A characteristic class of Homeo(X ) 0 -bundles and an abelian extension of the homeomorphism group
aa r X i v : . [ m a t h . G T ] S e p A CHARACTERISTIC CLASS OF
Homeo( X ) -BUNDLES ANDAN ABELIAN EXTENSION OF THE HOMEOMORPHISMGROUP SHUHEI MARUYAMA
Abstract.
A Homeo( X ) -bundle is a fiber bundle with fiber X whose structuregroup reduces to the identity component Homeo( X ) of the homeomorphism groupof X . We construct a characteristic class of Homeo( X ) -bundles as a third coho-mology class with coefficients in Z . We also investigate the relation between theuniversal characteristic class of flat fiber bundles and the gauge group extension ofthe homeomorphism group. Furthermore, under some assumptions, we constructand study the central S -extension and the corresponding group two-cocycle ofHomeo( X ) . Introduction and Main Theorem
Introduction.
Let X be a topological space ∗ and c ∈ H ∗ ( X ; Z ) a non-trivialcohomology class of X . Let Homeo( X ) denote the homeomorphism group of X withthe compact-open topology and G = Homeo( X ) its identity component. Let G δ denote the group G with the discrete topology. A fiber bundle X → E → B iscalled a G -bundle if the structure group reduces to G . A G -bundle is called flat ifthe structure group reduces to the discrete group G δ . In [2], a characteristic class e c ( E ) ∈ H ( B ; Z ) of the G -bundle is defined if c is in the first cohomology group H ( X ; Z ). The universal characteristic class e c is defined as an element in the secondcohomology H ( BG ; Z ) of the classifying space BG with coefficients in Z . Let ι ∗ : H ∗ ( BG ; Z ) −→ H ∗ ( BG δ ; Z )(1.1)be the map induced from the canonical map G δ → G . Then the cohomology class ι ∗ e c ∈ H ( BG δ ; Z ) gives the universal characteristic class of flat G -bundles. Since thecohomology H ∗ ( BG δ ; Z ) of the classifying space of the discrete group is isomorphicto the group cohomology H ∗ grp ( G ; Z ), the universal characteristic class ι ∗ e c givesthe second group cohomology class in H ( G ; Z ). On the other hand, the secondgroup cohomology H ( G ; Z ) is isomorphic to the equivalence classes of central Z -extensions of G . In [2], Fujitani constructed a central Z -extension of G thatcorresponds to the characteristic class ι ∗ e c . The central extension is given as ahomeomorphism group of the regular Z -covering of X . ∗ In this paper, we deal with topological spaces whose homeomorphism groups are topologicalgroups with respect to the compact-open topology. For example, the homeomorphism group oflocally compact, locally connected topological space is a topological group.
In the present paper, we consider the case when the first cohomology H ( X ; Z )is trivial and the second cohomology H ( X ; Z ) is non-trivial. In this case, thecharacteristic class e c above cannot be defined since H ( X ; Z ) = 0. On the otherhand, we can define a characteristic class e c ( E ) in H ( B ; Z ) by using a secondcohomology class c ∈ H ( X ; Z ). More precisely, the class e c is obtained as thetransgression image of the class c by the Serre spectral sequence of the G -bundle X → E → B (see section 2). Let e c ∈ H ( BG ; Z ) denote the universal characteristicclass. By the canonical map (1.1) and the isomorphism H ( BG δ ; Z ) ∼ = H ( G ; Z ),we obtain the third group cohomology class ι ∗ e c in H ( G ; Z ).1.2. Main results.
In this paper, we regard the circle S as the quotient group R / Z . Take a cohomology class c ∈ H ( X ; Z ). Let P → X be a principal S -bundlesuch that the first Chern class is equal to c . Then there is the following exactsequence 0 −→ Gau( P ) −→ A G ( P ) −→ G −→ , (1.2)where the group Gau( P ) is the gauge group of P and A G ( P ) is the bundle automor-phisms that cover elements in G . Since the group S is abelian, the gauge groupis also abelian. So the exact sequence (1.2) is an abelian extension. In general,an abelian extension 1 → A → Γ → G → e (Γ) ∈ H ( G ; A ). Thus the abelian extension (1.2) defines a groupcohomology class e ( A G ( P )) in H ( G ; Gau( P )).Here we obtain two group cohomology classes e ( A G ( P )) ∈ H ( G ; Gau( P )) andthe universal characteristic class ι ∗ e c ∈ H ( G ; Z ). The following theorem describesthe relation between these two classes. Theorem 1.1.
