A group three-cocycle of the Symplectomorphism group and the Dixmier-Douady class of Symplectic fibrations
aa r X i v : . [ m a t h . S G ] S e p A CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONSAND THE PREQUANTIZATION
SHUHEI MARUYAMA
Abstract.
We construct a characteristic class of Hamiltonian fibrations and flatHamiltonian fibrations as a third cohomology class with coefficients in Z . We alsoinvestigate the relation between the universal characteristic class of flat Hamil-tonian fibrations and the prequantization of the symplectic manifold appearingas the fiber. Moreover, we construct a group three-cocycle of Hamiltonian dif-feomorphism group and show that its cohomology class is equal to the universalcharacteristic class above. Introduction and Main Theorem
Introduction.
Let (
M, ω ) be a closed connected symplectic manifold. A fiberbundle (
M, ω ) → E → B is called a Hamiltonian fibration if the structure group re-duces to the Hamiltonian diffeomorphism group Ham(
M, ω ) with C ∞ -topology. LetHam( M, ω ) δ denote the Hamiltonian diffeomorphism group with discrete topology.A Hamiltonian fibration is called flat if the structure group reduces to the discretegroup Ham( M, ω ) δ .Several characteristic classes of Hamiltonian fibrations have been constructed. In[10], Reznikov constructed characteristic classes by an analog of the Chern-Weiltheory. The characteristic classes defined in [10] can also be constructed by usingthe coupling class and fiber integration (see [8]). There are other characteristicclasses explained in [8] and calculated for symplectic toric manifolds in [5], that areconstructed by using the Chern classes of the vertical tangent bundle, the couplingclass, and fiber integration. The characteristic classes above are all even degreecohomology classes with coefficients in R .In the present paper, we construct a characteristic class of Hamiltonian fibrationsas a third cohomology class with coefficients in Z . Note that this characteristic classis equal to zero as the cohomology class with coefficients in R .1.2. Construction of the characteristic class.
Let (
M, ω ) → E → B be aHamiltonian fibration with connected base space B . At first, we consider the Serrespectral sequence E p,qr with coefficients in R . It is known that the derivations d , : E , → E , and d , : E , → E , are equal to the 0-map (see [7]). So thecohomology class obtained as the transgression of the symplectic form is trivialand therefore we cannot get characteristic classes in this way.Next we consider the Serre spectral sequence E p,qr with coefficients in Z . Thenthe derivations d , and d , are not necessarily equal to the 0-map. Let ( M, ω ) bean integral symplectic manifold, that is, the class [ ω ] ∈ H ( M ; R ) is in the image of the canonical map H ( M ; Z ) → H ( M ; R ). We also assume that the symplecticmanifold M is simply-connected. In this case, the canonical map H ( M ; Z ) → H ( M ; R ) is injective. Let [ ω ] Z ∈ H ( M ; Z ) denote the unique element that is equalto [ ω ] in H ( M ; R ). The group E p,q in the Serre spectral sequence of ( M, ω ) → E → B is isomorphic to H p ( B ; H q ( M ; Z )). Since the Hamiltonian diffeomorphismgroup is connected with respect to the C ∞ -topology, the local system H ∗ ( M ; Z ) istrivial. Note that, by H ( M ; Z ) = 0, we have E p, = 0 for any p . So we have E , = E , = H ( M ; Z ) and E , = E , = H ( B ; Z ), and there is the transgressionmap d , : H ( M ; Z ) = E , → E , = H ( M ; Z ) . By the naturality of the Serre spectral sequence, the cohomology class d , [ ω ] Z ∈ H ( B ; Z ) also has the naturality. Thus the cohomology class d , [ ω ] Z gives rise to acharacteristic class of Hamiltonian fibrations. Definition 1.1.
Let (
M, ω ) be a connected, simply-connected integral symplecticmanifold. Let [ ω ] Z ∈ H ( M ; Z ) denote the class that is equal to [ ω ] in H ( M ; R ).For a Hamiltonian fibration ( M, ω ) → E → B with connected base space B , wedefine the characteristic class e ω ( E ) ∈ H ( B ; Z ) by e ω ( E ) = − d , [ ω ] Z .Let E Ham(
M, ω ) → B Ham(
M, ω ) be the universal Ham(
M, ω )-bundle. Thenthe universal Hamiltonian fibration (
M, ω ) → E → B Ham(
M, ω ) is given as E = E Ham(
M, ω ) × Ham(
M,ω ) M . By the naturality, the characteristic class e ω = e ω ( E ) ∈ H ( B Ham(
M, ω ); Z ) gives rise to the universal characteristic class of Hamiltonianfibrations.Let B Ham(
M, ω ) δ be the classifying space of the discrete group Ham( M, ω ) δ ,then there is the canonical map ι : B Ham(
M, ω ) δ → B Ham(
M, ω ). The co-homology class ι ∗ e ω ∈ H ( B Ham(
M, ω ) δ ; Z ) gives rise to the universal charac-teristic class of flat Hamiltonian fibrations. Note that the singular cohomology H ( B Ham(
M, ω ) δ ; Z ) is isomorphic to the group cohomology H (Ham( M, ω ); Z ).Thus the universal characteristic class ι ∗ e ω ∈ H ( B Ham(
M, ω ) δ ; Z ) of flat Hamilton-ian fibrations can be regarded as a group cohomology class in H (Ham( M, ω ); Z ).By abuse of notation, we use the symbol ι ∗ e ω to denote the corresponding groupcohomology class.In section 4, we will give examples of Hamiltonian fibrations for which the char-acteristic classes e ω and ι ∗ e ω are non-zero. In particular, we have the followingtheorem. Theorem 1.2.
