefore the averaged two form will be still closed. □

2.2. Symplectic M-bundles over ℂP^{2} with structure group SU(2). By a fiber bundle π : P_{ϕ} → ℂP^{2}, with fiber M and structure group SU(2) we mean a group homomorphism ϕ : SU(2) → Symp(M, ω), where the latter is the group of symplectomorphisms of the symplectic manifold (M, ω) and a principal SU(2)-bundle P → ℂP^{2} such that P_{ϕ} is obtained from P via the representation ϕ in the usual way. The classifying space for SU(2)-bundles is ℍP^{∞}, whose 7th skeleton is ℍP^{1} = S^{4} and therefore any principal SU(2)-bundle over a closed 4-manifold N is obtained form the universal bundle L → S^{4} by pulling it over N by a map ξ : N → S^{4}. Since e(L) = c_{2}(L) ∈ H^{4}(S^{4}, ℤ) is a generator we have e(ξ^{∗}(L)) = c_{2}(ξ^{∗}(L)) = deg(ξ).

Let ξ : ℂP^{2} → S^{4} be the map given by the formula

ξ([z_{0} : z_{1} : z_{2}]) = (\frac{2z̄_{0}z_{1}}{||z||^{2}} , \frac{2z̄_{0}z_{2}}{||z||^{2}} , \frac{|z_{1}|^{2} + |z_{2}|^{2} − |z_{0}|^{2}}{||z||^{2}} )

where ||z||^{2} = |z_{0}|^{2} + |z_{1}|^{2} + |z_{2}|^{2}, for [z_{0} : z_{1} : z_{2}] ∈ ℂP^{2}. Here we consider the 4-sphere as

S^{4} = {(w_{1}, w_{2}, t) ∈ ℂ × ℂ × ℝ | |w_{1}|^{2} + |w_{2}|^{2} + t^{2} = 1}.

Note that ξ([0 : z_{1} : z_{2}]) = (0, 0, 1), the North pole and ξ([0 : 0 : 1]) = (0, 0, −1), the South pole. So, the map ξ maps the complex line z_{0} = 0 to the North pole and is a diffeomorphism onto its image outside the line z_{0} = 0. In particular its degree is one.

Consider the linear SU(2)-action on ℂP^{2} given as

[z_{0} : z_{1} : z_{2}] ↦ [z_{0} : a′z_{1} + b′z_{2} : c′z_{1} + d′z_{2}]

where ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ SU(2) is the inverse of ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SU(2). Writing

w = (w_{1}, w_{2}) = (z_{1}/z_{0}, z_{2}/z_{0})

we get

ξ([z_{0} : z_{1} : z_{2}]) = (\frac{2w_{1}}{1 + ||w||^{2}} , \frac{2w_{2}}{1 + ||w||^{2}} , 1 − \frac{2}{1 + ||w||^{2}} ).

Hence, ξ becomes equivariant if we endow S^{4} with the SU(2)-action given by

(w_{1}, w_{2}, t) ↦ (a′w_{1} + b′w_{2}, c′w_{1} + d′w_{2}, t).

However, the latter is just the action of Lemma 2.1. Similarly, ξ is S^{1}-equivariant where the S^{1}-action ℂP^{2} is given by

[z_{0} : z_{1} : z_{2}] ↦ [z_{0} : e^{−iθ}z_{1} : e^{−iθ}z_{2}].

Let P_{ϕ} = ξ^{∗}(P_{ϕ}^{0}), the pull back of the M-bundle P_{ϕ}^{0} → S^{4} via the SU(2) and the S^{1}-equivariant map ξ : ℂP^{2} → S^{4}. Since ξ is an equivariant map the bundle π : P_{ϕ} → ℂP^{2} gets both SU(2) and S^{1}-actions, for which the projection map π is equivariant. Moreover, the pull back cohomology class ξ^{∗}(ω_{S^{4}}) is invariant with respect to both actions and restricts to ω on each fiber. On the other hand, the cohomology class [ξ^{∗}(ω_{S^{4}})]^{n+1} = ξ^{∗}([ω_{S^{4}}]^{n+1}) = 0.

