Obstruction of C ∞ -algebra models and characteristic classes
aa r X i v : . [ m a t h . A T ] M a y OBSTRUCTION OF C ∞ -ALGEBRA MODELS ANDCHARACTERISTIC CLASSES TAKAHIRO MATSUYUKI
Abstract.
In this paper, we consider an obstruction-theoretical constructionof characteristic classes of fiber bundles by simplicial method. We can get acertain obstruction class for a deformation of C ∞ -algebra models of fibers anda characteristic map from the exterior algebra of a vector space of derivations.Applying this construction for a surface bundle, we obtain the Euler class of asphere bundle and the Morita-Miller-Mumford classes of a bundle with positivegenus fiber. Introduction
Our purpose of the paper is to construct characteristic classes of a smooth fiberbundle X → E → B by obstruction theory for a certain simplicial bundle Q • ( X ) →Q • ( E ) → S • ( B ) obtained from the original bundle as follows. • The n -th base set S n ( B ) is the set of singular simplices ∆ n → B , • The n -th fiber set Q n ( E ) σ over an n -simplex σ ∈ S n ( B ) is the set ofChen’s formal homology connections [4, 5] on σ ∗ E . The total simplicialset Q • ( E ) is defined by the disjoint sum Q n ( E ) = a σ ∈ S n ( B ) Q n ( E ) σ . • The n -th typical fiber Q n ( X ) is the set of Chen’s formal homology con-nections ( ω, δ ) on X × ∆ n . It has the decomposition with respect todifferentials of formal homology connections Q • ( X ) = a δ Q • ( X, δ ) . A formal homology connection is well-known as a main tool of the de Rham homo-topy theory. It has homotopical information of X , which is equivalent to a minimal C ∞ -algebra model f : ( H, m ) → A ([9]). Here A is the de Rham complex of X ,and H is the de Rham cohomology of X .To apply the obstruction theory to the simplicial bundle, we need to calculatethe homotopy group of Q • ( X ). This simplicial set is very close to the Maurer-Cartan simplicial set MC • ( ˆ LW ⊗ A ) of the DGL ˆ LW ⊗ A , where ( ˆ LW, δ ) is thedual of the bar-construction of the C ∞ -algebra ( H, m ). The homotopy group ofthe Maurer-Cartan simplicial set is known in [7, 1, 2]. Using the results, we shallprove the homotopy groups of Q • ( X ) are described as vector spaces by π n ( Q • ( X ) , τ ) ≃ H n (Der( ˆ LW ) , ad( δ )) =: H n ( δ )for a formal homology connection τ = ( ω, δ ) on X and n ≥
1. The set π ( Q • ( X, δ ))of connected components can also be identified with a certain subspace H (1)0 ( δ ) of the 0-th homology H ( δ ) of the DGL (Der( ˆ LW ) , ad( δ )). These calculations of thehomotopy groups of Q • ( X ) are shown in Section 3.If Q • ( X ) is ( n − n -skeleton of Q • ( E ) → S • ( B ) o n ∈ H n +1 ( B ; Π n ) , where Π n is the local system of the n -th homotopy groups of fibers of Q • ( E ) → S • ( B ). Then, by contracting coefficients of o n , we can also the characteristic map(Λ p H n ( δ ) ∗ ) G → H p ( n +1) ( B ; R )for any p ≥
1. Here G is the structure group of E → B . For example, the imageof characteristic map for the sphere bundle associated to the Hopf fibration isgenerated by the Euler class of this bundle. Their definitions are written in Section5.1, and the example is in Section 5.2.On the other hand, if Q • ( X ) is not connected, instead of o n , there exists i ≥ o ( i ) ∈ H ( B ; gr i ( QA + ( E ))). Here QA + ( E ) is a certainlocal system of groups with a filtration. The local system Π of the 0-th homotopyset of fibers of Q • ( E ) → S • ( B ) has a structure of QA + ( E )-torsor, i.e., a free andtransitive action of QA + ( E ). So we can also construct the characteristic map(Λ • gr i ( H ( δ )) ∗ ) G → H • ( B ; R ) . Applying for a surface bundle, we get the obstruction class o (1) , and it is equal tothe 1st twisted Morita-Miller-Mumford class. It means that the characteristic mapgives Morita-Miller-Mumford classes. Their definitions are written in Section 5.3,and the application for a surface bundle is in Section 5.4.The paper is organized as follows. • In Section 2, we define terms used in the paper. • In Section 3, we define the simplicial set of formal homology connections,and calculate its homotopy groups. • In Section 4, we describe obstruction theory for general simplicial sets inorder to use in Section 5. This section is independent of other sections. • In Section 5, we apply the discussions in Section 4 for Q • ( E ) → S • ( B )and get obstruction classes. We also calculate the obstruction classes forspecific bundles. Acknowledgment.
I would like to thank my supervisor Y. Terashima for manyhelpful comments. This work was supported by Grant-in-Aid for JSPS ResearchFellow (No.17J01757). 2.
Preliminary
In this paper, all vector spaces are over the real number field R . The standard n -simplex is described by∆ n = ( ( t i ) ni =0 ∈ R n +1 ; n X i =0 t i = 1 ) . Fix its base point { (1 , , . . . , } = δ n · · · δ (∆ n ), where δ i : ∆ n − → ∆ n is the i -thcoface operator. BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 3 Throughout the paper, we consider a smooth fiber bundle X → E → B whosefiber X is a manifold with a base point, i.e., a smooth fiber bundle with a smoothsection B → E . We always suppose that • a manifold X is connected, • its base point ∗ is fixed, and • its rational homology group is finite-dimensional.The structure group of the bundle, which is a subgroup of the diffeomorphismgroup Diff( X ), acts on the homology group of X . We call its image G in theautomorphism group of H • ( X ; R ) the homological structure group .2.1. Graded vector space.
Let V be a Z -graded vector space. We denote V i the subspace of elements of V of cohomological degree i and V i = V − i thesubspace of elements of homological degree i . Remark that the linear dual V ∗ = Hom( V, R ) of V is graded by ( V ∗ ) i = Hom( V i , R ). For an element v ∈ V , itshomological degree is denoted by | v | .The p -fold suspension V [ p ] of V for an integer p is defined by V [ p ] i := V i + p . In the case of p = 1, elements of V [1] i are written by s x for x ∈ V i +1 usingthe symbol s of cohomological degree −
1. The symbol s is also regarded as themap s : V → V [1] with cohomological degree −
1. We often use the inverse map s − : V [1] → V of the suspension s too.2.2. FDGL.
In Section 2.3 and 2.4, we use the following structures:
Definition 2.1.
Let L be a Z -graded Lie algebra and F = {F ( i ) } ∞ i =0 a decreasingfiltration of L . The pair ( L, F ) is called a Z -graded filtered Lie algebra , FGL for short, if it satisfies [ F ( i ) , F ( j ) ] ⊂ F ( i + j ) for integers i, j ≥
0. Moreover, giventhe differential δ on L with homological degree −
1, the triple (
L, δ, F ) is calleda filtered DGL , FDGL for short, if it satisfies δ ( F ( i ) ) ⊂ F ( i +1) for integers i ≥
0. Then the homology H • ( L, δ ) has the canonical FGL structure whose filtration¯ F = { ¯ F ( i ) } ∞ i =0 is defined by¯ F ( i ) := Im(Ker( δ ) ∩ F ( i ) → H • ( L, δ )) . Then the Lie algebra of derivations on a FGL (resp. FDGL) is a FGL (resp.FDGL) as follows:
Definition 2.2.
Let ( L, F ) is a FGL. PutDer( L ) n := { D ∈ End( L ); D ([ x, y ]) = [ D ( x ) , y ]+( − | x | n [ x, D ( y )] , D ( L p ) ⊂ L p + n } , Der( L ) := M n Der( L ) n , D = {D ( i ) } i ≥ , D ( i ) := { D ∈ Der( L ); D ( F ( q ) ) ⊂ F ( q + i ) } . Then the space Der( L ) of derivations on L is a Z -graded Lie subalgebra of End( L ),and (Der( L ) , D ) is a FGL. If ( L, δ, F ) is a FDGL, then so is (Der( L ) , ad( δ ) , D ). TAKAHIRO MATSUYUKI
Free Lie algebra.