Let δ : H ( G ; Gau( P )) → H ( G ; Z ) denote the connecting ho-momorphism (defined below). Then the following holds; δe (Aut( P )) = ι ∗ e c . The connecting homomorphism δ is given as follows. Since the group S is abelian,the gauge group Gau( P ) is isomorphic to the group C ( X, S ) = { f : X → S : continuous } . Let C ( X, R ) denote the group of all continuous maps from X to R . By the assump-tion H ( X ; Z ) = 0 and the isomorphism H ( X ; Z ) ∼ = Hom( π ( X ); Z ), all elementsin C ( X, S ) can be lifted to elements in C ( X, R ). So we have the following exactsequence 0 −→ Z −→ C ( X, R ) −→ C ( X, S ) ∼ = Gau( P ) −→ . (1.3)By the exact sequence (1.3) and the identification Gau( P ) ∼ = C ( X, S ), we have theconnecting homomorphism δ : H ( G ; Gau( P )) −→ H ( G ; Z ) . This is the map in Theorem 1.1.
CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 3 From here we assume that the cohomology class c ∈ H ( X ; Z ) is equal to zeroin H ( X ; R ). Let δ : H ( G ; S ) → H ( G ; Z ) be the connecting homomorphismwith respect to the exact sequence 0 → Z → R → S → · · · −→ H ( X ; S ) δ −→ H ( X ; Z ) −→ H ( X ; R ) −→ · · · , we take a cohomology class ρ ∈ H ( X ; S ) such that δ ( ρ ) = c . Then we canconstruct a central S -extension1 −→ S −→ A G ( P δρ ) −→ G −→ S ) δ -bundle P δρ over X (see section 5). This central S -extension determines a second group cohomology class e ( A G ( P δρ )) in H ( G ; S ).For this class e ( A G ( P δρ )), we have the following theorem, that is analogous to The-orem 1.1. Theorem 1.2.
Let δ : H ( G ; S ) → H ( G ; Z ) be the connecting homomorphismwith respect to the exact sequence 0 → Z → R → S → δe ( A G ( P δρ )) = ι ∗ e c . Moreover, we construct a group two-cocycle G x,α ∈ C ( G ; S ) that representsthe class e ( A G ( P δρ )) (for definition, see (5.5) in section 5).2. Construction of the characteristic class
Let X be a connected topological space satisfying H ( X ; Z ) = 0. Take a non-zero cohomology class c in H ( X ; Z ). Let X → E → B denote a G -bundle withconnected base space B . Let us consider the Serre spectral sequence E p,qr of thebundle X → E → B . Since the structure group G is connected, the local system H ∗ ( X ; Z ) is trivial. By the assumption H ( X ; Z ) = 0, we have E , = E , = H ( B ; H ( X ; Z )) = H ( X ; Z )and E , = E , = H ( B ; H ( X ; Z )) = H ( B ; Z ) . By the transgression map d , : H ( X ; Z ) = E , −→ E , = H ( B ; Z ) , we obtain the cohomology class d , c in H ( B ; Z ). Definition 2.1.
For a G -bundle X → E → B above, we put e c ( E ) = − d , c ∈ H ( B ; Z ) . By the naturality of the Serre spectral sequence, the cohomology class d , c hasalso the naturality with respect to bundle maps. Thus the class e c ( E ) gives rise toa characteristic class of G -bundles. Let e c ∈ H ( BG ; Z ) denote the universal char-acteristic class of G -bundles. By the canonical map ι ∗ : H ( BG ; Z ) → H ( BG δ ; Z ),we obtain the universal characteristic class ι ∗ e c of flat G -bundles. SHUHEI MARUYAMA
In section 6, we give examples of G -bundle that the characteristic class e c and ι ∗ e c are non-zero. Remark 2.2.
Let Homeo(
X, c ) denote the group of homeomorphisms that preservethe cohomology class c . Let X → E → B be a fiber bundle with the structure groupHomeo( X, c ) and E p,qr the Serre spectral sequence of the bundle. Then we have E , = E , = H ( B ; H ( X ; Z )) = H ( X ; Z ) π ( B ) , where H ( X ; Z ) π ( B ) is the π ( B )-invariant cohomology classes. Since the cohomol-ogy class c is in E , = H ( X ; Z ) π ( B ) , a characteristic class of Homeo( X, c )-bundlescan be defined in the same way.3.
Preliminaries
Group cohomology and Hochschild-Serre spectral sequence.
Let G bea group and A a right G -module. The group p -cochain c : G p → A is a functionfrom p -fold product G p to A . Let C p ( G ; A ) denote the set of all group p -cochains.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 ) · g p +1 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 with coefficients in A and denoted by H ∗ grp ( G ; A ).For a group G , the group cohomology H ∗ grp ( G ; A ) is isomorphic to the singular co-homology H ∗ ( BG δ ; A ) of the classifying space BG δ , where G δ denote the group G with discrete topology and A is the local system on BG δ (see [1]).For an exact sequence 1 → K → Γ π −→ G → Theorem 3.1. [4] There exists the spectral sequence with E p,q ∼ = H p grp ( G ; H q grp ( K ; A ))which converges to H ∗ grp (Γ; A ).Note that the group Γ acts on H q grp ( K ; A ) by conjugation and this action inducesthe G -action. By this action, we consider H q grp ( K ; A ) as a right G -module.3.2. Abelian extension and second group cohomology.