Let n be a positive integer. For the complex projective space C P n with the Fubini-Study form ω F S , the universal characteristic classes e ω ∈ H ( B Ham( C P n , ω F S ); Z ) , ι ∗ e ω ∈ H ( B Ham( C P n , ω F S ) δ ; Z )are non-zero.1.3. Main Theorem.
In this paper, we regard S as the quotient group R / Z . Let( M, ω ) be a symplectic manifold as above. Then there is a prequantization S -bundle P → M , that is, the total space P is a contact manifold such that the contact form CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 3 θ is a connection form and its curvature form is equal to ω . The quantomorphismgroup Quant(
P, θ ) is defined byQuant(
P, θ ) = { ϕ : P → P | ϕ ∗ θ = θ } . Then the identity component Q = Quant( P, θ ) of the quantomorphism group givesthe following central S -extension (see [6])0 → S → Q → Ham(
M, ω ) → . (1.1)In general, a central S -extension of G determines a second group cohomology classin H ( G ; S ). Thus, the above central S -extension (1.1) defines a group cohomol-ogy class e ( Q ) in H (Ham( M, ω ); S ). Let us consider the connecting homomor-phism δ : H (Ham( M, ω ); S ) → H (Ham( M, ω ); Z )(1.2)with respect to the short exact sequence 0 → Z → R → S →
0. Then weobtain the group cohomology class δe ( Q ) ∈ H (Ham( M, ω ); Z ). The followingtheorem clarifies the relation between δe ( Q ) and the universal characteristic class ι ∗ e ω ∈ H (Ham( M, ω ); Z ) of Hamiltonian fibrations. Theorem 1.3.
The universal characteristic class ι ∗ e ω is equal to δe ( Q ).Moreover, we also give an explicit group three-cocycle of the class δe ( Q ). Thecocycle is defined in a similar way to the two-cocycle defined in [4] (see (5.1) insubsection 5.2).Let H b (Ham( M, ω ); Z ) denote the third bounded cohomology of the Hamiltoniandiffeomorphism group Ham( M, ω ). Since the connecting homomorphism (1.2) fac-tors the map δ b : H (Ham( M, ω ); S ) → H b (Ham( M, ω ); Z ) and the comparisonmap r : H b (Ham( M, ω ); Z ) → H (Ham( M, ω ); Z ) (see Lemma 2.2), the universalcharacteristic class ι ∗ e ω is bounded. By the non-triviality of the universal charac-teristic class ι ∗ e ω (Theorem 1.2), we obtain the following corollary. Corollary 1.4.
The bounded cohomology class δ b e ( Q ) ∈ H b (Ham( C P n , ω F S ); Z ) isnon-zero. In particular, the third bounded cohomology H b (Ham( C P n , ω F S ); Z ) isnon-trivial. 2. Preliminaries
Group cohomology and bounded cohomology.