Let ω_{F S} denote the Fubuni-Study symplectic form on ℂP^{2}. The form π^{∗}(ω_{F S}) is invariant under the SU(2) and the S^{1}-action on the complex projective plane and is identically zero when restricted to each fiber {pt} × M. Hence for any positive large enough constant κ ≫ 0 the 2-form ω_{ϕ} = ξ^{∗}(ω_{S^{4}}) + κ π^{∗}(ω_{F S}) is a symplectic form on P_{ϕ}. Moreover, both actions on P_{ϕ} are Hamiltonian. This is obvious for SU(2) since it is simply connected. For the S^{1}-action one can argue as follows: Since averaging commutes with exterior derivative locally we have ω_{ϕ} = π_{i}^{∗}(ω) + dv + κ π^{∗}(ω_{F S}) for some equivariant one form v on P_{ϕ}. Let χ be a vector field generated by the S^{1}- action. Since the form is invariant the Lie derivative of dv along χ will be zero. Now by the Cartan formula we get i_{χ^{♯}}dv = −d(i_{χ^{♯}}v) and hence the S^{1}-action on P_{ϕ} is also Hamiltonian.

Remark 2.3. 1) By multiplying the last coordinate of the map ξ : ℂP^{2} → S^{4} by −1, if necessary, we can arrange so that the pull back SU(2)-bundle ξ^{∗}(L) → ℂP^{2} has c_{2} = −1. Since SU(2)-bundles are determined by c_{2} we see that ξ^{∗}(L) is smoothly isomorphic to O(1) ⊕ O(−1). Therefore, the construction of P_{ϕ} could be made just over ℂP^{2} using the bundle O(1) ⊕ O(−1) with appropriate SU(2) and S^{1}-actions.

2) We will orient the bundles P_{ϕ}^{0} and P_{ϕ} as follows: The manifold P_{ϕ} is oriented with the orientation coming from the symplectic form ω_{ϕ}. Since P_{ϕ} is the pull back of P_{ϕ}^{0} via the map ξ : ℂP^{2} → S^{4}, whose degree is chosen as above, the orientation on P_{ϕ} induces one on P_{ϕ}^{0}.

2.3. Hamiltonians and symplectic reduction. Let μ : P_{ϕ} → ℝ be a Hamiltonian for the S^{1}-action on P_{ϕ}. Recall that the S^{1}-equivariant map ξ : ℂP^{2} → S^{4} maps the line z_{0} = 0 to the North pole and sends the point [0 : 0 : 1] to the South pole of the sphere. So, over some small S^{1}-invariant disjoint tubular neighborhoods U and V of the line z_{0} = 0 and the point [0 : 0 : 1], respectively, the bundle π : P_{ϕ} → ℂP^{2} is isomorphic to the product bundles U × M → U and V × M → V, where the S^{1}- action on the M-factor is trivial. Moreover, by the construction, ω_{ϕ} when restricted to π^{−1}(U) and π^{−1}(V), is just (ω_{ϕ})_{|} = ω + κ π^{∗}(ω_{F S}). Since the action on the Mfactor is trivial we see that the moment map restricted to π^{−1}(U) and π^{−1}(V) is just a multiple of the moment map μ_{0} : ℂP^{2} → ℝ of the S^{1}-action on ℂP^{2} plus a constant, which depends only on the open set U or V; i.e., μ(p) = κ π(μ_{0}(p)) + C(p), for all p ∈ π^{−1}(U) ∪ π^{−1}(V), where C is a locally function on the union π^{−1}(U) ∪ π^{−1}(V).

We are ready now to prove the main theorem.