Let W be a Z -graded vector space. In this paper, W isalways homologically non-negatively graded.The graded free Lie algebra LW and the completed free Lie algebra ˆ LW gener-ated by W have the canonical FGL structures as follows: • (grading) These two Lie algebras LW and ˆ LW can be defined as the prim-itive part of the tensor algebra T W and the completed tensor algebra ˆ
T W : LW = Prim T W = { x ∈ T W ; ∆( x ) = 1 ⊗ ∆( x ) + ∆( x ) ⊗ } , ˆ LW = Prim ˆ T W = { x ∈ ˆ T W ; ∆( x ) = 1 ⊗ ∆( x ) + ∆( x ) ⊗ } , where ∆ is the (completed) coproduct. Since the algebra T W is Z -gradedby ( T W ) n := M p ≥ M i + ··· + i p = n ( W i ⊗ · · · ⊗ W i p ) , so is the Lie algebra T W :( LW ) n := LW ∩ ( T W ) n . The Lie algebra ˆ LW ⊂ ˆ T W is Z -graded in the same way. • (filtration) Let Γ = { Γ n } ∞ n =1 be the lower central series of LW and ˆΓ = { ˆΓ n } ∞ n =1 the completed lower central series of ˆ LW . It is described byΓ n = LW ∩ M m>n W ⊗ m , ˆΓ n = ˆ LW ∩ Y m>n W ⊗ m . Note that ˆΓ n = lim ←− k Γ n / Γ n + k , ˆ LW = lim ←− k LW/ Γ k +1 . Then (
LW,
Γ) and ( ˆ
LW, ˆΓ) are FGLs.2.4.
Derivations.
Fix a differential δ on ˆ LW satisfying | δ | = − δ ( W ) ⊂ ˆΓ .Then, from Section 2.2, the triples ( ˆ LW, δ, ˆΓ) and (Der( ˆ LW ) , ad( δ ) , D ) are FDGLs.Its homology H • ( δ ) := H • (Der( ˆ LW ) , ad( δ )) has the induced filtration ¯ D as inDefinition 2.2. Especially, we have the Lie algebra H ( δ ) filtered by H ( i )0 ( δ ) := ¯ D ( i )0 . Definition 2.3 (exponential map) . Consider the group of automorphisms Aut( ˆ LW )of the completed Lie algebra ˆ LW filtered by A ( i ) := Ker(Aut( ˆ LW ) → Aut( ˆ
LW/ ˆΓ i +1 )) . Then the bijection exp : D (1)0 → A (1) preserving their filtrations, which is calledthe exponential map , defined byexp( D ) = ∞ X n =0 D n n ! ∈ End( ˆ LW ) . The map has the inverse map log : A (1) → D (1)0 . The product of the group A (1) andthe Lie bracket of D (1)0 are related by the Baker-Campbell-Hausdorff formula. Wecan also consider the group of automorphisms Aut( δ ) of the completed dgl ( ˆ LW, δ )filtered by A ( i ) ( δ ) := A ( i ) ∩ Aut( δ ) and the Lie algebraDer( δ ) := Ker(ad( δ ) : Der( ˆ LW ) → Der( ˆ LW )) , BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 5 which is filtered by D ( i ) ( δ ) := D ( i ) ∩ Der( δ ). Then we can get the restrictionexp : D (1)0 ( δ ) → A (1) ( δ ) of the exponential map D (1)0 → A (1) . So we can get thequotient group QAut( δ ) := Aut( δ ) / exp(ad( δ )(Der( ˆ LW ) )) , which is filtered by the image QA ( δ ) of the filtration {A ( i ) ( δ ) } i ≥ , and the inducedexponential map exp : H (1)0 ( δ ) → QA (1) ( δ ). Definition 2.4.
Let G be a group with a decreasing filtration { G ( i ) } i ≥ of normalsubgroups satisfying [ G ( i ) , G ( j ) ] ⊂ G ( i + j ) , G (0) = G. Then gr i ( G ) := G ( i ) /G ( i +1) is an abelian group with respect to the sum inducedby the product of G , and gr( G ) := ∞ M i =0 gr i ( G )is a Lie algebra with respect to the Lie bracket defined by the commutator of G .Similarly, for a filtered Lie algebra ( L, F ), we get the Lie algebragr( L ) := ∞ M i =0 gr i ( L ) , gr i ( L ) := F ( i ) / F ( i +1) . Using the notations above, the exponential map induces the isomorphismgr i ( H ( δ )) ≃ gr i ( QA ( δ ))for i ≥ δ ( W ) ⊂ [ W, W ], we can define another grading of Der( ˆ LW ) byDer i ( ˆ LW ) := { D ∈ Der( ˆ LW ); D ( W ) ⊂ W ⊗ ( i +1) } . Then ad( δ ) has the degree 1 with respect to the grading. So we have the canonicalidentification H i ( δ ) := H i (Der( ˆ LW ) , ad( δ )) ≃ gr i ( H ( δ )) , where i is the second grading of Der( ˆ LW ).2.5. Formal homology connections.
In this subsection, we shall review the def-inition of a formal homology connection on a manifold X . We denote the de Rhamcomplex on X by A • ( X ), the reduced de Rham complex and cohomology by A = ˜ A • ( X ) := Ker( A • ( X ) → A • ( ∗ ) = R ) , H = ˜ H • DR ( X )and the suspension of the reduced real homology by W = ˜ H • ( X ; R )[ − Definition 2.5 (Chen [4, 5]) . A formal homology connection on X is a pair( ω, δ ) satisfying the following conditions:(i) an ˆ LW -coefficient differential form ω ∈ A ⊗ ˆ LW with cohomological degree1 is described by ω = ∞ X k =1 X i ,...,i k ω i ··· i k x i · · · x i k , TAKAHIRO MATSUYUKI where x , . . . , x n is a homogeneous basis of W , such that Z x p ω p = 1 . (ii) a linear map δ : ˆ LW → ˆ LW is a differential with homological degree − LW such that δ ( W ) ⊂ ˆΓ . (iii) the form ω is a Maurer-Cartan element of ( A ⊗ ˆ LW, d + δ ), i.e., the flatnesscondition δω + dω + [ ω, ω ] = 0 holds. (Though the sign notation may bedifferent from Chen’s original definition, they are equivalent.)We call such a differential δ Chen differential of X . If X is simply connected, wecan replace the free Lie algebra LW and its derivation δ : LW → LW with ˆ LW and δ : ˆ LW → ˆ LW respectively.According to Chen [4, 5], a Riemannian metric on X defines the canonical formalomology connection on X . If X is a formal manifold (e.g. K¨ahler manifold),there exists a Chen differential δ satisfying δ ( W ) ⊂ [ W, W ]. This differential iscorresponding to the usual product of the cohomology H by method of Section 2.6.We can pull-back a formal homology connection by a diffeomorphism as follows.So the diffeomorphism group of X acts on the set of formal homology connectionson X . Definition 2.6 (pull-back of formal homology connections) . Let ϕ : Y → X be adiffeomorphism preserving base points, and ( ω, δ ) be a formal homology connectionon X . Then we can define the formal homology connection on Yϕ ∗ ( ω, δ ) := (( ϕ ∗ ⊗ | ϕ | ) ω, | ϕ | ◦ δ ◦ | ϕ | − ) , where | ϕ | : ˆ LW ( X ) → ˆ LW ( Y ) is the induced map by ϕ , W ( X ) = ˜ H • ( X ; R )[ − , W ( Y ) := ˜ H • ( Y ; R )[ − . C ∞ -algebras and formal homology connections. In the subsection, weshall review the definition of a C ∞ -algebra, and mention the relation between aformal homology connection and a C ∞ -algebra ([8]). The definitions and discussionsin the subsection are used in the proofs of Theorem 3.3 and 3.4. Definition 2.7 ( C ∞ -algebra) . Let A be a vector space and m = { m i } ∞ i =1 be afamily of linear maps m i : A ⊗ i → A with degree 2 − i . The pair ( A, m ) satisfyingthe following conditions is called a C ∞ -algebra : • ( A ∞ -relation) X k + l = i +1 k − X j =0 ( − ( j +1)( l +1) m k ◦ (id ⊗ jA ⊗ m l ⊗ id ⊗ ( i − j − l ) A ) = 0for i ≥
1, and • (commutativity) X σ ∈ Sh( j,i − j ) ǫ · m i ( a σ (1) , · · · , a σ ( i ) ) = 0for i > j > a , . . . , a i ∈ A , where Sh( i, i − j )is the set of ( i, i − j )-shuffles and ǫ is the Koszul sign. BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 7 If m = 0, ( A, m ) is called minimal . If higher products are all zero, i.e. m = m = · · · = 0, ( A, m ) can be regarded as differential graded commutative algebra(DGcA).