Let G be a group and A an abelian group. Then the set of all functions C p grp ( G ; A ) = { c : G p → A } from p -fold product G p to A is called the p -cochain group of G . The coboundary map δ : C p grp ( G ; A ) → C p +1grp ( G ; A ) is defined by δc ( g , . . . , g p +1 ) = c ( g , . . . , g p +1 ) + p X i =1 ( − i c ( g , . . . , g i g i +1 , . . . , g p +1 )+ ( − p +1 c ( g , . . . , g p ) SHUHEI MARUYAMA for p > δ = 0 for p = 0. The cohomology of the cochain complex ( C ∗ grp ( G ; A ) , δ )is called the group cohomology of G and denoted by H ∗ grp ( G ; A ).For a group G , the group cohomology H ∗ grp ( G ; A ) and the singular cohomology H ∗ ( BG δ ; A ) of the classifying space BG δ are isomorphic, where G δ denote the group G with discrete topology. If we take a model of the classifying space BG δ as a fatrealization of the nerve N G δ , then the isomorphism of cohomologies are obtainedfrom the isomorphism between the cell complex C ∗ cell ( BG δ ; A ) of BG δ and the groupcomplex C ∗ grp ( G ; A ) (see, for example, [2, Chapter 5]).Let G be a topological group. The canonical map G δ → G induces the contin-uous map ι : BG δ → BG . Thus we obtain the induced map ι ∗ : H ∗ ( BG ; Z ) → H ∗ ( BG δ ; Z ). About this map ι ∗ , the following theorem is known, which we will usein the proof of the non-triviality of cohomology classes. Theorem 2.1. [9] Let G be a finite-dimensional Lie group with finite connectedcomponents. Then the map ι ∗ : H ∗ ( BG ; Z ) → H ∗ ( BG δ ; Z ) is injective.Let A be Z or R . The all bounded functions C pb ( G ; A ) = { c : G p → A : bounded } defines a subcomplex of ( C ∗ grp ( G ; A ) , δ ) since the above coboundary map preservesthe boundedness. The cohomology of the above subcomplex is called the bounded co-homology of G and denoted by H ∗ b ( G ; A ). The inclusion map C ∗ b ( G ; A ) ֒ → C ∗ grp ( G ; A )induces a map r : H ∗ b ( G ; A ) → H ∗ grp ( G ; A ) from the bounded cohomology to the or-dinary group cohomology, which is called the comparison map .Let us consider the short exact sequence 0 → Z → R → S →
0, Then we havethe cohomology long exact sequences · · · → H p grp ( G ; Z ) → H p grp ( G ; R ) → H p grp ( G ; S ) δ −→ H p +1grp ( G ; Z ) → · · · and · · · → H pb ( G ; Z ) → H pb ( G ; R ) → H p grp ( G ; S ) δ b −→ H p +1 b ( G ; Z ) → · · · , where δ and δ b are the connecting homomorphisms. By the definition of connectinghomomorphisms and the comparison map, we have the following lemma. Lemma 2.2.
Let G be a group. Then the connecting homomorphism δ factors themap δ b : H p grp ( G ; S ) → H p +1 b ( G ; Z ) and the comparison map r : H p +1 b ( G ; Z ) → H p +1grp ( G ; Z ).2.2. Hochschild-Serre spectral sequence.
For a group extension 1 → K → Γ π −→ G →
1, there is the following spectral sequence called the Hochschild-Serre spectralsequence.
Theorem 2.3. [3] There exists the spectral sequence with E p,q ∼ = H p grp ( G ; H q grp ( K ))which converges to H ∗ grp (Γ).To describe the derivations, let us recall the filtration ( A ∗ np ) of C n grp (Γ) whichdefines the Hochschild-Serre spectral sequence. For p ≤
0, define A ∗ np = C n grp (Γ) andfor p > n , define A ∗ np = 0. For 0 < p ≤ n , the group A ∗ np is defined as the groupof all functions c in C n grp (Γ) which satisfy that the value c ( γ , . . . , γ n ) depends only CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 5 on γ , . . . , γ n − p ∈ Γ and π ( γ n − p +1 ) , . . . , π ( γ n ) ∈ G . Put Z p,qr = { c ∈ A ∗ p + qp | δc ∈ A ∗ p + q +1 p + r } , then the Hochschild-Serre spectral sequence E p,qr is given by E p,qr = Z p,qr / ( Z p − ,q +1 r − + δZ p +1 − r,q + r − r − ) . The derivations d p,qr : E p,qr → E p + r,q − r +1 r are induced from the coboundary map δ .2.3. Central extension and second group cohomology.
An exact sequence1 → A i −→ Γ p −→ G → central A -extension of G if the image i ( A ) is in the center of Γ. The second group cohomology H ( G ; A ) is isomorphicto the equivalence classes of central A -extensions of G (see [1]). For a central A -extension Γ, the corresponding cohomology class e (Γ) is defined as follows. Takea section s : G → Γ of the projection p : Γ → G . For any g, h ∈ G , the value s ( g ) s ( h ) s ( gh ) − is in i ( A ) ∼ = A . Thus we obtain a cochain c ∈ C ( G ; A ) by putting c ( g, h ) = s ( g ) s ( h ) s ( gh ) − . (2.1)It can be seen that the cochain c is a cocycle, and its cohomology class [ c ] does notdepend on the section. We put e (Γ) = [ c ]. This e (Γ) is the class that correspondsto the central A -extension Γ.Using the Hochschild-Serre spectral sequence, the cohomology class e (Γ) can bedescribed as follows. Lemma 2.4.
Let 1 → A → Γ π −→ G → A -extension of G and E p,qr theHochschild-Serre spectral sequence of the central extension. Then the correspondingcohomology class e (Γ) is equal to the negative of d , (id A ), where d , : H ( A ; A ) = E , → E , = H ( G ; A ) is the derivation of the spectral sequence. Proof.