Proof of Theorem 1.1. Replacing the Hamiltonians by adding constants if necessary we can assume that 0 is a regular value for μ and hence for μ_{0} such that μ_{0}^{−1}(0) = S^{3} lies in U. Note that this S^{3} divides ℂP^{2} into two pieces. By multiplying all the symplectic forms with −1 if necessary we can assume that ℂP _{0}^{2} = μ_{0}^{−1}((−∞, 0]) is a closed tubular neighborhood of the line z_{0} = 0 and D^{4} = μ_{0}^{−1}([0, ∞)) is a closed 4-ball with common boundary S^{3} = μ^{−1}(0). The M-fiber bundle over these pieces are just products and

P_{ϕ} = ℂP _{0}^{2} × M ∪ D^{4} × M/(g, x) ∼ (g, ϕ(g, x)), for (g, x) ∈ ∂(ℂP _{0}^{2}) × M.

Let α ∈ H_{k}(M, ℚ) be a class represented by a cycle a : A → M. We need to show that the class ∂_{ϕ}(α), represented by the cycle S^{3} × A → M, (g, x) ↦ ϕ(g, a(x)), is trivial in H_{k+3}(M, ℚ). We can clearly view S^{3} × A as a subset of the boundary of ℂP _{0}^{2} × M. The identification in the above decomposition of P_{ϕ} maps S^{3} × A into the other piece by the map (g, x) ↦ ϕ(g, a(x)). On the other hand, the radial contraction of D^{4} to its center {0} induces a radial contraction of D^{4}×M to {0}×M. Moreover, the composition of the identification map with the contraction will map S^{3} × A into M exactly via the map (g, x) ↦ ϕ(g, a(x)). Since S^{3} × A = ∂(ℂP _{0}^{2} × A) the class ∂_{ϕ}(α) is trivial in H_{k+3}(P_{ϕ}, ℚ).

Now consider the symplectic quotient μ^{−1}(0)/S^{1}, which is equal to the product S^{3}/S^{1} × M = S^{2} × M, because S^{1} acts trivially on M by the construction of the S^{1}-action. By the Kirwan’s Surjectivity Theorem ([K]) the map, induced by the inclusion μ^{−1}(0) ⊆ P_{ϕ}, K : H^{i}(P_{ϕ}, ℚ) → H^{i}(S^{2} × M, ℚ) is onto, for all i. So the restriction map H^{i}(P_{ϕ}, ℚ) → H^{i}(M, ℚ) is surjective. Hence the map in homology induced by the inclusion of a fiber H_{i}(M, ℚ) → H_{i}(P_{ϕ}, ℚ) is injective. Finally, by the above paragraph ∂_{ϕ}(α) is trivial in H_{k+3}(M, ℚ). □

3. Some invariants of the SU(2)-action

In this section we will study the sections of the bundles P_{ϕ}^{0} and P_{ϕ}, define some invariants of the SU(2)-action on (M, ω), make some computations and mention some applications to the study of the topology of real algebraic varieties.

The orientations on the manifolds P_{ϕ}^{0} and P_{ϕ}, which we will need when we consider integrals over them, are the ones described in Remark 2.3.

3.1. Sections of P_{ϕ}^{0}. The lemma below describes the SU(2)-equivariant (with respect to the SU(2)-action on P_{ϕ}^{0} and on S^{4} described in Subsection 2.1) sections of the bundle P_{ϕ}^{0} → S^{4} up to homotopy.

Lemma 3.1. There is an SU(2)-equivariant section s : S^{4} → P_{ϕ}^{0} if and only if the SU(2)-action on M has a fixed point. Moreover, if s_{i} : S^{4} → P_{ϕ}^{0}, i = 1, 2, are two such sections then the difference of (s_{2})_{∗}([S^{4}]) − (s_{2})_{∗}([S^{4}]) as a homology class is in the image of the map π_{4}(M) → π_{4}(P_{ϕ}^{0}), induced by the inclusion of a fiber.

Proof. Let l(t) : [−1, 1] → S^{4} be a one to one geodesic arc from the point (0, 0, −1) to the point (0, 0, 1). If s : S^{4} → P_{ϕ}^{0} is an equivariant section then s is determined completely by its values s(l(t)), t ∈ [−1, 1]. On the other hand, the points (0, 0, ±1) are the fixed points of the action on the sphere and hence the points s(0, 0, ±1) are in the fixed point set of action on M. Moreover, since the action on S^{4} is free outside the poles, any section s defined on the arc l(t) with s(0, 0, ±1) ∈ M fixed points of the SU(2)-action, extends uniquely to a section. Indeed, the section s : S^{4} → P_{ϕ}^{0} is just the trace of the section s(l(t)), t ∈ [−1, 1], under the SU(2)-action on P_{ϕ}^{0}.