Remark 2.8 (Bar construction of a C ∞ -algebra) . Let (
A, m ) be a C ∞ -algebraand s : A → A [1] be the suspension map. We denote the tensor coalgebra T c ( A [1])generated by A [1] by BA . It is a bialgebra by the tensor coproduct ∆ and the shuffleproduct µ . Defining the suspension of m i by ¯ m i := s ◦ m i ◦ ( s − ) ⊗ i for all i ≥ m i : A [1] ⊗ n → A [1] is degree 1 and satisfies the commutativity condition.Thus extending the unique coderivation m i : BA → BA by the co-Leibniz rule∆ ◦ m i = ( m i ⊗ id + id ⊗ m i ) ◦ ∆, then we have the Hopf derivation m := ∞ X i =1 m i . Furthermore m is a degree 1 codifferential, i.e. m = 0, from the A ∞ -relations of m . Definition 2.9 ( C ∞ -morphism) . Let (
A, m ) and ( A ′ , m ′ ) be two C ∞ -algebras and f = { f i } ∞ i =1 be a family of linear maps f i : A ⊗ i → A ′ with degree 1 − i satisfyingthe following conditions: • ( A ∞ -morphism) X l ≥ ,k + ··· + k l = k ( − P lj =1 k j ( l − j )+ P ν<µ k ν k µ m ′ l ◦ ( f k ⊗ · · · ⊗ f k l )= X s +1+ t = i,s + l + t = k ( − k +( s +1)( l +1) f i ◦ (id ⊗ sA ⊗ m l ⊗ id ⊗ tA )for k ≥
1, and • (commutativity) X σ ∈ Sh( j,i − j ) ǫ · f i ( a σ (1) , · · · , a σ ( i ) ) = 0for i > j > a , . . . , a i ∈ A .Then f is called a C ∞ -morphism . If f is a quasi-isomorphism, f is called a C ∞ -quasi-morphism . Definition 2.10.
Given C ∞ -algebra ( A, m A ), a pair f : ( H, m ) → ( A, m A ) ofa C ∞ -algebra structure m on the cohomology H := H ( A, m A ) and a C ∞ -quasi-isomorphism f such that f induces the identity map on the cohomology H iscalled C ∞ -algebra model of A . Remark 2.11 (Bar construction of a C ∞ -morphism) . Let f : ( A, m ) → ( A ′ , m ′ ) bea C ∞ -morphism. Defining the suspension of f i by ¯ f i := s ◦ f i ◦ ( s − ) ⊗ i : A [1] ⊗ i → A ′ [1] for all i ≥
1, then the degree of ¯ f i is 0. Constructing the coalgebra map BA → BA ′ f := ∞ X k =1 X i ≥ ,k + ··· + k i = k ¯ f k ⊗ · · · ⊗ ¯ f k i from maps ¯ f n , we have the equations f ◦ m = m ′ ◦ f , f ◦ µ = µ ◦ ( f ⊗ f ) . TAKAHIRO MATSUYUKI So f is a differential bialgebra map ( BA, m ) → ( BA ′ , m ′ ) between bar constructions.According to [9], a formal homology connection ( ω, δ ) on X is equivalent to aminimal C ∞ -algebra model f : ( H, m ) → A of A . It is verified as follows: put ω = − X i ,...,i k ( − ǫ s − ¯ f n ( x i , . . . , x i k ) x i · · · x i k ,δ = m ∗ , where ǫ = | x i | ( | x i | + · · · + | x i k | ) + · · · + | x i k − || x i k | , ¯ f n = s f n ( s − ) ⊗ n : H [1] ⊗ n → A [1], x i is the dual basis of x i , and m is the bar-construction of m . Then the differential δ on the dual ( BH ) ∗ = ˆ T W of thebar-construction BH can be restricted on ˆ LW since δ is a coderivation. So thepair ( ω, δ ) is a formal homology connection on X . Conversely we can recover f : ( H, m ) → A from ( ω, δ ). Note that the condition that f is an A ∞ -morphismcorresponds to the flatness.For a diffeomorphism ϕ : Y → X , the formal homology connection ϕ ∗ ( ω, δ ) on Y is corresponding to the C ∞ -algebra model ϕ ∗ ◦ f ◦ ( ϕ ∗ ) − : ( H ( Y ) , ϕ ∗ ◦ m ◦ ( ϕ ∗ ) − ) → A ( Y ) . Here H ( Y ) is the reduced de Rham cohomology of Y , and A ( Y ) is the reduce deRham complex of Y .3. The set of formal homology connections
In this section, we shall introduce the main tool of this paper, the simplicialbundle Q • ( X ) → Q • ( E ) → S • ( B ) associated to a fiber bundle X → E → B . • In subsection 3.1, we shall define the Kan complex Q • ( X ) and calculateits homotopy group. • In subsection 3.2, we shall define the bundle Q • ( E ) → S • ( B ) and give howto fix differentials of fibers.The notions and the theorems in this section is used in Section 5.3.1. The simplicial set of formal homology connections.
Let X be a mani-fold. The set of formal homology connections on X is denoted by Q ( X ).We define the simplicial de Rham DGA A • = { A n } ∞ n =0 on X by A n := ˜ A • ( X × ∆ n ) . Its face maps and degeneracy maps are induced by the coface maps and the code-generacy maps of the cosimplicial space ∆ • = { ∆ n } ∞ n =0 .Put Q n ( X ) := Q ( X × ∆ n ). The face map Q n ( X ) → Q n − ( X ) and the de-generacy map Q n ( X ) → Q n +1 ( X ) are induced by those of the simplicial DGA A • .Thus the family Q • ( X ) = { Q n ( X ) } ∞ n =0 of sets is a simplicial set. Given a Chendifferential δ on X , the set of formal homology connections ( ω, δ ) on X × ∆ n isdenoted by Q n ( X, δ ). Then Q • ( X, δ ) is also a simplicial set.We denote the set of Maurer-Cartan elements of ( A n ⊗ ˆ LW, d + δ ) by MC n ( X, δ ).We obtain the simplicial set MC • ( X, δ ), and then Q • ( X, δ ) is a simplicial subset ofMC • ( X, δ ). Lemma 3.1.
For any n -th simplicial Maurer-Cartan element α ∈ MC n ( X, δ ), if ∂ i α ∈ Q n − ( X ) for some 0 ≤ i ≤ n , then α ∈ Q n ( X, δ ). BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 9 Proof.
We can identify α with a C ∞ -map f : H → A n . The face map ∂ i : A n → A n − for any i gives the standard identification by H • ( X × ∆ n ) ≃ H • ( X × ∆ n − )and H ( ∂ i f ) : H → H ( A n − ) is the identity map under the assumption. Thereforewe have the commutative diagram H H ( f ) / / id= H ( ∂ i f ) ❍❍❍❍❍❍❍❍❍ H ( A n ) H ( ∂ i )=id (cid:15) (cid:15) HH ( A n − ) H and it leads H ( f ) = id : H → H ( A n ) = H . (cid:3) Since the simplicial set MC • ( X, δ ) is a Kan complex (proved in Section 4 of [7]),the following lemma is obtained immediately from Lemma 3.1:
Lemma 3.2.
The simplicial set Q • ( X ) is a Kan complex. Furthermore the mapinduced by the inclusion π ( Q • ( X, δ )) → π (MC • ( X, δ ))is injective, and the map π n ( Q • ( X, δ ) , τ ) → π n (MC • ( X, δ ) , τ )for τ ∈ Q ( X, δ ) and n ≥ Theorem 3.3.
The homotopy groups of the simplicial set Q • ( X ) are described by π n ( Q • ( X ) , τ ) ≃ H n ( δ )for n ≥ τ = ( ω, δ ) on X , where H ( δ ) isequipped with the Baker-Campbell-Hausdorff product of H ( A ⊗ ˆ LW ). Proof.