Take a section s : G → Γ. Define a cochain x ∈ C (Γ; A ) by putting x ( γ ) = γ · ( sπ ( γ )) − . Then x | A is equal to id A and δx ∈ C (Γ; A ) defines a cocyclein C ( G ; A ) which is equal to the negative cocycle − c of the central extension.Thus, by definition of the derivation of the Hochschild-Serre spectral sequence, wehave d (id A ) = − e (Γ). (cid:3) Proofs
For the proof of Theorem 1.3, we prepare the following lemmas.
Lemma 3.1.
Let 1 → A → Γ → G → A -extension of G and BA δ → B Γ δ → BG δ be the corresponding fibration of classifying spaces of discretegroups. Then the obstruction class o (id , B Γ δ ) ∈ H ( BG δ ; A ) coincides with theclass e (Γ) ∈ H ( G ; A ) under the isomorphism H ( BG δ ; A ) ∼ = H ( G ; A ). Proof.
Let s : G → Γ be a section. Take the model of classifying space BG δ asthe fat realization k N G δ k . For each 2-cell ( g, h ) : ∆ → BG δ , we obtain the lift( g, h ) | ∂ ∆ : ∂ ∆ → B Γ δ by using the section s : G → Γ. From the contractibilityof the standard simplex ∆ , we obtain the homotopy class h ( g, h ) ∈ π ( BA δ ) = A which corresponds to c ( g, h ) = s ( g ) s ( h ) s ( gh ) − ∈ A . Since h ∈ C ( BG δ ; A ) is the SHUHEI MARUYAMA obstruction cocycle and c is the cocycle of e (Γ), we have o (id , B Γ δ ) = [ h ] = [ c ] = e (Γ). (cid:3) Lemma 3.2.
Let H ( S ; S ) δ −→ H ( S ; Z ) be the connecting homomorphism,then the class δ (id S ) is equal to the negative of the class e ( R ) in H ( S ; Z ) thatcorresponds to the central Z -extension 0 → Z → R → S → Proof.
Let l : S = R / Z → [0 , ⊂ R be the section of the projection R → S . By the definition of the connecting homomorphism, we have δ (id S ) = [ δl ] ∈ H ( S ; Z ). By definition of the cocycle (2.1) of e ( R ), the class − [ δl ] is equal to e ( R ). (cid:3) Lemma 3.3.
Let 0 → S → Γ π −→ G → S -extension. Then thefollowing diagram H ( S ; S ) d , / / δ (cid:15) (cid:15) H ( G ; S ) δ (cid:15) (cid:15) H ( S ; Z ) − d , / / H ( G ; Z )commutes, where δ are the connecting homomorphisms and d , and d , are thederivations of the Hochschild-Serre spectral sequence of the central S -extension. Proof.
Take an element ϕ ∈ H ( S ; S ) and section s : G → Γ such that s (1 G ) = 1 Γ where 1 G and 1 Γ are the identity elements of G and Γ respectively. Put ϕ s : Γ → S by ϕ s ( γ ) = ϕ ( γ · ( sπ ( γ )) − ). Then ϕ s | S = ϕ and δϕ s ∈ C (Γ; A ) defines acocycle δϕ s in C ( G ; A ), that is, δϕ s = π ∗ δϕ s . Thus we have d , ( ϕ ) = [ δϕ s ] ∈ H ( G ; S ). By definition of the connecting homomorphism, we have δd , ( ϕ ) = δ [ δϕ s ] = [ δ ( lδϕ s )] ∈ H ( G ; Z ). Next, we calculate d , δ ( ϕ ). By definition ofthe connecting homomorphism, δ ( ϕ ) is equal to [ δ ( lϕ )] ∈ H ( S ; Z ). Define c ∈ C (Γ , Z ) by c = δ ( lϕ s ) − π ∗ ( lδϕ s ). Then c | S × S = δ ( lϕ ) ∈ C ( S ; Z ) and δc = − δ ( π ∗ ( lδϕ s )) = − π ∗ ( δ ( lδϕ s )). Thus we have d , δ ( ϕ ) = − [ δ ( lδϕ s )] ∈ H ( G ; Z )and the lemma follows. (cid:3) Lemma 3.4.
Let (
M, ω ) be an integral symplectic manifold and (
M, ω ) → E → B Ham(
M, ω ) the universal Hamiltonian fibration. Let BS → BQ → B Ham(
M, ω )be the fibration that corresponds to the central S -extension (1.1). Then, there isthe following commuting diagram of fibrations M / / f (cid:15) (cid:15) E / / φ (cid:15) (cid:15) B Ham(
M, ω ) BS / / BQ / / B Ham(
M, ω ) , (3.1)where the map f : M → BS is a classifying map of the prequantization S -bundleover M . CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 7
Proof.