The second statement follows the long exact sequence corresponding to the fibration M → P_{ϕ}^{0} → S^{4},

⋯ → π_{4}(M) → π_{4}(P_{ϕ}^{0}) → π_{4}(S^{4}) = ℤ → · · · .
□

Theorem 1.1 implies the following result.

Proposition 3.2. If the SU(2) action on M is symplectic then the homology class s_{∗}([S^{4}]) of an equivariant section s : S^{4} → P_{ϕ}^{0} is determined only by the connected components of the fixed point set containing the fixed points s(0, 0, −1) and s(0, 0, 1).

Proof. Assume the set up in the proof of the Lemma 3.1. If s_{1} and s_{2} are two such sections with s_{1}(0, 0, −1) = s_{2}(0, 0, −1) and s_{1}(0, 0, 1) = s_{2}(0, 0, 1) then the difference homology class can be identified with the trace of a loop in M based at one of these two fixed points. However, by Theorem 1.1 the latter is trivial. Now assume that s_{1}(0, 0, −1), s_{2}(0, 0, −1) ∈ F_{0} and s_{1}(0, 0, 1), s_{2}(0, 0, 1) ∈ F_{1}, for some connected components F_{0} and F_{1} of the fixed point set. Join the fixed points in F_{0} and F_{1} by arcs contained completely in the fixed point sets. The trace of a path that lies in the fixed point set is just the path itself and hence these arcs do not contribute to the difference of the homology classes. This finishes the proof. □

We will call a cohomology class u ∈ H^{4}(M, ℚ) monotone if it vanishes on spherical classes, i.e., on the image of π_{4}(M) → H_{4}(M, ℚ).

We know that the cohomology class [ω_{S^{4}}] ∈ H^{2}(P_{ϕ}^{0}, ℚ), which restricts to [ω] on each fiber with ω_{S^{4}}^{n+1} = 0. Following [LMP, S], we denote the Chern classes of the vertical tangent bundle

T_{∗}^{vert} = ker(π_{∗} : T_{∗}P_{ϕ}^{0} → T_{∗}S^{4})

by c_{i}^{ϕ}. These classes are clearly invariants of the action ϕ and hence, so is any integral of the form

I^{0}(k, k_{1}, ⋯, k_{n}) = ∫_{P_{ϕ}^{0}} ω_{S^{4}}^{k} (c_{1}^{ϕ})^{k_{1}} ⋯ (c_{n}^{ϕ})^{k_{n}}

where k, k_{i} are non negative integers with 2k + 2k_{1} + ⋯ + 2nk_{n} = 2n + 4 = dim(P_{ϕ}^{0}).

Let x ∈ M be any fixed point of the SU(2)-action. Then the function v ↦ (v, x), v ∈ S^{4}, defines a section, say s_{x} : S^{4} → P_{ϕ}^{0}. Note that the pull back bundle over S^{4} of the vertical bundle via the section s_{x} is nothing but the associated complex vector bundle of the principal SU(2)-bundle U(ℍ^{2}) → S^{4} (see Subsection 2.1) corresponding to the representation of SU(2) on tangent space T_{x}M. We have then the following result about the representations of SU(2) on tangent spaces of the fixed points, which follows easily from Lemma 3.1.

Corollary 3.3. Let (M, ω) and ϕ be as above. Assume that c_{2}(M) is a monotone class. Then for any two fixed points x_{1} and x_{2} of the action on M, the vector bundles over S^{4}, corresponding to the SU(2)-representations at the tangent spaces T_{x_{i}}, i = 1, 2, have the same second Chern class.

Remark 3.4. Lemma 3.1 and the above corollary are valid indeed for any smooth action on M since we do not make use of the symplectic form at all.