An exact sequence1 → A i −→ Γ p −→ G → abelian A -extension of G if the group A is abelian. The group Γ acts on A by conjugation, that is, for γ ∈ Γ and a ∈ A , weput γ · a = γ − aγ . This action induces the right G -action on A . So we consider theabelian group A as a right G -module. It is known that the second group cohomology H ( G ; A ) is isomorphic to the equivalence classes of abelian A -extensions of G (see[1]). For an abelian A -extension Γ, the corresponding cohomology class e (Γ) isdefined as follows. Take a section s : G → Γ of the projection p : Γ → G . For CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 5 any g, h ∈ G , the value s ( g ) s ( h ) s ( gh ) − is in i ( A ) ∼ = A . Thus we obtain a grouptwo-cochain c ∈ C ( G ; A ) by putting c ( g, h ) = s ( g ) s ( h ) s ( gh ) − . (3.1)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 cohomology class e (Γ) is the classthat corresponds to the abelian A -extension Γ.By definition of the derivations of the Hochschild-Serre spectral sequence, thecohomology class e (Γ) can be described as follows (see, for example, [6]). Lemma 3.2.
Let 1 → A → Γ π −→ G → A -extension of G and E p,qr the Hochschild-Serre spectral sequence of the abelian extension. Then thecorresponding cohomology class e (Γ) is equal to the negative of d , (id A ), where d , : H ( A ; A ) G = E , → E , = H ( G ; A ) is the derivation of the spectralsequence and H ( A ; A ) G is the G -equivariant homomorphisms.3.3. Cohomology of the gauge group.
Since the exact sequence0 −→ Z −→ C ( X, R ) π −→ C ( X, S ) ∼ = Gau( P ) −→ C ( X, R ) is contractible, the gauge group Gau( P ) isthe ( Z , K ( Z , B Gau( Z )is the ( Z , H ( B Gau( P ); Z ) = 0 , H ( B Gau( P ); Z ) = Z . Let i : R → C ( X, R ) and j : S → Gau( P ) = C ( X, S ) be inclusions, and ev R x : C ( X, R ) → R and ev x : C ( X, S ) → S the evaluation maps at x ∈ X . Then wehave the following commuting diagram of fibrations B Z / / B R / / i (cid:15) (cid:15) BS j (cid:15) (cid:15) B Z / / BC ( X, R ) / / ev R x (cid:15) (cid:15) B Gau( P ) ev x (cid:15) (cid:15) B Z / / B R / / BS , where we also use the symbols i, j, ev R x , and ev x for induced maps between theclassifying spaces due to the abuse of the notation. Since the composition j and ev x is equal to the identity, the composition ev ∗ x : H ∗ ( BS ; Z ) → H ∗ ( B Gau( P ); Z ) and j ∗ : H ∗ ( B Gau( P ); Z ) → H ∗ ( BS ; Z ) is also equal to the identity. Thus, the firstChern class of B Z → BC ( X, R ) → B Gau( P ) is equal to ev ∗ x c and non-zero, where c ∈ H ( BS ; Z ) is the universal first Chern class.Next we consider the first group cohomology H (Gau( P ); Z ) = Hom(Gau( P ) , Z ).For a homomorphism ψ : Gau( P ) → Z , the composition ψ ◦ π : C ( X, R ) → Z isalso a homomorphism. For any element f ∈ C ( X, R ), the map g = f / ∈ C ( X, R ) SHUHEI MARUYAMA satisfies f = g + g . Thus any homomorphism φ : C ( X, R ) → Z must be trivial.Since π is surjective, we have ψ = 0 and therefore H (Gau( P ); Z ) = 0.4. Proofs
For a cohomology class c ∈ H ( X ; Z ), let Homeo( X, c ) denote the group of c -preserving homeomorphisms. Lemma 4.1.
Let P → X be a principal S -bundle with the first Chern class c ∈ H ( X ; Z ). Let Aut( P ) denote the bundle automorphisms of P . Then the canonicalprojection p : Aut( P ) → Homeo( X ) gives the surjection Aut( P ) → Homeo(
X, c ).In particular, we have the following exact sequence0 −→ Gau( P ) −→ Aut( P ) −→ Homeo(
X, c ) −→ . Remark 4.2.
The exact sequence above gives rise to a Serre fibration with respectto the compact-open topology (see [8]).
Proof of lemma 4.1.