The bundle E = E Ham(
M, ω ) × Q P → B Ham(
M, ω )gives one of a model of the universal Hamiltonian fibation. Then there is a principal Q -bundle E Ham(
M, ω ) × P → E . We take a bundle map to the universal Q -bundle E Ham(
M, ω ) × P Ψ / / (cid:15) (cid:15) EQ (cid:15) (cid:15) E ψ / / BQ.
The Q -bundle E Ham(
M, ω ) × EQ → E Ham(
M, ω ) × Q EQ gives another model ofthe universal Q -bundle. Then the mapΦ : E Ham(
M, ω ) × P → E Ham(
M, ω ) × EQ ; ( a, p ) ( a, Ψ( a, p ))gives a bundle map. Let φ : E → E Ham(
M, ω ) × Q EQ = BQ denote the classifyingmap that is covered by Φ. Then it can be seen that the map φ covers the identityand the restriction f : M → BS to the fiber gives rise to the classifying map of thebundle P → M . (cid:3) Proof of Theorem 1.3.
Take a commuting diagram (3.1). Consider the Serre spectralsequences E p,qr and E ′ p,qr of the fibrations M → E → B Ham(
M, ω ) and BS → BQ → B Ham(
M, ω ) respectively. Since H ( BS ; Z ) = 0, we have E ′ , = E ′ , = H ( BS ; Z ) and E ′ , = E ′ , = H ( B Ham(
M, ω ); Z ). By the naturality of theSerre spectral sequence, we have the commuting diagram H ( BS ; Z ) d ′ , / / f ∗ (cid:15) (cid:15) H ( B Ham(
M, ω ); Z ) H ( M ; Z ) d , / / H ( B Ham(
M, ω ); Z ) . Since [ ω ] Z is equal to the first Chern class of the prequantization bundle P → M ,we have [ ω ] Z = f ∗ ( c ), where c ∈ H ( BS ; Z ) is the universal first Chern class.By the commuting diagram above, we have d , ([ ω ] Z ) = d , f ∗ ( c ) = d ′ , ( c ) ∈ H ( B Ham(
M, ω ); Z ). Let E ′′ p,qr denote the Serre spectral sequence of the fibration BS δ → BQ δ → B Ham(
M, ω ) δ (or, equivalently, the Hochschild-Serre spectralsequence of the central S -extension 0 → S → Q → Ham(
M, ω ) → H ( BS δ ; Z ) = 0, we have E ′′ , = E ′′ , = H ( BS δ ; Z ). By the naturality of theSerre spectral sequence for the fibrations BS δ / / (cid:15) (cid:15) BQ δ / / (cid:15) (cid:15) B Ham(
M, ω ) δ (cid:15) (cid:15) BS / / BQ / / B Ham(
M, ω ) , SHUHEI MARUYAMA we have the commuting diagram H ( BS ; Z ) d ′ , / / ι ∗ (cid:15) (cid:15) H ( B Ham(
M, ω ); Z ) ι ∗ (cid:15) (cid:15) H ( BS δ ; Z ) d ′′ , / / H ( B Ham(
M, ω ) δ ; Z ) . Thus we have ι ∗ d , ([ ω ] Z ) = ι ∗ d ′ , ( c ) = d ′′ , ι ∗ ( c ). Since the map ι : BS δ → BS is a classifying map of the S -bundle B Z = S → B R δ → BS δ , the class ι ∗ ( c )is the first Chern class of the S -bundle, and thus equal to the obstruction class o (id , B R δ ). Applying Lemma 3.1 to the central Z -extension 0 → Z → R → S → o (id , B R δ ) = e ( R ). Together with Lemma 2.4, Lemma 3.2, and Lemma 3.3,we have ι ∗ e ω = − ι ∗ d , ( c ( Q )) = − d ′′ , ( o (id , BQ δ ))= − d ′′ , ( e ( R )) = d ′′ , δ (id S ) = − δd ′′ , (id S ) = δ ( e ( Q )) . (cid:3) By similar arguments, we have the following.
Theorem 3.5.
Let 0 → S → Γ → G → S -extension such that theprojection Γ → G gives a principal S -bundle. Let c (Γ) ∈ H ( G ; Z ) denote thefirst Chern class of the S -bundle and E p,qr denote the Serre spectral sequence of theuniversal bundle G → EG → BG . Assume that G is connected. Then we havei) c (Γ) ∈ E , ,ii) − ι ∗ d , ( c (Γ)) = δ ( e (Γ)). 4. Examples
In this section, we show that the universal characteristic class e ω and ι ∗ e ω are non-zero for (flat) Hamiltonian fibrations whose fiber is the complex projective space.Let us consider the central S -extension of the projective unitary group(4.1) 0 → S → U ( n ) → P U ( n ) → , where we regard S as the unitary group U (1). Lemma 4.1.