Example 3.5. 1) If the action ϕ : SU(2) × M → M is trivial then the integrals

I^{0}(k, k_{1}, ⋯, k_{n}) = ∫_{P_{ϕ}^{0}} ω_{S^{4}}^{k} (c_{1}^{ϕ})^{k_{1}} ⋯ (c_{n}^{ϕ})^{k_{n}}

are all zero, because in this case P_{ϕ}^{0} = S^{4} × M and all the forms in the integral are trivial on T_{∗}S^{4}.

2) Consider the standard action of SU(2) on (ℂP^{2}, ω), where ω = ω_{F S}, and consider

I^{0}(2, 0, 1) = ∫_{P_{ϕ}^{0}} ω_{S^{4}}^{2} c_{2}^{ϕ}.

Note that c_{2}(ℂP^{2}) is a monotone class because π_{4}(ℂP^{2}) = 0. It follows from the Wang sequence of the fibration that c_{2}^{ϕ} = λ[ω_{S^{4}}]^{2} + π_{S^{4}}^{∗}(v) for some v ∈ H^{4}(S^{4}, ℚ) and real number λ. Let x_{0} = [1 : 0 : 0], the only fixed point of the action, and denote the corresponding section of P_{ϕ}^{0} → S^{4} by s_{x_{0}}. Then

c_{2}^{ϕ}([s_{x_{0}}]) = λ[ω_{S^{4}}]^{2}([s_{x_{0}}]) + π_{S^{4}}^{∗}(v)([s_{x_{0}}]) = π_{S^{4}}^{∗}(v)([s_{x_{0}}]) = v([S^{4}]),

where the first equality follows from the fact s_{x_{0}}^{∗}([ω_{S^{4}}]) = 0 and the second one from that π_{S^{4}} ∘s_{x_{0}} = id_{S^{4}}. It follows from the representation of SU(2) on the tangent space T_{x_{0}}ℂP^{2} that c_{2}^{ϕ}([s_{x_{0}}]) = c_{2}(L) = −1, where L → S^{4} is the canonical SU(2)-bundle (see Section 2.1).

Now, I^{0}(2, 0, 1) = ∫_{P_{ϕ}^{0}} ω_{S^{4}}^{2} c_{2}^{ϕ} = ∫_{P_{ϕ}^{0}} ω_{S^{4}}^{2} π_{S^{4}}^{∗}(v)

= ∫_{P_{ϕ}^{0}−(one fiber)} ω_{S^{4}}^{2} π_{S^{4}}^{∗}(v) = ∫_{ℝ^{4}×ℂP^{2}} ω_{S^{4}}^{2} π_{S^{4}}^{∗}(v)
= (∫_{ℝ^{4}} v) (∫_{ℂP_{2}} ω_{S^{4}}^{2}) = v([S^{4}]) ω^{2}([ℂP^{2}])
= −c_{1}^{2}([ℂP^{2}]) = −9.

Note that we can define analogous integrals over P_{ϕ}: There is a unique cohomology class u_{ϕ} = ξ^{∗}([ω_{S^{4}}]) ∈ H^{2}(P_{ϕ}, ℚ), which restricts to [ω] on each fiber with u_{ϕ}^{n+1} = 0. Similarly, we define

I(k, k_{1}, ⋯, k_{n}) = ∫_{P_{ϕ}} u_{ϕ}^{k} (c_{1}^{ϕ})^{k_{1}} ⋯ (c_{n}^{ϕ})^{k_{n}}

where k, k_{i} are non negative integers with 2k + 2k_{1} + ⋯ + 2nk_{n} = 2n + 4 = dim(P_{ϕ}). Note that the two invariants are indeed equal, where the first one may be more suitable for computations. However, if the action has no fixed points then the bundle over S^{4} may have no section, as the next example shows.