From Proposition 5.4 and Theorem 5.5 in [1], we have π n ( Q • ( X ) , τ ) ≃ π n (MC • ( X, δ ) , τ ) ≃ H n − ( A ⊗ ˆ LW, d + δ + [ ω, − ]) . Let F : BH → BA be the bar-construction F : BH → BA of the C ∞ -morphism H → A corresponding to τ . An endomorphism D : BH → BA is called a Hopfderivation over F if it satisfies D ∇ = ∇ ( D ⊗ F + F ⊗ D ) , ∆ D = ( D ⊗ F + F ⊗ D )∆ , where ∇ is the product and ∆ is the coproduct. The vector space of Hopf derivationsover F is denoted by Der F ( BH, BA ). A Hopf derivation D over F is uniquelydetermined by the lowest term BH D → BA proj. → A [1]. So there exists the naturalinclusion Der F ( BH, BA ) ⊂ Hom(
BH, A [1]).We shall prove the suspension of ( A ⊗ ˆ LW, d + δ + [ ω, − ]) and the chain complex(Der F ( BH, BA ) , D ) are isomorphic. Here the differential D is defined by D ( D ) = m A ◦ D − ( − D D ◦ m , where m A and m are the bar-constructions of the C ∞ -algebra structures m A and m on A and H respectively. Here m is the corresponding C ∞ -structure on H to δ .Through the embedding ˆ LW ⊂ ˆ T W = ( BH ) ∗ , consider the linear isomorphismΦ : A [1] ⊗ ˆ LW → Der F ( BH, BA ) ⊂ Hom(
BH, A [1]) defined byΦ( α ⊗ f )( x ) = f ( x ) α for x ∈ BH . Here the differential on A [1] ⊗ ˆ LW is equal to s ( d + δ + [ ω, − ]) s − .Then, using F = Φ( s ω ), we haveΦ( s ( d + δ + [ ω, − ]) s − ( α ⊗ f ))( x )= dαf ( x ) + ( − α +1 αδf ( x ) + s [ ω, s − α ⊗ f ]( x )= dαf ( x ) − ( − α + f αf m ( x ) + m A ◦ ( F ⊗ Φ( αf ))( x ) + m A ◦ (Φ( αf ) ⊗ F )( x )= D Φ( αf )( x ) . Thus the map Φ is a chain isomorphism.On the other hand, the map F ◦ − : (Der( ˆ LW ) , ad( δ )) = (Der( BH ) , ad( m )) → (Der F ( BH, BA ) , D )is a quasi-isomorphism because F is a quasi-isomorphism. So we get the isomor-phism H n − ( A ⊗ ˆ LW, d + δ + [ ω, − ]) ≃ H n (Der( ˆ LW ) , ad( δ )) . (cid:3) The next theorem is also proved in [11].
Theorem 3.4 (Kajiura-M.-Terashima) . The set π ( Q • ( X, δ )) has the standardright action of QA (1) ( δ ), and the action is free and transitive. Proof.
There are the identifications obtained in Section 2.6: π ( Q • ( X, δ )) = { C ∞ -algebra model f : ( H, m ) → A } / ( C ∞ -homotopic) , QA (1) ( δ ) = { C ∞ -isom f : ( H, m ) → A ; f = id H : H → H } / ( C ∞ -homotopic) , where m is the minimal C ∞ -algebra structure on H corresponding to δ . So QA (1) ( δ )acts on the right of π ( Q • ( X, δ )) by composition. Since any C ∞ -quasi-isomorphismhas a homotopy inverse, the action is free and transitive. (cid:3) The simplicial bundle of formal homology connections.
Let X → E → B be a smooth fiber bundle. In the section, we shall define the simplicial bundle offormal homology connections on fibers. Definition 3.5.
We define the simplicial bundle Q • ( E ) → S • ( B ) over the simplicialset S • ( B ) of smooth singular simplices ∆ n → B as follows: • the fiber over an n -simplex σ ∈ S n ( B ) is Q n ( E ) σ := Q ( σ ∗ E ), and • the face maps and the degeneracy maps are the induced maps Q n ( E ) σ →Q n − ( E ) ∂ i σ and Q n ( E ) σ → Q n +1 ( E ) s i σ by the coface maps and the code-generacy maps of ∆ • respectively.We can check that Q • ( E ) → S • ( B ) is a bundle of simplicial sets in the sense ofMay [15]. Proposition 3.6.
The simplicial map Q • ( E ) → S • ( B ) is a simplicial bundle withfiber Q • ( X ). BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 11 Proof.
For an n -simplex σ ∈ S n ( B ) and a trivialization ϕ σ : ∆ n × X ≃ σ ∗ E , weobtain the trivialization ϕ σ,P : ∆ i × X ≃ σ ( P ) ∗ E for P ∈ ∆[ n ] i by the diagram∆ n × X ϕ σ / / σ ∗ E ∆ i × X f P × id X O O ϕ σ,P / / σ ( P ) ∗ E O O regarding σ as a simplicial map σ : ∆[ n ] → S • ( B ). Here the map f P : ∆ i → ∆ n isthe induced map P : ∆[ i ] → ∆[ n ].Then we obtain the simplicial trivializationˆ ϕ σ : σ ∗ Q • ( E ) ≃ ∆[ n ] × Q • ( X ) . by ( P, α ) ( P, ϕ ∗ σ,P α ), where σ ∗ Q i ( E ) = { ( P, α ) ∈ ∆[ n ] i × Q ( σ ( P ) ∗ E ) } . (cid:3) We need to define a section of differentials δ to restrict the simplicial fiber Q • ( X )to Q • ( X, δ ). Definition 3.7.
Fix a Chen differential δ ∈ Der( ˆ LW ) − of X . Suppose δ is G -invariant with respect to the action of the homological structure group G onDer( ˆ LW ) (induced by the action on W ). Then it gives the section ˆ δ of the surjectivemap between sets CD ( E ) := a b ∈ B CD ( E ) b → B, where CD ( E ) b := { Chen differential of E b } for b ∈ B . Explicitly, ˆ δ : B → CD ( E )is described byˆ δ ( b ) = | ϕ b | − ◦ δ ◦ | ϕ b | ∈ Der( ˆ LW ( E b )) − , W ( E b ) := ˜ H • ( E b ; R )[ − | ϕ b | : ˆ LW ( E b ) ≃ ˆ LW is the isomorphism induced by a trivialization ϕ b : E b ≃ X . We call ˆ δ a section of Chen differentials . Given this, we can considerthe simplicial bundle Q • ( E, ˆ δ ) → S • ( B ) defined by Q n ( E, ˆ δ ) σ := Q ( σ ∗ E, ˆ δ ( σ ))for σ ∈ S n ( B ). Here ˆ δ ( σ ) is the Chen differential of σ ∗ E defined by ˆ δ ( σ ) throughthe isomorphism ˜ H • ( σ ∗ E ) ≃ ˜ H • ( E σ ) on homologies. Here σ = ∂ · · · ∂ n σ is theimage of the base point of ∆ n .For example, if X is formal, the differential δ corresponding to the cohomologyring structure of X is Diff( X )-invariant.4. Obstruction theory
Obstruction theory for simplicial sets is studied in [3, 6]. We shall review a partof them and rewrite briefly obstruction theory as in Steenrod [19] for simplicial setsin order to fit our use. • In Section 4.1, we define the fundamental groupoid, a local system, andthe cochain complex with local coefficient. • In Section 4.2, we define obstruction classes to extend a section of a sim-plicial bundle over the n -skeleton for n ≥ • In Section 4.3, we suppose the local system Π of connected componentsof fibers has a free and transitive action of a local system G of filteredgroups. Under the assumption, we introduce obstruction classes to extenda section of a simplicial bundle over the 0-skeleton stepwisely using thefiltration of G .We shall apply these constructions to the simplicial bundle Q • ( X ) → Q • ( E ) → S • ( B ) in Section 5.4.1. Local system.
We shall define cohomology with local coefficients briefly. Wecan see definitions in [3, 6].
Definition 4.1.
Let X be a Kan complex. We define the fundamental groupoid Π ( X ) of X such that the set of objects is X and the set of morphisms from x to y is the set of homotopy classes of γ ∈ X satisfying ∂ γ = x and ∂ γ = y . Acovariant functor Π ( X ) → Ab is called a local system on X . Here Ab is thecategory of abelian groups.Let E → B be a Kan simplicial bundle with n -simple fiber X , i.e., X is a Kancomplex and π ( X , x ) acts on π n ( X , x ) trivially. Definition 4.2.