For a bundle automorphism F : P → P , the homeomorphism p ( F ) : X → X preserves the class c by the naturality of the Chern class. Thus thehomeomorphism p ( F ) is in Homeo( X, c ). For a homeomorphism f ∈ Homeo(
X, c ),let f ∗ P → X denote the pullback bundle of P by f . Since the first Chern classof f ∗ P → X is equal to c and the first Chern class completely determines theisomorphic class of principal S -bundles, the bundles f ∗ P → X and P → X areisomorphic. Thus we have the following diagram P ∼ = / / (cid:15) (cid:15) f ∗ P / / (cid:15) (cid:15) P (cid:15) (cid:15) X X f / / X and this gives a bundle automorphism of P that covers the homeomorphism f . (cid:3) Let A G ( P ) denote the group of bundle automorphisms that cover elements in G .Then, by lemma 4.1, we have the following exact sequence0 −→ Gau( P ) −→ A G ( P ) −→ G −→ . (4.1)Let E ′ p,qr and E ′′ p,qr denote the Hochschild-Serre spectral sequence of the exact se-quence (4.1) with coefficients in Gau( P ) and Z respectively. Then we have E ′ , = H ( G ; H (Gau( P ); Gau( P ))) = H (Gau( P ); Gau( P )) G , where the group H (Gau( P ); Gau( P )) G is the G -equivariant self-homomorphismon Gau( P ). Note that the G -action on C ( X, S ) = Gau( P ) induced from the A G ( P )-action is given by the pullback, that is, λ · g = g ∗ λ = λ ◦ g for λ ∈ C ( X, S )and g ∈ G . Since the first group cohomology H (Gau( P ); Z ) is trivial, we have E ′′ , = E ′′ , = H ( G ; H (Gau( P ); Z )) = H (Gau( P ); Z ) G , where the group H (Gau( P ); Z ) G is the G -invariant cohomology classes. CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 7 Lemma 4.3.
The connecting homomorphism δ : H (Gau( P ); Gau( P )) → H (Gau( P ); Z )with respect to the exact sequence (1.3) of coefficients induces the map δ : H (Gau( P ); Gau( P )) G −→ H (Gau( P ); Z ) G . Proof.
Let [ φ ] be an element in H (Gau( P ); Gau( P )) G . Note that the cocycle φ isa G -equivariant homomorphism, that is, the following holds φ ( λ · g ) = φ ( λ ) · g for any λ ∈ Gau( P ) and g ∈ G . For a point x in X , let s x ( λ ) : X → R denotethe lift of λ : X → S satisfying s x ( λ )( x ) ∈ [0 , s x defines a section s x : Gau( P ) = C ( X, S ) → C ( X, R ). Then a cocycle c x ∈ C (Gau( P ); Z ) of δ [ φ ] ∈ H (Gau( P ); Z ) is given by c x ( λ , λ ) = δ ( s x ◦ φ )( λ , λ )= s x ( φ ( λ )) − s x ( φ ( λ λ )) + s x ( φ ( λ ))for λ , λ ∈ Gau( P ). For g ∈ G , we put y = g ( x ). Since s x ( λ · g ) = ( s y ( λ )) ◦ g forany λ ∈ Gau( P ), we have c x · g ( λ , λ ) = c x ( λ · g, λ · g )= s x ( φ ( λ · g )) − s x ( φ (( λ λ ) · g )) + s x ( φ ( λ · g ))= s x ( φ ( λ ) · g ) − s x ( φ ( λ λ ) · g ) + s x ( φ ( λ ) · g )= ( s y ( φ ( λ )) − s y ( φ ( λ λ )) + s y ( φ ( λ ))) ◦ g = s y ( φ ( λ )) − s y ( φ ( λ λ )) + s y ( φ ( λ ))= c y ( λ , λ ) . Since the image δ [ φ ] of the connecting homomorphism is independent of the choiceof the section s x , we have ( δ [ φ ]) · g = [ c x · g ] = [ c y ] = δ [ φ ] . Thus the cohomology class δ [ φ ] is in H (Gau( P ); Z ) G . (cid:3) Lemma 4.4.
The following diagram H (Gau( P ); Gau( P )) G = E ′ , d ′ , / / δ (cid:15) (cid:15) H ( G ; Gau( P )) δ (cid:15) (cid:15) H (Gau( P ); Z ) G = E ′′ , − d ′′ , / / H ( G ; Z )(4.2)commutes, where the maps δ are the connecting homomorphisms and d ′ , and d ′′ , are the derivations of the Hochschild-Serre spectral sequences E ′ p,qr and E ′′ p,qr above. SHUHEI MARUYAMA
Proof.