Let e ( P U ( n )) ∈ H ( P U ( n ); S ) denote the class correspondingto the central extension (4.1). Then the class δe ( P U ( n )) ∈ H ( P U ( n ); Z ) and δ b e ( P U ( n )) ∈ H b ( P U ( n ); Z ) are non-zero. Proof.
The S -bundle (4.1) is non-trivial since the fundamental groups of U ( n ) and S × P U ( n ) are different. So the first Chern class c ∈ E , ⊂ H ( P U ( n ); Z )of the bundle (4.1) is non-zero. Note that the derivation d , : E , → E , ∼ = H ( BP U ( n ); Z ) and the map ι ∗ : H ( BP U ( n ); Z ) → H ( BP U ( n ) δ ; Z ) are injec-tive by Theorem 2.1. Thus the class − δe ( P U ( n )) = ιd , c ∈ H ( P U ( n ); Z ) isnon-zero. Since δe ( P U ( n )) = rδ b e ( P U ( n )) by lemma 2.2, the class δ b e ( P U ( n )) ∈ H b ( P U ( n ); Z ) is also non-zero. (cid:3) CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 9
Let (
M, ω ) be the complex projective space C P n with the Fubini-Study form ω FS .For this symplectic manifold ( C P n , ω F S ), its prequantization bundle is the Hopffibration S → S n +1 p −→ C P n with the connection form θ = z dz + · · · + z n dz n , where we consider the sphere S n +1 as the subspace in C n +1 with coordinate system ( z , . . . , z n ). Let us recallthat there is a central S -extension (1.1)0 → S → Q → Ham( C P n , ω F S ) → , where the group Q is the identity component of the quantomorphism group. Proof of Theorem 1.2.
By theorem 1.3, it is enough to show that the cohomologyclass δe ( Q ) ∈ H (Ham( C P n , ω F S ); Z ) is non-zero.Since the action on S n +1 by U ( n +1) preserves the connection form θ , the unitarygroup is included in Q . Since the inclusion is S -equivariant, we have the commutingdiagram 1 / / S / / U ( n + 1) / / (cid:15) (cid:15) P U ( n + 1) f (cid:15) (cid:15) / / / / S / / Q / / Ham( C P n , ω F S ) / / e ( P U ( n + 1)) = f ∗ e ( Q ) ∈ H ( P U ( n + 1); S ). Let us consider thecommuting diagram H (Ham( C P n , ω F S ); S ) f ∗ / / δ (cid:15) (cid:15) H ( P U ( n + 1); S ) δ (cid:15) (cid:15) H (Ham( C P n , ω F S ); Z ) f ∗ O O H ( P U ( n + 1); Z ) , where δ denote the connecting homomorphisms. Then we have f ∗ δe ( Q ) = δf ∗ e ( Q ) = δe ( P U ( n + 1)) and, by lemma 4.1, the last term δe ( P U ( n + 1)) is non-zero. Thusthe classes δe ( Q ) ∈ H (Ham( C P n , ω F S ); Z ) is non-zero. (cid:3) Cocycles
In this section, we give explicit cocycles of the group cohomology classes e ( Q ) ∈ H (Ham( M, ω ); S ) and ι ∗ e ω = δe ( Q ) ∈ H (Ham( M, ω ); Z ). These cocycles aredefined in a similar way to the two-cocycle defined in [4].5.1. A cocycle of e ( Q ) . Let (
M, ω ) be a simply-connected integral symplecticmanifold. Let (Ω ∗ ( M ) , d ) denote the deRham complex and ( C ∗ ( M ; R ) , δ ) the C ∞ -singular cochain complex with coefficients in R . Then there is the canonical cochainmap I : Ω n ( M ) → C n ( M ; R ); η → I η defined by I η ( σ ) = R σ η , where η is an n -form in Ω n ( M ) and σ is a C ∞ -singular n -simplex. By deRham theorem, the map I induces the isomorphism of the coho-mologies. By straight forward calculation, we have the following. Lemma 5.1.