We define the local system Π n ( E / B ) on B as follows: for a vertex v ∈ B , Π n ( E / B ) v := π n ( v ∗ E ) . Note that we need not to choose a base point of v ∗ E because it is n -simple. For apath γ ∈ B such that v = ∂ γ and v = ∂ γ , take a trivialization ϕ γ : ∆[1] × v ∗ E ≃ γ ∗ E such that ∆[1] × v ∗ E ϕ γ / / γ ∗ E v ∗ E δ O O v ∗ E . incl. O O Here δ i : ∆[0] → ∆[1] is the coface maps. Then we have the isomorphism g γ : v ∗ E → v ∗ E , which is called the holonomy along γ , defined by∆[1] × v ∗ E ϕ γ / / γ ∗ E v ∗ E δ O O g γ / / v ∗ E . incl. O O So we put Π n ( E / B )( γ ) := ( g − γ ) ∗ : π n ( v ∗ E ) → π n ( v ∗ E ) . We can prove that it is depend on only the homotopy class of γ since E → B is Kanfibration. In fact, for another path γ ′ homotopic to γ by a homotopy σ ∈ B , there BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 13 exists a homotopy h satisfying the commutative diagramΛ [2] × v ∗ E ϕ γ ∪ ϕ γ ′ / / (cid:15) (cid:15) σ ∗ E (cid:15) (cid:15) ∆[2] × v ∗ E h / / ∆[2]by Theorem 7.8 in [15]. Here Λ [2] is the (2 , Definition 4.3.
Let X be a Kan complex, A a simplicial subset of X , and M :Π ( X ) → Ab a local system on X . We define the cochain complex with coef-ficient M by C n ( X , A ; M ) := ( c : X n → a v ∈X M ( v ); c ( x ) ∈ M ( x ) , c |A = 0 ) , where x = ∂ · · · ∂ n x , and its normalized version by N n ( X , A ; M ) := n \ i =0 Ker( s ∗ i : C n ( X , A ; M ) → C n − ( X , A ; M )) . The differential δ : C n ( X , A ; M ) → C n +1 ( X , A ; M ) is defined by δc ( x ) = M ( x ) − c ( ∂ x ) − c ( ∂ x ) + · · · + ( − n +1 c ( ∂ n +1 x ) , where x = ∂ · · · ∂ n x . Its cohomology is denoted by H n ( X , A ; M ).4.2. Obstruction cocycles and difference cochains.
Let A be a simplicialsubset of B . We call a simplicial map s satisfying the following diagram an n -partial section relative to A : E (cid:15) (cid:15) sk n ( B ) ∪ A s : : tttttttttt / / B Given an n -partial section s : sk n ( B ) ∪ A → E relative to A , we shall constructthe obstruction cocycle of sc ( s ) ∈ N n +1 ( B , A ; Π n ( E / B ))to extend a partial section sk n +1 ( B ) ∪ A → E as follows: for an ( n + 1)-simplex σ ∈ B n +1 , we get the induced section s σ such that σ ∗ E / / E sk n (∆[ n + 1]) s σ O O sk n ( σ ) / / sk n ( B ) . s O O So we put c ( s )( σ ) := g − σ [ s σ ] ∈ π n ( σ ∗ E ) , where g σ : π n ( σ ∗ E ) → π n ( σ ∗ E ) is an isomorphism induced by the inclusion σ ∗ E → σ ∗ E . Proposition 4.4.
The cochain c ( s ) is a cocycle. Proof.
For an ( n + 2)-simplex σ ∈ B n +2 , we have( ∂ i σ ) ∗ E / / σ ∗ E / / E sk n (∆[ n + 1]) sk n ( δ i ) / / s ∂iσ O O sk n (∆[ n + 2]) sk n ( σ ) / / s σ O O sk n ( B ) . s O O So the commutative diagrams for i = 0 σ ∗ E ❍❍❍❍❍❍❍❍❍ (cid:15) (cid:15) ( ∂ i σ ) ∗ E / / σ ∗ E σ ∗ E (cid:15) (cid:15) σ ∗ E (cid:15) (cid:15) g σ o o ( ∂ σ ) ∗ E / / σ ∗ E imply the equations g − ∂ i σ [ s ∂ i σ ] = g − σ ( s σ ) ∗ [sk n ( δ i )] , g − σ g − ∂ σ [ s ∂ σ ] = g − σ ( s σ ) ∗ ( σ ) ∗ [sk n ( δ )] . Here note that [sk n ( δ i )] ∈ π n (sk n (∆[ n + 2]) ,
0) and [sk n ( δ )] ∈ π n (sk n (∆[ n + 2]) , δc ( s ))( σ ) = g − σ ( s σ ) ∗ ( σ ) ∗ [sk n ( δ )] + X i =0 ( − i [sk n ( δ i )] = 0 , using the relation ( σ ) ∗ [sk n ( δ )] + P i =0 ( − i [sk n ( δ i )] = 0 in π n (sk n (∆[ n + 2]) , (cid:3) We shall define the difference cochain for n -partial sections s , s : sk n ( B ) → E and a fiberwise homotopy h : sk n − ( B ) × ∆[1] → E × ∆[1] between their restrictionon sk n − ( B ). Gluing these maps, we have the map h (cid:3) : (sk n ( B ) × sk (∆[1])) ∪ (sk n − ( B ) × ∆[1]) → E × ∆[1] . We consider the obstruction cocycle c ( h (cid:3) ) ∈ N n +1 (sk n ( B ) × ∆[1] , (sk n ( B ) × sk (∆[1])) ∪ (sk n − ( B ) × ∆[1]); Π (cid:3) n ) , where Π (cid:3) n = Π n ( E × ∆[1] / B × ∆[1]). Note that faces of non-degenerate simplicesof sk n ( B ) × ∆[1] are in (sk n ( B ) × sk (∆[1])) ∪ (sk n − ( B ) × ∆[1]). Through theEilenberg-Zilber map × : N n ( B ) ⊗ N (∆[1]) → N n +1 (sk n ( B ) × ∆[1] , (sk n ( B ) × sk (∆[1])) ∪ (sk n − ( B ) × ∆[1])) , we can define the cochain d ( s , h, s ) ∈ N n ( B ; Π n ( E / B )) by d ( s , h, s )( σ ) := ( − n c ( h (cid:3) )( σ × I )for σ ∈ B n . Here I is the unique non-degenerate simplex in ∆[1] . Proposition 4.5.
The cochain d ( s , h, s ) satisfies δd ( s , h, s ) = c ( s ) − c ( s ) . BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 15 Proof.
It is proved by the equations δd ( s , h, s )( σ ) = g − σ d ( s , h, s )( ∂ σ ) + X i =0 ( − i d ( s , h, s )( ∂ i σ )= ( − n g − σ c ( h (cid:3) )( ∂ σ ⊗ I ) + X i =0 ( − n + i c ( h (cid:3) )( ∂ i σ ⊗ I )= c ( h (cid:3) )( σ ⊗ ∂I ) − δc ( h (cid:3) )( σ ⊗ I )= c ( s ) − c ( s ) . (cid:3) The next two propositions hold in the same way as in obstruction theory [19].
Proposition 4.6. An n -partial section s : sk n ( B ) → E extends to an ( n +1)-partialsection sk n +1 ( B ) → E if and only if c ( s ) = 0. Proposition 4.7.
For n -partial sections s, s ′ : sk n ( B ) → E , if obstruction cocycles c ( s ) and c ( s ′ ) are cohomologous, there is a homotopy between s | sk n − ( B ) and s ′ | sk n − ( B ).Suppose a fiber X of a Kan fiber bundle E → B is ( n − π ( X , x ) is abelian if n = 1). Then we can get an n -partial section s : sk n ( B ) → E .If we get another n -partial section s ′ , these is a homotopy between s | sk n − ( B ) and s ′ | sk n − ( B ). So we obtain an invariant o n ( E ) := [ c ( s )] ∈ H n +1 ( B ; Π n ( E / B )) . It is called the obstruction class of E → B .4.3.
Obstruction for n = 0 . We consider an extension of a 0-partial section underthe following situation: for a simplicial bundle
E → B , suppose that the local systemΠ ( E / B ) of sets has a free and transitive right action of a local system G of groupson B .At first, we define the non-abelian obstruction class of a 0-partial section. Forthat, we remark the definition of the non-abelian cohomology with values in a localsystem of non-abelian groups. Here “non-abelian cohomology” is in the sense of[10]. Definition 4.8.