Example 3.6. The bundle corresponding to the standard action of SU(2) on ℂP^{1}, P_{ϕ}^{0} → S^{4}, has no section. Otherwise, by deforming the section to a constant, say [1 : 0], over D^{4} we would arrive at the contradiction that the map SU(2) → ℂP^{1} given by

g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ [ā : b̄] = ϕ(g, [1 : 0])

is homotopically trivial, which is nothing but basically the Hopf map (see Section 2.2).

On the other hand, as we will see in the next section that the bundle π : P_{ϕ} → ℂP^{2} has always a section.

3.2. Sections of P_{ϕ}. Since the SU(2) action on ℂP^{2} has a fixed point, an equivariant section exists if and only if the SU(2)-action on M has a fixed point. Note also that, any equivariant section of P_{ϕ}^{0} → S^{4} pulls back to an equivariant section of P_{ϕ} → ℂP^{2}. Indeed, these pull back sections are those equivariant sections s : ℂP^{2} → P_{ϕ} such that s(z_{0} = 0) and s([1 : 0 : 0]) are two fixed points of the action on M (since the map ξ : ℂP^{2} → S^{4} maps the line z_{0} = 0 to a pole of S^{4} the bundle P_{ϕ} restricted to the line z_{0} = 0 is trivial). In particular, for this class of equivariant sections of P_{ϕ} the analogues of Lemma 3.1, Proposition 3.2 and Corollary 3.3 will hold.

Another source of equivariant sections of the bundle is the set of points of M whose stabilizers is a circle. Namely, let H = Stab_{SU(2)}([0 : 1 : 1]) and consider the path r(t) = [1 − t : t : t], t ∈ [0, 1], in ℂP^{2}. Let x_{0}, x_{1} be points in M with Stab_{SU(2)}(x_{0}) = SU(2) (x_{0} is a fixed point) and H ⩽ Stab_{SU(2)}(x_{1}). Choose a section of the bundle over the path r(t) with s([1 : 0 : 0]) = x_{0} and s([0 : 1 : 1]) = x_{1}. Then this extends uniquely to an equivariant section s : ℂP^{2} → P_{ϕ}. Moreover, any equivariant section is of this form and the analogues of Lemma 3.1 and Corollary 3.3 will hold in this case also.

Unlike the bundle P_{ϕ}^{0} → S^{4} the bundle over ℂP^{2} has always a section.

Lemma 3.7. Let p_{0} be any point in the fiber π^{−1}([1 : 0 : 0]) = M. Then the bundle π : P_{ϕ} → ℂP^{2} has a section s : ℂP^{2} → P_{ϕ} with s([1 : 0 : 0]) = p_{0}.

Proof. Recall the decomposition of P_{ϕ} from the proof of the Theorem 1.1

P_{ϕ} = ℂP _{0}^{2} × M ∪ D^{4} × M/(g, x) ∼ (g, ϕ(g, x)), for (g, x) ∈ ∂(ℂP _{0}^{2}) × M.

We define a section s : ℂP^{2} → P_{ϕ} as follows: For v ∈ D^{4} let s(v) = (v, p_{0}). Over ∂(D^{4}) the section looks like v ↦ (v, p_{0}) and over ∂(ℂP _{0}^{2}) it is given by the formula v ↦ (v, ϕ(v^{−1}, p_{0})).

Since the action of the maximal torus H = Stab_{SU(2)}([0 : 1 : 1]) on M is also Hamiltonian it has a fixed point, say p_{1} ∈ M. Hence, Stab_{SU(2)}(p_{1}) contains H. Let σ : [0, 1] → M be a path from p_{0} to p_{1}. The SU(2)-orbit of this path,

SU(2) × [0, 1] → M, (g, t) ↦ ϕ(g, σ(t))

gives a map β : ℂP _{0}^{2} → M, whose restriction to the boundary ∂(ℂP _{0}^{2}) is the orbit of p_{0}. Now we can extend the section over ℂP _{0}^{2} as v ↦ (v, β(v)). □

Remark 3.8. i) For the last part of the above proof we could use also the chain constructed in Theorem 1.1 bounded by the cycle ∂_{ϕ}(p_{0}).

ii) Note that if the point p_{0} ∈ M is not a fixed point of the SU(2)-action then the section of the above lemma is not equivariant. In particular, if M = ℂP^{1} with the SU(2)-action as in Example 3.6, then neither bundles have an equivariant section.