The cochain map I is compatible with the pullbacks. In particular,all elements in Ham( M, ω ) preserve the cocycle I ω .Let j : R → S denote the projection. Let us consider the cohomology long exactsequence · · · −→ H ( M ; Z ) −→ H ( M ; R ) j ∗ −→ H ( M ; S ) −→ · · · . Since the symplectic form ω is integral, the cohomology class j ∗ [ I ω ] = [ jI ω ] is equal tozero. We take a singular one-cochain α ∈ C ( M ; S ) such that δα = jI ω . By lemma5.1, the cochain α − g ∗ α is a cocycle for any g ∈ Ham(
M, ω ). Since M is simply-connected, the one-cocycle α − g ∗ α ∈ C ( M ; S ) is a coboundary. Take a point x ∈ M and a coboundary K α ( g ) ∈ C ( M ; S ) of α − g ∗ α . For g, h ∈ Ham(
M, ω ), weput G x,α ( g, h ) = Z hxx α − g ∗ α = K α ( g )( h ( k )) − K α ( g )( x ) . Here the symbol R hxx denotes the pairing of the cocycle α − g ∗ α and a path from x to hx . Then G x,α is a group two-cochain in C (Ham( M, ω ); S ). By the samearguments in [4, Theorem 3.1] we have the following. Proposition 5.2.
The cochain G x,α is a cocycle. Moreover, the cohomology class[ G x,α ] does not depend on the choice of x and α .Then the following theorem holds. Theorem 5.3.
The cohomology class [ G x,α ] is equal to e ( Q ).For the proof of theorem 5.3, we show the following lemmas. Let us recall that p : P → M is the prequantization S -bundle with the contact form θ ∈ Ω ( P )satisfying dθ = p ∗ ω . Lemma 5.4.
The cochain jI θ − p ∗ α ∈ C ( P ; S ) is a coboundary, that is, there isa zero-cochain β ∈ C ( P ; S ) such that δβ = jI θ − p ∗ α . Proof.
By lemma 5.1 and the definition of α , we have δ ( jI θ − p ∗ α ) = jI p ∗ α − p ∗ jI ω = jp ∗ I ω − jp ∗ I ω = 0 . Thus the cochain jI θ − p ∗ α is a cocycle. Let γ ′ be a loop in P . Since M is simply-connected, the map π ( S ) → π ( P ) is surjective. Thus there is a loop γ in a fiberof P → M that homotopic to the loop γ ′ . Thus we have Z γ ′ jI θ − p ∗ α = Z γ jI θ − p ∗ α = Z γ jI θ − Z pγ α = Z γ jI θ , where the symbol R γ denotes the pairing of a cocycle and a cycle. Moreover, thelast term is equal to the projection of the value R γ θ ∈ R to S . Since the form θ is CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 11 a connection form, the value R γ θ is in Z , that is, R γ jI θ = 0 holds. Thus the cocycle jI θ − p ∗ α is cohomologous to 0 and the lemma follows. (cid:3) By lemma 5.4, we take a singular cochain β ∈ C ( X ; S ) satisfying δβ = jI θ − p ∗ α .Take a base point y ∈ P such that p ( y ) = x . Define a group cochain τ ∈ C ( Q ; S )by putting τ ( ϕ ) = Z ϕyy jI θ − p ∗ α = β ( ϕy ) − β ( y )for ϕ ∈ Q . Lemma 5.5.
The restriction τ | S : S → S is equal to the identity. Proof.
For u ∈ S ⊂ Q , take a path γ from y to y · u in the fiber over p ( y ) = x .Then we have τ ( u ) = Z y · uy jI θ − p ∗ α = j Z γ θ = u and the lemma follows. (cid:3) Lemma 5.6.
The equation − δτ = π ∗ G x,α ∈ C ( Q ; S )holds, where π : Q → Ham(
M, ω ) is the projection.
Proof.
Take elements ϕ, ψ in Q and put g = π ( ϕ ) and h = π ( ψ ). Note that, bydefinition of the quantomorphism group Q and lemma 5.1, we have ϕ ∗ jI θ = jI θ and ψ ∗ jI θ = jI θ . Thus we obtain − δτ ( ϕ, ψ ) = Z ϕψyy − Z ϕyy − Z ψyy jI θ − p ∗ α = Z ϕψyϕy − Z ψyy jI θ − p ∗ α = Z ψyy ϕ ∗ ( jI θ − p ∗ α ) − ( jI θ − p ∗ α )= Z ψyy − p ∗ g ∗ α + p ∗ α = Z hxx α − g ∗ α = G x,α ( g, h ) = π ∗ G x,α ( ϕ, ψ )and the lemma follows. (cid:3) Proof of theorem 5.3.