Let X be a simplicial set and G a local system of groups on X .Define the (non-abelian) cochain complex of X with coefficient G C n ( X ; G ) := ( c : X n → a v ∈X G ( v ); c ( x ) ∈ G ( x ) ) for 0 ≤ n ≤ ϕ of C ( X ; G ) on C ( X ; G ):( ϕ ( f ) c )( γ ) = f ( ∂ γ ) c ( γ )( G ( γ ) − f ( ∂ γ ) − )for f ∈ C ( X ; G ) and c ∈ C ( X ; G ),(ii) the action ψ of C ( X ; G ) on C ( X ; G ):( ψ ( f ) c )( σ ) = Ad( G ( ∂ σ ) − f ( ∂ ∂ σ ))( c ( σ ))for f ∈ C ( X ; G ) and c ∈ C ( X ; G ), (iii) the map δ : C ( X ; G ) → C ( X ; G ) satisfying δ (1) = 1 and δ ( ϕ ( f ) c ) = ψ ( f ) c for f ∈ C ( X ; G ) and c ∈ C ( X ; G ): δc ( σ ) = ( G ( ∂ σ ) − c ( ∂ σ )) c ( ∂ σ ) − c ( ∂ σ )for c ∈ C ( X ; G ) and σ ∈ X .The we get the H ( X ; G ) := Ker( C ( X ; G ) → Aut( C ( X ; G )) ⋉ C ( X ; G ) → C ( X ; G ))and the H ( X ; G ) := δ − (1) /C ( X ; G ) . Given a 0-partial section s : sk ( B ) → E , put c ( s )( γ ) = [ s ( ∂ γ )] − (Π ( γ ) − [ s ( ∂ γ )]) ∈ G γ for γ ∈ B , i.e., c ( s )( γ ) ∈ G γ is the unique element satisfying[ s ( ∂ γ )] c ( s )( γ ) = Π ( γ ) − [ s ( ∂ γ )] . By definition, c ( s ) ∈ C ( B ; G ) is a cocycle. For another section s ′ : sk ( B ) → E , ifwe can get f ∈ C ( B ; G ) uniquely such that s ′ ( x ) = s ( x ) f ( x )for x ∈ X , then c ( s ′ ) = ϕ ( f ) c ( s ) holds. We denote f by d ( s, s ′ ) as in Section 4.2.Especially the cohomology class o ( E ) := [ c ( s )] ∈ H ( B ; G )is independent of a choice of a 0-partial section s : sk ( B ) → E . As usual obstructiontheory, o ( E ) = 1 if and only if there is a 1-partial section sk ( B ) → E . It followsfrom the following proposition: Proposition 4.9. If o ( E ) = 1, there exists a 0-partial section s : sk ( B ) → E suchthat c ( s ) = 1. Proof.
If [ c ( s )] = 1, there exists f ∈ C ( B ; G ) such that c ( s ) = ϕ ( f )(1). So replacing s with sf − , we get the proposition. (cid:3) The non-abelian obstruction o ( E ) is hard to deal with. So we shall consider toget a certain abelian cocycle using a filtration {F ( i ) G} ∞ i =1 of G such that G b = F (1) G b ⊲ F (2) G b ⊲ · · · , [ F ( i ) G b , F ( j ) G b ] ⊂ F ( i + j ) G b for b ∈ B , and the map G ( γ ) for γ ∈ B preserves the filtration. Given such afiltration, we have the local system of abelian groups defined bygr i ( G ) := F ( i ) G / F ( i +1) G . It is also written by gr i ( G ) b = gr i ( G b ) for b ∈ B .If the image of c ( s ) to C ( B ; G / F ( i ) G ) is trivial, i.e., c ( s )( γ ) ∈ F ( i ) G γ for γ ∈ B ,we get its image c i ( s ) to the (abelian) chain complex C ( B ; gr i ( G )). For anotherpartial section s ′ : sk ( B ) → E satisfying the same condition, we can also get theimage d i ( s, s ′ ) of d ( s, s ′ ) to C ( B ; gr i ( G )). Then it satisfies the equation c i ( s ′ ) − c i ( s ) = δd i ( s, s ′ ) . It means o ( i ) ( E ) := [ c i ( s )] ∈ H ( B ; gr i ( G )) is obtained uniquely. BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 17 Proposition 4.10. If o ( i ) ( E ) is defined and trivial, there exists a partial section s : sk ( B ) → E such that c ( s )( γ ) ∈ F ( i +1) G γ for γ ∈ B . Proof.
Supposing o ( i ) ( E ) = [ c i ( s )] = 1, we have 1 = [ c ( s )] ∈ H ( B ; G / F ( i +1) G ).Then there exists a 0-partial section s ′ : sk ( B ) → E such that c ( s ′ ) = 1 ∈ C ( B ; G / F ( i +1) G ). This section satisfies the required condition. (cid:3) Obstruction of the bundles of formal homology connections
Let X → E → B be a smooth fiber bundle with homological structure group G .Fix a G -invariant Chen differential δ on ˆ LW , where W = ˜ H • ( X ; R )[ − Q • ( E ) → S • ( B ) and get characteristic classes using the obstruction. • In Section 5.1, we consider the case that Q • ( X ) is connected. • In Section 5.2, we give a non-trivial example of the construction in Section5.1. • In Section 5.3, we consider the case that Q • ( X ) is not connected. • In Section 5.4, we give a non-trivial example of the construction in Section5.3. We shall prove that the our obstruction class for a surface bundle isequivalent to the 1st Morita-Miller-Mumford class.5.1.
Connected cases.
Suppose H (1)0 ( δ ) = 0 and H i ( δ ) = 0 for n > i >
0. Inaddition, suppose, if n = 1, H ( δ ) ≃ H ( ˆ LW ⊗ A, d + δ + [ τ, − ]) is abelian withrespect to the Baker-Campbell-Hausdorff product. Then we get the obstructionclass of the simplicial bundle Q • ( E, ˆ δ ) → S • ( B ) o = o n ( Q • ( E, ˆ δ )) ∈ H n +1 ( B ; Π n ) , where Π n = Π n ( Q • ( E, ˆ δ ) /S • ( B )), and the characteristic maps of a fiber bundle E → B (Λ p H n ( δ ) ∗ ) G → H p ( n +1) ( B ; R )by ψ ψ ( o , . . . , o ) for p ≥
1. Strictly speaking, the map is defined as follows.First take a family { ϕ i : E | U i ≃ U i × X } of local trivializations of E over an opencovering { U i } of B such that the images of transition functions is included in thestructure group. For a simplex σ ∈ S n +1 ( B ), choose an open set U i containing avertex b = σ . Then the trivialization ϕ i induces the isomorphism g b : Π n ( b ) = H n (ˆ δ ( b )) ≃ H n ( δ ). If p = 1, put ψ ( o )( σ ) := [ ψ ( g b c ( σ ))] when o is represented by acochain c . For a generic integer p >
1, the value of ψ is defined by the cup product ψ ( o , . . . , o ) := P ψ j ( o ) · · · ψ j p ( o ) when ψ is described by the sum P ψ j · · · ψ j p for ψ j ∈ H n ( δ ) ∗ . It is independent of choices of open sets and a family of trivializationsbecause of G -invariance of ψ .5.2. Example of a sphere bundle.