We believe that J-holomorphic sections of P_{ϕ} → ℂP^{2} deserve some attention also.

3.3. Algebraic actions on real algebraic varieties. The analogous constructions in case of Hamiltonian circle actions on closed symplectic manifolds can be done (cf. see also [LMP, S]). This result has the following consequence. Let M = X_{ℂ} be a nonsingular projective complexification of a nonsingular real algebraic variety X and assume that we have a linear (hence Hamiltonian) S^{1}-action on M = X_{ℂ}. Assume that the action leaves the real part invariant and the action on X is free. Then the S^{1}-analogue of Theorem 1.1 implies that we can map D^{2} × X into P_{ϕ} via an equivariant map, which extends the action ϕ : S^{1} × X → X. Moreover, the map descends to a map

D^{2} ×_{S^{1}} X → P_{ϕ}

which shows that X bounds, in the manifold P_{Φ}, the mapping cylinder of the quotient map p : X → X/S^{1} = B. In particular, this implies that all the homology classes in X represented by cycles of the form p^{−1}(A), where A is a cycle in B, are homologous to zero in the complexification X_{ℂ}, a fact proven in [O1] previously.

Lastly, we will mention another application along the same lines. The result mentioned in the introduction that the natural action of the homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold (M, ω) on the homology groups H_{∗}(M, ℚ),

H_{k}(Ham(M, ω), ℚ) × H_{l}(M, ℚ) → H_{k+l}(M, ℚ)

is trivial ([LM, LMP]) has an immediate consequence in the study of topology of real algebraic varieties: Let X be a nonsingular compact real algebraic variety with a nonsingular projective complexification i : X → X_{ℂ}. Clearly X_{ℂ} carries a Kähler and hence a symplectic structure such that X is a Lagrangian submanifold. Define KH_{i}(X, ℚ) as the kernel of the homomorphism i_{∗} : H_{i}(X, ℚ) → H_{i}(X_{ℂ}, ℚ) and ImH^{i}(X, ℝ) as the image of the homomorphism i^{∗} : H^{i}(X_{ℂ}, ℚ) → H^{i}(X, ℚ). In [O1, O2] it is shown that both KH_{i}(X, ℚ) and ImH^{i}(X, ℚ) are independent of the projective complexification i : X → X_{ℂ} and thus (entire rational) isomorphism invariants of X. We know also that the natural linear action of a unitary group on a complex projective variety is Hamiltonian. We have then the following corollary.

Corollary 3.9. Let X and X_{ℂ} be as above and G be a compact Lie group acting unitarily on X_{ℂ}, leaving the real part X invariant. Then the image of the trace map

H_{k}(G, ℚ) × H_{l}(X, ℚ) → H_{k+l}(X, ℚ)

lies in KH_{k+l}(X, ℚ).

References

[K] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes (31), Princeton University Press, Princeton, New Nersey, 1984.

[LM] F. Lalonde, D. McDuff, Symplectic structures on fiber bundles, Topology 42 (2003), 309-347.

[LMP] F. Lalonde, D. McDuff, L. Polterovich, Topological rigidity of Hamiltonian loops and Quantum homology, Invent. Math. 135 (1999), 369-385.

[MS] D. McDuff, D. Salamon, Introduction to symplectic topology, Oxford University Press, New York, 1997.

[O1] Y. Ozan, On homology of real algebraic varieties, Proc. Amer. Math. Soc. 129 (2001), 3167-3175.

[O2] Y. Ozan, Homology of non orientable real algebraic varieties, preprint.

[S] P. Seidel, π_{1} of symplectic automorphism groups and invertibles in quantum cohomology rings, Geom. Functional Anal. 7 (1997) 1046-1095.

Yıldıray Ozan, Mathematics Department, Middle East Technical University, 06531, Ankara, Turkey

  E-mail address: ozan@metu.edu.tr