Let φ be an element in H (Gau( P ); Gau( P )) G and s : G → A G ( P ) a sectionsuch that s (id) = id. At first, we give a cocycle of the class δd ′ , [ φ ]. Let us definea cochain φ s : A G ( P ) → Gau( P ) by putting φ s ( F ) = φ ( F ◦ ( s ( p ( F ))) − ) ∈ Gau( P ) , where p : A G ( P ) → G is the projection. We can see that the cochain φ s restricts to φ on Gau( P ) and the coboundary δφ s defines a cocycle in C ( G ; Gau( P )). Thus,by definition of the derivation of Hochschild-Serre spectral sequence, the cocycle of d ′ , [ φ ] is given by δφ s ∈ C ( G ; Gau( P )). Thus, the cocycle of δd ′ , [ φ ] is given by δ ( s x ◦ ( δφ s )) ∈ C ( G ; Z ) , (4.3)where s x : Gau( P ) → C ( X, R ) is the section defined in Lemma 4.3.Next, we give a cocycle of the class d ′′ , δ [ φ ]. A cocycle of δ [ φ ] is given by thecoboundary c x = δ ( s x ◦ φ ) ∈ C (Gau( P ); Z ). Put c ′ x = δ ( s x ◦ φ s ) − p ∗ ( s x ◦ ( δφ s )) , then it can be seen that the cochain c ′ x is in C ( A G ( P ); Z ) and the restrictionof c ′ x on Gau( P ) is equal to c x . Moreover, since the coboundary δc ′ x is equal to − p ∗ δ ( s x ◦ ( δφ s )), the cocycle of d ′′ , δ [ φ ] is given by − δ ( s x ◦ ( δφ s )) ∈ C ( G ; Z ) . (4.4)By (4.3) and (4.4), we have δd ′ , [ φ ] = − d ′′ , δ [ φ ] and the lemma follows. (cid:3) Lemma 4.5.
Let X → E → BG be the universal G -bundle and B Gau( P ) → BA G ( P ) → BG the fibration that corresponds to the exact sequence (1.2). Thenthere is the following commuting diagram X / / f (cid:15) (cid:15) E / / φ (cid:15) (cid:15) BGB
Gau( P ) / / BA G ( P ) / / BG, (4.5)where the map f → B Gau( P ) is the composition of the classifying map X → BS and the map BS → B Gau( P ) induced from the inclusion S → Gau( P ). Proof.
Let P G = P × Gau( P ) Gau( P ) → X denote the associated bundle. Then thebundle E = EG × A G ( P ) P G → BG gives one of a model of the universal G -bundle with fiber X . Since EG × P G → E isa principal A G ( P )-bundle, there is a bundle map to the universal principal A G ( P )-bundle EG × P G Ψ / / (cid:15) (cid:15) EA G ( P ) (cid:15) (cid:15) E ψ / / BA G ( P ) . CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 9 Define the map Φ : EG × P G → EG × EA G ( P ) by Φ( a, p ) = ( a, Ψ( a, p )). Then themap Φ gives a bundle map from EG × P G → E to EG × EA G ( P ) → EG × A G ( P ) EA G ( P ) = BA G ( P ). Let φ : E → BA G ( P ) denote the classifying map that iscovered by Φ. Then it can be seen that the map φ covers the identity on BG .Moreover, the restriction f : X → B Gau( P ) to the fiber gives rise to the classifyingmap of the bundle P G → M . (cid:3) Proof of Theorem 1.1.
Take a commuting diagram (4.5). Let us consider the Serrespectral sequences E p,qr and E ′′′ p,qr of the fibrations X → E → BG and B Gau( P ) → BA G ( P ) → BG respectively. Since H ( B Gau( P ); Z ) = 0, we have E ′′′ , = E ′′′ , = H ( B Gau( P ); Z ) and E ′′′ , = E ′′′ , = H ( BG ; Z ). By the naturality of the Serrespectral sequence, we have the commuting diagram H ( B Gau( P ); Z ) d ′′′ , / / f ∗ (cid:15) (cid:15) H ( BG ; Z ) H ( X ; Z ) d , / / H ( BG ; Z ) . Then we have f ∗ ev ∗ x c = c ∈ H ( X ; Z ) since f : X → B Gau( P ) is the compositionof the classifying map X → BS and the map j : BS → B Gau( P ) (see section 3.3).Thus, the universal characteristic class e c ∈ H ( BG ; Z ) is equal to − d ′′′ , ev ∗ x c . Let E ′′ p,qr denote the Serre spectral sequence of the fibration B Gau( P ) δ → BA G ( P ) δ → BG δ (or, equivalently, the Hochschild-Serre spectral sequence of 1 → Gau( P ) → A G ( P ) → G → B Gau( P ) δ / / (cid:15) (cid:15) BA G ( P ) δ / / (cid:15) (cid:15) BG δ (cid:15) (cid:15) B Gau( P ) / / BA G ( P ) / / BG and the naturality, we have the commuting diagram of cohomologies H ( B Gau( P ); Z ) d ′′′ , / / ι ∗ (cid:15) (cid:15) H ( BG ; Z ) ι ∗ (cid:15) (cid:15) H (Gau( P ); Z ) d ′′ , / / H ( G ; Z ) , here we identify the group cohomology and the singular cohomology of classifyingspace of discrete groups. Then we have ι ∗ e c = − ι ∗ d ′′′ , ev ∗ x c = − d ′′ , ι ∗ ev ∗ x c . On the other hand, by the commuting diagram (4.2) and lemma 3.2, we have δe ( A G ( P )) = − δd ′ , (id Gau( P ) ) = d ′′ , δ (id Gau( P ) ) . Since the class ι ∗ ev ∗ x c is equal to − δ (id Gau( P ) ) (see [6, Lemma 2.4]), we have ι ∗ e c = δe ( A G ( P ))and the theorem follows. (cid:3) Central S -extension of G In this section, we assume that the topological space X admits the universalcovering space e X . We fix a non-zero cohomology class c ∈ H ( X ; Z ) such that it isequal to zero in H ( X ; R ).5.1. Construction of the central S -extension. By the cohomology long exactsequence · · · −→ H ( X ; R ) −→ H ( X ; S ) δ −→ H ( X ; Z ) −→ H ( X ; R ) −→ · · · , there is an element ρ ∈ H ( X ; S ) such that δ ( ρ ) = c . By the isomorphism H ( X ; S ) ∼ = Hom( π ( X ); S ), we regard the class ρ as a homomorphism ρ : π ( X ) → S . Let P δρ → X be a principal S δ -bundle with the holonomy homomorphism ρ ,that is, we put P δρ = e X × ρ S δ . Lemma 5.1.