We consider the sphere bundle S → E = S × S S → S associated to the Hopf fibration S → S → S , where U (1) = S acts on S = C ∪ {∞} by rotations. Here S = { ( z , z ) ∈ C ; | z | + | z | = 1 } ,E = S × S S = S × S / (( z, w ) ∼ ( ζz, ζ − w ) , ζ ∈ S ) , where S = { ζ ∈ C ; | ζ | = 1 } . The image of ( z, w ) ∈ S × S in E is denoted by[ z, w ]. We use the identification S /S ≃ S = C ∪{∞} defined by [ z , z ] z /z . Since the action of S on S has two fixed points 0 and ∞ , this fiber bundle hasa section S = S /S → S × S S defined by [ b ] [ b, ∞ ]. We fix the section.Denote the volume form on the fiber S = C ∪ {∞} by v = √− π dwd ¯ w (1 + | w | ) , where w is the complex coordinate of the Riemann sphere S , and the desuspensionof the fundamental class by x = s − [ v ] ∈ W = H ( S )[ − LW, δ ) of S is given by LW = L ( x ) ( | x | = 1) , δx = 0and its Lie algebra of derivationsDer( LW ) = (cid:28) x ∂∂x , [ x, x ] ∂∂x (cid:29) . Here h v , . . . , v n i is the vector space generated by v , . . . , v n , and L ( v , . . . , v n ) isthe Lie algebra generated by v , . . . , v n . We also get H ( δ ) = Der( LW ) = (cid:28) [ x, x ] ∂∂x (cid:29) . For simplicity, we restrict the bundle Q • ( E ) → S • ( S ) to the Kan complexdefined by K n = { (∆ n , sk ∆ n ) → ( S , ∞ ) } ⊂ S n ( S ) . If n ≤ K n is described by K = { p ∞ } , K = { γ ∞ } , where p ∞ : ∆ → S and γ ∞ : ∆ → S are constant maps to the point ∞ . Weput Q • := Q • ( E ) | K • .Put D = { z ∈ C ; | z | ≤ } ⊂ C . We use the map ρ : D → S defined by ρ ( z ) = ( z/ (1 − | z | ) ( | z | < ∞ ( | z | = 1) , the trivialization ϕ ρ : D × S → ρ ∗ E defined by ϕ ρ ( z, w ) = (cid:18) z, (cid:20)(cid:18) z | z | , − | z | | z | (cid:19) , w (cid:21)(cid:19) . Choose an orientation-preserving diffeomorphism h : ∆ / ( δ ∆ ∪ δ ∆ ) ≃ D suchthat [0 ,
1] = ∆ δ → ∆ / ( δ ∆ ∪ δ ∆ ) h → D is given by t e π √− t . Here the standard 1-simplex ∆ is identified with theinterval [0 ,
1] by the isomorphism [0 , ≃ ∆ : t ( t, − t ). By composing theprojection ∆ → ∆ / ( δ ∆ ∪ δ ∆ ), we get ¯ h : ∆ → D and the 2-simplex σ = ρ ¯ h : ∆ → S in K • . Then we have the trivialization ϕ σ : ∆ × S ≃ σ ∗ E and BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 19 its restriction ϕ σ,∂ : ∆ × S ≃ ( ∂ σ ) ∗ E = γ ∗∞ E satisfying∆ × S ϕ σ,∂ (cid:15) (cid:15) δ × id / / ∆ × S ϕ σ (cid:15) (cid:15) ¯ h × id / / D × S ϕ ρ (cid:15) (cid:15) γ ∗∞ E / / (cid:15) (cid:15) σ ∗ E / / (cid:15) (cid:15) ρ ∗ E / / (cid:15) (cid:15) E (cid:15) (cid:15) ∆ δ / / ∆ h / / D ρ / / S . On the other hand, we have the trivialization ϕ γ ∞ : ∆ × S ≃ ∆ × E ∞ = γ ∗∞ E defined by ( t, w ) ( t, [(1 , , w ]). Then the transition function g = ϕ − γ ∞ ◦ ϕ σ,∂ :∆ × S ≃ ∆ × S is described by g ( t, w ) = ϕ − γ ∞ ( t, [( e π √− t , , w ]) = ( t, e − π √− t w ) . We can also define the trivialization ϕ p ∞ : S ≃ p ∗∞ E = E ∞ on ∞ ∈ S ϕ p ∞ ( w ) = [(1 , , w ] . The partial section s : sk K → Q is defined as follows: s ( p ∞ ) := v x ∈ Q ( E ) p ∞ , s ( γ ∞ ) := v x ∈ Q ( E ) γ ∞ , where v := ( ϕ − p ∞ ) ∗ v ∈ A ( E ∞ ) and v := ( ϕ − γ ∞ ) ∗ v ∈ A ( γ ∗∞ E ). Remark that v ∈ A ( S ) ⊂ A (∆ × S ). According to Section 4.2, we get s σ : sk (∆[2]) → σ ∗ Q • ( E ). Its homotopy class is equal to [ s σ ] = [ v x ] ∈ π ( σ ∗ Q • ( E ) , v x ). So wehave c ( s )( σ ) = g ∗ [ s σ ] = g ∗ [ v x ] = [ g ∗ ( v ) x ] ∈ π ( Q • ( S ) , vx )under the identification ϕ ∗ p ∞ : π ( Q • ( E ∞ ) , v x ) ≃ π ( Q • ( S ) , vx ). By direct calcu-lation, we get g ∗ ( v ) = v + ξdt, where ξ = − ¯ wdw + wd ¯ w (1 + | w | ) = df, f ( w ) = 12 11 + | w | . Then the 2-form Ξ = t ξdt − t ξdt + 2 f dt dt , satisfies the equation( v + Ξ) = 2 v Ξ = 4 f vdt dt = − f vdt dt = − d ( f v ( t dt − t dt )) . So we obtain the formal homology connection α = ( v +Ξ) x − f v ( t dt − t dt )[ x, x ] ∈ Q ( S ) satisfying ∂ α = ( v + ξdt ) x, ∂ α = vx + 4 f vdt [ x, x ] , ∂ α = vx. Therefore the equation[ g ∗ ( v ) x ] = [( v + ξdt ) x ] = [ vx + 4 f vdt [ x, x ]] ∈ π ( Q • ( S ) , vx )holds. Furthermore Z S f v = Z S √− π dwd ¯ w (1 + | w | ) = 1 π Z ∞ rdr (1 + r ) Z π dθ = 2 Z ∞ dx (1 + x ) = 1 means that the de Rham cohomology class [4 f v ] ∈ H ( S ) is non-trivial. Accordingto Theorem 4.10 of [1], we have c ( s )( σ ) = 0 and o = [ c ( s )] = 0 ∈ H ( K ; H ( δ )) . Finally evaluating the class with the dual basis ν of [ x, x ] ∂/∂x ∈ Der( LW ) , weget the non-trivial characteristic class ν ( o ) ∈ H ( K ) = H ( S ) , which is the Euler class of the sphere bundle E → S (given in [16]).5.3. Non-connected cases. If H (1)0 ( δ ) = 0, we can apply the construction in Sec-tion 4.3. Put Π = Π ( Q ( E, ˆ δ ) /S • ( B )). From Theorem 3.4, the group QA (1) (ˆ δ ( b ))acts on Π ( b ) freely and transitively.The local system QA + ( E ) of groups is defined by QA + ( E ) b := QA (1) (ˆ δ ( b )) , γ ∗ ( f ) := ( g − γ ) ∗ ◦ f ◦ ( g γ ) ∗ for b ∈ B , γ ∈ S ( B ) and f ∈ QA + ( E ) γ (0) , where g γ : E γ (0) → E γ (1) is theholonomy along γ . Then we get the non-abelian obstruction class o = o ( Q • ( E )) ∈ H ( B ; QA + ( E ))in Section 4.3.Furthermore we have the filtration {QA ( i ) ( E ) } ∞ i =1 of QA + ( E ) defined in Section2.4. By the observations in Section 2.4, there exists the identification as local systemof vector spaces gr i ( QA + ( E )) ≃ gr i ( H +0 ( E )) , where the local system H +0 ( E ) of Lie algebras is defined in the same way as QA + ( E ).Here note that gr( H +0 ( E )) is defined similarly to gr( QA + ( E )) using its filtration.Suppose we get the obstruction class o ( i ) = 0 ∈ H ( B ; gr i ( H +0 ( E ))). In the sameway as in Section 5.1, the characteristic map(Λ • gr i ( H ( δ )) ∗ ) G → H • ( B ; R )is obtained by ψ ψ ( o ( i ) , . . . , o ( i ) ).Especially, if X is formal and δ corresponds to the product of the cohomology H of X , we obtain the characteristic map(Λ • H i ( δ ) ∗ ) G → H • ( B ; R ) . When X is an oriented closed manifold, we shall clarify a relation between thecharacteristic map constructed in [11] and the construction above. According toChen’s theorem, given a metric of the fiber bundle E → B , we have the map s : B → Q ( E ) as follows: for b ∈ B , the metric on E b gives the formal homologyconnection s ( b ) on E b compatible with the Hodge decomposition of E b .Composing the natural projection Q ( E ) → CD ( E ) with s , we get the sectionˆ δ : B → CD ( E ). Theorem 5.1.
Let X be an oriented closed manifold and E → B be a smoothbundle with section and metric. Suppose the metric gives a section ˆ δ of Chen BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 21 differentials corresponding to a G -invariant Chen differential δ of X . Then we havethe commutative diagram of chain complexes C • CE ( H (1)0 ( δ )) G Φ / / (cid:15) (cid:15) A • ( B ) R (cid:15) (cid:15) (Λ • gr ( H ( δ )) ∗ ) G Φ / / C • ( B ; R ) , where the first row map Φ is the characteristic map in [11], the second row Φ isthe characteristic map defined byΦ ( ζ ) := X ζ i ( c ( s )) · · · ζ i p ( c ( s )) ,ζ i ( c ( s ))( γ ) := ζ i ( c ( s )( γ )) ( γ ∈ S ( B ))for ζ = P ζ i · · · ζ i p ∈ (Λ p gr ( H ( δ )) ∗ ) G , the first column is the natural projectionand the second column R is the de Rham map. Proof.