Let E p,qr denote the Hochschild-Serre spectral sequence of 0 → S → Q → G → S . Then there is the derivation d , : H ( S ; S ) = E , → E , = H ( G ; S ) . By lemma 5.5, lemma 5.6, and the definition of the derivation of the spectral se-quence, we have d , (id S ) = − [ G x,α ]. On the other hand, by lemma 2.4, we have d , = − e ( Q ). Thus we have [ G x,α ] = e ( Q ). (cid:3) A cocycle of ι ∗ e ω = δe ( Q ) . In this subsection, we define a group three-coycle H x,σ,w of Ham( M, ω ) and show that the cohomology class [ H x,σ,w ] coincides with theclass ι ∗ e ω = δe ( Q ) in H (Ham( M, ω ); Z ).For x ∈ M , let P x = { γ : [0 , → M | γ (0) = x } denote the based path spaceof M . Take a section σ : M → P x of the projection P x → M ; γ → γ (1). Bysimply-connectedness of M , we take a disk D ( g, h ) whose boundary is the loop σ ( gx ) − σ ( ghx ) + gσ ( hx ) for g, h ∈ Ham(
M, ω ), Let w ∈ C ( M ; Z ) be a cocycle of[ ω ] Z ∈ H ( M ; Z ). Then we define a group cochain H x,σ,w ∈ C (Ham( M, ω ); Z ) byputting H x,σ,w ( f, g, h ) = Z D ( g,h ) f ∗ w − w (5.1)for f, g, h ∈ Ham(
M, ω ). Since the cocycle f ∗ w − w is a coboundary, the value H x,σ,w ( f, g, h ) dose not depend on the choice of D ( g, h ). The proof of the followingproposition is straight forward. Proposition 5.7.
The group cochain H x,σ,w is a cocycle. Lemma 5.8.
The cohomology class [ H x,σ,w ] in H (Ham( M, ω ); Z ) is independentof the choice of x, σ, w . Proof.
Let y ∈ M be another point, γ a path from x to y , and σ ′ : M → P y asection. Let D ′ ( g, h ) be a disk whose boundary is σ ′ ( gx ) − σ ′ ( ghx ) + gσ ′ ( hx ) and S ( g ) a disk whose boundary is γ + σ ′ ( gy ) − gγ − σ ( gx ). We put I ( g, h ) = Z S ( h ) g ∗ w − w, then we have Z D ( g,h ) f ∗ w − w − Z D ′ ( g,h ) f ∗ w − w = δ I ( f, g, h ) . This implies that the class is independent of x and σ . Let w ′ ∈ C ( M ; Z ) be anothercocycle of [ w ] Z and take a cochain v ∈ C ( M ; Z ) such that δv = w ′ − w . We put I ′ ( g, h ) = Z σ ( hx ) g ∗ v − v, then we have Z D ( g,h ) f ∗ w − w − Z D ( g,h ) f ∗ w ′ − w ′ = δ I ′ ( f, g, h ) . This implies that the class is independent of w . (cid:3) The following theorem holds.
Theorem 5.9.
The cohomology class [ H x,σ,w ] is equal to ι ∗ e ω = δe ( Q ). CHARACTERISTIC CLASS OF HAMILTONIAN FIBRATIONS 13
Proof.
By theorem 5.3, a cocycle of e ( Q ) is given by G x,α . Let α ∈ C ( M ; R ) bea lift of α , that is, the cochain α satisfies jα = α under the map j : C ( M ; R ) → C ( M ; S ). We put G x,α ( g, h ) = Z σ ( hx ) α − g ∗ α. Since the group cochain G x,α in C (Ham( M, ω ); R ) is a lift of G x,α , a cocycle of δe ( Q ) is given by δ G x,α ∈ C (Ham( M, ω ); Z ) . For f, g, h ∈ Ham(
M, ω ), we have δ G x,α ( f, g, h )= Z γ ( hx ) α − g ∗ α − Z γ ( hx ) α − g ∗ f ∗ α + Z γ ( ghx ) α − f ∗ α − Z γ ( gx ) α − f ∗ α = Z γ ( ghx ) − γ ( gx ) − gγ ( hx ) α − f ∗ α = Z D ( g,h ) δ ( α − f ∗ α )= Z D ( g,h ) ( δα − I ω ) − f ∗ ( δα − I ω ) = Z D ( g,h ) f ∗ ( I ω − δα ) − ( I ω − δα ) , where the third equality follows from f ∗ I ω = I ω (lemma 5.1). For the projection j : C ∗ ( M ; R ) → C ∗ ( M ; S ), we have j ( I ω − δα ) = jI ω − δjα = jI ω − jI ω = 0 . So the cocycle I ω − δα is in C ( M ; Z ) and this cocycle represents [ ω ] Z . Thus, if weput w = I ω − δα , we have H x,σ,w = δ G x,α and the theorem follows. (cid:3) Remark 5.10.
Let l : S → [0 , ⊂ R be the section of R → S and we put α = lα .For this lift α of α , the group two-cochain G x,α gives rise to a bounded two-cochain.Thus, when w = I ω − δα = I ω − lα , the group three-cocycle H x,σ,w is a boundedcocycle that represents δ b e ( Q ). Remark 5.11.
By Theorem 1.2, the cohomology class δe ( Q ) and the boundedcohomology class δ b e ( Q ) are non-trivial for the complex projective space ( C P n , ω F S ).This gives an example that the (bounded) cocycle H x,σ,w is cohomologically non-trivial. References
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