Let Aut( P δρ ) denote the bundle automorphisms of P δρ and Homeo( X, ρ )the group of ρ -preserving homeomorphisms. Then the canonical projection π :Aut( P δρ ) → Homeo( X ) gives the surjection Aut( P δρ ) → Homeo(
X, ρ ) and its kernelis isomorphic to S , that is, there is the following exact sequence of groups1 → S → Aut( P δρ ) → Homeo(
X, ρ ) → . (5.1) Proof.
At first, we show that the image π (Aut( P δρ )) is contained in Homeo( X, ρ ).Take a bundle automorphism F ∈ Aut( P δρ ) that covers a homeomorphism f : X → X . Let γ : [0 , → X be a loop and e γ : [0 , → P δρ a lift of γ . Then F e γ : [0 , → P δρ is a lift of the loop f γ . Since ρ : π ( X ) → S δ is the holonomy of P δ , the value ρ ( f γ ) ∈ S δ is given by F ( e γ (1)) = F ( e γ (0)) · ρ ( f γ ) . (5.2)Since the left-hand side of (5.2) is equal to F ( e γ (0) · ρ ( γ )) = F ( e γ (0)) · ρ ( γ ), we have f ∗ ρ ( γ ) = ρ ( f γ ) = ρ ( γ ). Thus the homeomorphism f preserves the holonomy ρ andwe have π (Aut( P δρ )) ⊂ Homeo(
X, ρ ).Next we show that the map π : Aut( P δρ ) → Homeo(
X, ρ ) is surjective. For ahomeomorphism f : X → X that preserves ρ , we take a lift e f : e X → e X . Then themap e X × S δ → e X × S δ defined by ( x, u ) → ( e f ( x ) , u ) induces a bundle automor-phism F : P δρ → P δρ that covers f . Thus the map π : Aut( P δρ ) → Homeo(
X, ρ ) issurjective.The kernel of π : Aut( P δρ ) → Homeo(
X, ρ ) is the gauge group Gau( P δρ ) and thisis isomorphic to C ( X, S δ ) ∼ = S δ since the fiber of P δρ is the discrete group S δ . (cid:3) CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 11 By restricting the above exact sequence (5.1) to the subgroup G of Homeo( X, ρ ),we have the abelian S -extension1 −→ S −→ A G ( P δρ ) −→ G −→ . (5.3)Moreover, the group S is in the center of A G ( P δρ ), this is a central S -extension of G .Thus we obtain the corresponding group cohomology class e ( A G ( P δρ )) ∈ H ( G ; S ).Let P ρ → X be a principal S -bundle defined by P ρ = e X × ρ S . Then we have thefollowing commuting diagram of groups1 / / S / / j (cid:15) (cid:15) A G ( P δρ ) / / k (cid:15) (cid:15) G / / / / Gau( P ρ ) / / A G ( P ρ ) / / G / / , (5.4)where j : S → Gau( P ρ ) is the inclusion. Recall that e ( A G ( P δρ )) ∈ H ( G ; S ) and e ( A G ( P ρ )) ∈ H ( G ; Gau( P ρ )) denote the cohomology classes that correspond tothe central extension A G ( P δρ ) and the abelian extension A G ( P ρ ) respectively. By thecommuting diagram above, we have the following proposition. Proposition 5.2.
Let j ∗ : H ( G ; S ) → H ( G ; Gau( P ρ )) be the map of coeffi-cients change by j . Then we have j ∗ ( e ( A G ( P δρ ))) = e ( A G ( P ρ )) in H ( G ; Gau( P ρ )). Proof.
Recall that a cocycle c of e ( A G ( P δρ )) is given by c ( g, h ) = s ( g ) s ( h ) s ( gh ) − , where s : G → A G ( P δρ ) is a section. Since k ◦ s : G → A G ( P ρ ) is a section of A G ( P ρ ) → G , a cocycle c ′ of e ( A G ( P ρ )) is given by c ′ ( g, h ) = k ( s ( g )) k ( s ( h )) k ( s ( gh )) − = k ( s ( g ) s ( h ) s ( gh ) − ) = j ( c ( g, h )) . Thus we have j ∗ ( e ( A G ( P δρ ))) = e ( A G ( P ρ )). (cid:3) Proof of Theorem 1.2.
By the commuting diagram of coefficients0 / / Z / / (cid:15) (cid:15) R / / (cid:15) (cid:15) S / / j (cid:15) (cid:15) / / Z / / C ( X, R ) / / C ( X, S ) = Gau( P ρ ) / / , we have the following commuting diagram of cohomologies H ( G ; S ) δ / / j ∗ (cid:15) (cid:15) H ( G ; Z ) H ( G ; Gau( P ρ )) δ / / H ( G ; Z ) , where the maps δ are the connecting homomorphisms. Together with Theorem 1.1and proposition 5.2, we obtain δe ( A G ( P δρ )) = δj ∗ e ( A G ( P δρ )) = δe ( A G ( P ρ )) = ι ∗ e c . (cid:3) Cocycle description of e ( A G ( P δρ )) . By the similar construction explainedin [3], we obtain the group two-cocycle that represents the class e ( A G ( P δρ )). Let c ∈ H ( X ; Z ) and ρ ∈ H ( X ; S ) be a cohomology class as above. Let α ∈ C ( X ; S )be a cocycle that represents the cohomology class ρ , where ( C ∗ ( X ; S ) , d ) denotesthe singular cochain complex of X with coefficients in S . The singular cochain C ∗ ( X ; S ) and the cohomology H ∗ ( X ; S ) are the right G -modules by pullback.Since g ∗ ρ = ρ for any homeomorphism g ∈ G , the cochain g ∗ α − α is a coboundary.Thus there is a cochain K α ( g ) ∈ C ( X ; S ) such that g ∗ α − α = d K α ( g ) . Then a two-cochain G x,α in C ( G ; S ) is defined by G x,α ( g, h ) = Z hxx g ∗ α − α = K α ( g )( hx ) − K α ( g )( x ) , (5.5)where x ∈ X . Here the symbol R hxx denotes the pairing of the cocycle g ∗ α − α anda path from x to hx . By the same arguments in [5, Theorem 3,1], we have thefollowing proposition. Proposition 5.3.
The two-cochain G x,α ∈ C ( G ; S ) is a cocycle and the coho-mology class [ G x,α ] is independent of the choice of the point x ∈ X and the singularcochain α .Then the following theorem holds. Theorem 5.4.
The cohomology class [ G x,α ] is equal to e ( A G ( P ρ )).For the proof of Theorem 5.4, we show the following lemmas. Let us recall that p : P δρ → X is the principal S δ -bundle with the holonomy ρ . Lemma 5.5.
The pullback p ∗ ρ ∈ H ( P δρ ; S ) is equal to zero. Proof.
Let γ : [0 , → P δρ be a loop. Then γ is a lift of the loop pγ in X . Since ρ : π ( X ) → S is the holonomy homomorphism, we have π ∗ ρ ( γ ) = ρ ( pγ ) = γ (1) − γ (0) = 0and the lemma follows. (cid:3) By lemma 5.5, we take a singular cochain θ ′ ∈ C ( P δρ ; S ) satisfying dθ ′ = p ∗ α .Take a base point y ∈ P δρ and put p ( y ) = x . For any u ∈ S , let P δρ ( u ) denote theconnected component of P δρ that contains the point y · u . Then we put z ( u ) = θ ′ ( y ) − θ ′ ( y · u ) + u ∈ S . CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 13 By the straight forward calculation, we have the equality z ( u ) = z ( u + ρ ( γ )) for any γ ∈ π ( X ). Thus, the value z ( u ) defines the continuous function z : P δρ → S δ . Notethat z ( y · u ) = z ( u ) and z ( y ) = 0. We put θ = θ ′ + z ∈ C ( X ; Z ). Since dz = 0, wehave dθ = p ∗ α . Let τ ∈ C ( A G ( P δρ ); S ) denote a cochain defined by τ ( φ ) = Z φyy θ = θ ( φy ) − θ ( y ) . Lemma 5.6.
The restriction τ | S : S → S is equal to the identity. Proof.
By definition of τ , we have τ ( u ) = θ ( y · u ) − θ ( y ) = θ ′ ( y · u ) + z ( y · u ) − θ ′ ( y ) − z ( y ) = u for any u ∈ S . (cid:3) Lemma 5.7.
The equality − δτ = π ∗ G x,α ∈ C ( A G ( P δρ ); S )holds, where π : A G ( P δρ ) → G is the projection. Proof.
For the pullback p ∗ K α ( g ) ∈ C ( P δρ ; S ), we have d ( p ∗ K α ( g )) = p ∗ ( d K α ( g )) = p ∗ g ∗ α − p ∗ α = φ ∗ p ∗ α − p ∗ α = φ ∗ dθ − dθ = d ( φ ∗ θ − θ ) , where φ ∈ A G ( P δρ ) is a lift of g ∈ G . Thus, for φ, ψ ∈ A G ( P δρ ) that covers g, h ∈ G respectively, we have π ∗ G x,α ( φ, ψ ) = G p ( y ) ,α ( g, h ) = K α ( g )( hp ( y )) − K α ( g )( p ( y ))= K α ( g )( pψ ( y )) − K α ( g )( p ( y ))= p ∗ K α ( g )( ψ ( y )) − p ∗ K α ( g )( y )= ( φ ∗ θ − θ )( ψ ( y )) − ( φ ∗ θ − θ )( y )= ( θ ( φψ ( y )) − θ ( y )) − ( θ ( φ ( y )) − θ ( y )) − ( θ ( ψ ( y )) − θ ( y ))= − δτ ( φ, ψ )and the lemma follows. (cid:3) Proof of theorem 5.4.
Let E p,qr denote the Hochschild-Serre spectral sequence of 0 → S → A G ( P δρ ) → G → S . Then there is the derivation d , : H ( S ; S ) = E , → E , = H ( G ; S ) . By lemma 5.6, lemma 5.7, 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 3.2, we have d , (id S ) = − e ( A G ( P δρ )). Thus we have [ G x,α ] = e ( A G ( P δρ )). (cid:3) Examples
In this section, we give two examples that the universal characteristic classes e c and ι ∗ e c are non-trivial. The first example is the complex projective space C P n .The proof of the following theorem is the same as [6, Theorem 1.2], so I will omit it. Theorem 6.1.
Let n be a positive integer and c ∈ H ( C P n ; Z ) the generator ofcohomology. Let G denote the identity component Homeo( C P n ) of the homeo-morphism group. Then the universal characteristic classes e c ∈ H ( BG ; Z ) and ι ∗ e c ∈ H ( BG δ ; Z ) are non-zero.The cohomology class c in Theorem 6.1 is non-zero in H ( C P n ; R ). So this ex-ample does not satisfy the assumption in section 5. The following example satisfiesthe assumption in section 5. Theorem 6.2.
Let c be the non-zero element in H ( R P ; Z ) ∼ = Z / Z . Let G denote the identity component Homeo( R P ) of the homeomorphism group. Thenthe universal characteristic classes e c ∈ H ( BG ; Z ) and ι ∗ e c ∈ H ( BG δ ; Z ) are non-zero. Proof.
Since R P is homeomorphic to SO (3), the group SO (3) is included in G asthe left translations. Thus we have the commuting diagram SO (3) / / (cid:15) (cid:15) ESO (3) / / F (cid:15) (cid:15) BSO (3) f (cid:15) (cid:15) G / / EG / / BG.
Recall that the universal G -bundle R P → E → BG is given as E = EG × G R P .For the base point b ∈ R P that corresponds to the unit 1 ∈ SO (3), we define amap φ : ESO (3) → E by putting Φ( x ) = [ F ( x ) , b ] for x ∈ ESO (3). Then we havethe commuting diagram SO (3) / / ESO (3) / / Φ (cid:15) (cid:15) BSO (3) f (cid:15) (cid:15) SO (3) ∼ = R P / / E / / BG. (6.1)Let us consider the Serre spectral sequences of the two fibrations in (6.1). Bythe naturality of the Serre spectral sequence, the pullback f ∗ e c ∈ H ( BSO (3); Z )of the universal characteristic class e c is equal to the transgression image of c ∈ H ( R P ; Z ) ∼ = H ( SO (3); Z ) with respect to the fibration SO (3) → ESO (3) → BSO (3). Since
ESO (3) is contractible, the transgression map is injective and thusthe class f ∗ e c is non-zero. So the class e c is also non-zero. Next we consider the CHARACTERISTIC CLASS OF Homeo( X ) -BUNDLES 15 following commuting diagram H ( BG ; Z ) ι ∗ / / f ∗ (cid:15) (cid:15) H ( BG δ ; Z ) (cid:15) (cid:15) H ( BSO (3); Z ) ι ∗ / / H ( BSO (3) δ ; Z ) . Since the map ι ∗ : H ( BSO (3); Z ) → H ( BSO (3) δ ; Z ) is injective [7], the class ι ∗ f ∗ e c ∈ H ( BSO (3) δ ; Z ) is non-zero. Thus the class ι ∗ e c ∈ H ( BG δ ; Z ) is alsonon-zero. (cid:3) Remark 6.3.
By Theorem 1.2 and Theorem 6.2, the class e ( A G ( P δρ )) is also non-zero for G = Homeo( R P ) . Thus, this is an example that the cocycle (5.5) iscohomologically non-trivial. Remark 6.4.
For c = 0 ∈ H ( R P ; Z ) ∼ = Z / Z and G = Homeo( R P ) , the univer-sal characteristic class e c ∈ H ( BG ; Z ) is equal to zero. References
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