Take a base point ∗ of B and put the universal covering of B ˜ B = { γ : [0 , → B ; γ (0) = ∗} / (homotopy preserving boundary) . We identify the fiber E ∗ on ∗ with the typical fiber X .The smooth map µ : ˜ B → Q ( X, δ ) from the universal cover ˜ B of B to the modulispace Q ( X, δ ) := π ( Q • ( X, δ )) of C ∞ -algebra models of X is defined by µ ([ γ ]) = g − γ · [ s ( γ (1))] . Here g γ : E ∗ → E γ (1) is the holonomy along γ . The right-invariant Maurer-Cartanform η ∈ A ( Q ( X, δ ); H (1)0 ( δ )) . is defined by the right-action of QA + ( δ ) on Q ( X, δ ). So we get the flat connection η µ := µ ∗ η ∈ A ( ˜ B ; H (1)0 ( δ )) . On the other hand, we can regard s as the 0-partial section s : sk ( S • ( B )) → Q • ( E ).Its non-abelian obstruction cocycle is described by c ( s )( γ ) = [ s ( γ (0))] − g − γ [ s ( γ (1))] = g l ( µ ([ l ]) − µ ([ γl ])) , where l is a path from ∗ to γ (0) and a path ˜ γ : [0 , → ˜ B is the lift of γ such that˜ γ (0) = [ l ]. The map Ψ : Q ( X, δ ) → Q ( X, δ ) defined by Ψ( α ) = µ ([ l ]) − α satisfiesthe differential equation d Ψ = Ψ η . Thus, solving the equation over the path µ ˜ γ ,we have µ ([ l ]) − µ ([ γl ]) = Ψ( µ ˜ γ (1)) = X Z µ ˜ γ η · · · η. Therefore we get the description using iterated integrals c ( s )( γ ) = g l · X Z µ ˜ γ η · · · η = g l · X Z ˜ γ η µ · · · η µ . Its projection to gr ( H ( δ )) is equal to c ( s )( γ ) = g l R ˜ γ η µ and Z Φ( ξ ) = Z ξ ( η µ , . . . , η µ ) = ¯ ξ (cid:18)Z η µ , . . . , Z η µ (cid:19) = Φ ( ¯ ξ ) ∈ C p ( ˜ B )for ξ ∈ C pCE ( H (1)0 ( δ )) G , where ¯ ξ is the projection of ξ . Since the element is π ( B, ∗ )-invariant, we can regard it as element in C p ( B ). (cid:3) Furthermore if c ( s ) = · · · = c i − ( s ) = 0, we get the (cocycle-level) characteristicmap Φ i : (Λ • gr i ( H ( δ )) ∗ ) G → C • ( B ; R ) defined using c i ( s ) instead of c ( s ). Since η µ ∈ A ( B ; H ( i )0 ( δ )), the commutative diagram similar to Theorem 5.1 exists. Sothe construction above using obstructions is the “leading term” of the characteristicmap obtained in [11] Φ : C • CE ( H (1)0 ( δ )) G → A • ( B ) . Example of surface bundles.
We consider the case of X = Σ g , which is theclosed oriented surface with genus g ≥
2. This is a K¨ahler manifold, so δ = ω ∂∂v is a Chen differential. Here v ∈ W is the fundamental form of Σ g and ω ∈ [ W , W ]is the intersection form, i.e., ω = P gi =1 [ x i , y i ] for a symplectic basis { x i , y i } of W with respect to the intersection form of Σ g .5.4.1. The first obstruction for surface bundles.
For a oriented surface bundle (withsection), its homological structure group is in the symplectic group Sp( W ) of W . Proposition 5.2.
We have the identification as Sp( W )-vector space H ( δ ) ≃ Λ W . Proof.
An element D ∈ Der ( LW ) is described by the form D = D + [ v, z ] ∂∂v for D ∈ Der ( LW ) and z ∈ W . Then we can calculate the image by ad( δ ):[ δ, D ] = − D ( ω ) + [ ω, z ] ∂∂v . So, D is in the kernel if and only if D ( ω ) ∈ ( ω ), where ( ω ) is the Lie ideal in LW generated by ω . This condition is equivalent to the condition: D induces aderivation on LW / ( ω )On the other hand, an element P ∈ Der ( ˆ LW ) is described by P = X b i v ∂∂x i for b i ∈ R , where { x i } gi =1 is a basis of W . Its image of ad( δ ) is[ δ, P ] = X b i ω ∂∂x i − P ( ω ) ∂∂v . Since we can prove [ v, W ] = { P ( ω ); P ∈ Der ( ˆ LW ) } by direct calculus, for any D ∈ Der ( ˆ LW ) , there exists P ∈ Der ( ˆ LW ) such that D P := D + [ δ, P ] ∈ Der ( LW ) . Furthermore for another P ′ ∈ Der ( ˆ LW ) such that D P ′ = D +[ δ, P ′ ] ∈ Der ( LW ),their difference [ δ, P − P ′ ] is in Hom( W , R ω ) ⊂ Der ( LW ). So if D is in the ker-nel, D P and D P ′ induce the same derivation on LW / ( ω ). Therefore we get theisomorphism H ( δ ) ≃ Der ( LW / ( ω )) . According to [17], we have the isomorphism Der ( LW / ( ω )) ≃ Λ W . (cid:3) BSTRUCTION OF C ∞ -ALGEBRA MODELS AND CHARACTERISTIC CLASSES 23 By the proposition above, for a oriented surface bundle E → B with section, weget the obstruction class o (1) = o (1) ( Q ( E, ˆ δ )) ∈ H ( B ; Λ W ( E )) . Here Λ W ( E ) is the local system of vector spaces such that the space on b ∈ B isΛ W ( E ) b = Λ ˜ H ( E b ; R )[ − . This local system is defined in the same way as QA + ( E ) and H ( E ). Then we alsoget the characteristic map(Λ • (Λ W ) ∗ ) Sp( W ) → H • ( B ; R ) . Twisted Morita-Miller-Mumford class.
Let M g, ∗ be the mapping class groupof Σ g fixing a base point. The action of the diffeomorphism group on Q (Σ g , δ )induces the action of M g, ∗ on Q (Σ g , δ ) = π ( Q • (Σ g , δ )).We shall show that the obstruction o (1) of a surface bundle can be regarded asthe 1st twisted Morita-Miller-Mumford class m , = [ τ ] ∈ H ( M g, ∗ ; Λ W ) . Here the cross homomorphism τ : M g, ∗ → Λ W is the 1st Johnson map. This mapis defined as follows: for f ∈ Q (Σ g , δ ), the total Johnson map τ f : M g, ∗ → QA (1) ( δ )is given by the equation f ◦ τ f ( ϕ ) = ϕ · f for ϕ ∈ M g, ∗ . By composing the quotient map QA (1) ( δ ) → gr ( QA ( δ )) = H ( δ ) = Λ W , we get τ : M g, ∗ → Λ W . The map is independent of a choice of f .For any Σ g -bundle E → B , denote the 1st twisted Morita-Miller-Mumford classof E → B by m , ( E ) := ρ ∗ m , ∈ H ( π ( B, b ); Λ W ) ≃ H ( B ; Λ W ( E )) , where ρ : π ( B, b ) → M g, ∗ is the holonomy representation of the surface bundle E → B . Note that π ( B, b ) acts on Λ W through ρ . Theorem 5.3.
The obstruction class of a surface bundle E → B is equal to theminus of the 1st twisted Morita-Miller-Mumford class of E → B : o (1) ( E ) = − m , ( E ) ∈ H ( B ; Λ W ( E )) . Proof.
The following discussion is also used in [12] and Section 4 of [14].Fix a metric of a Σ g -bundle E → B . This metric µ of E → B gives a section s of Π . This defines the obstruction class o ( E ) = [ c ( s )] ∈ H ( B ; QA (1) ( E )) . According to the proof of Theorem 5.1, we have c ( s )( γ ) = X Z ˜ γ η µ · · · η µ = τ µ ( b ) ( ρ ( γ )) − ∈ QA (1) ( E ) b . for a loop γ with a base point b ∈ B . Here ˜ γ, µ, η µ are as in the proof of Theorem5.1. The equation in gr ( QA ( E )) b = gr ( QA ( δ )) ≃ Λ W implies c ( s )( γ ) = − τ µ ( b )1 ( ρ ( γ )) . So we get o (1) ( E ) = − ρ ∗ m , ∈ H ( π ( B, b ); Λ W ). (cid:3) So the obtained characteristic mapΛ • (Λ W ∗ ) Sp( W ) → H • ( B ; R )gives Morita-Miller-Mumford classes of E → B by the result of [13]. References [1] A. Berglund,
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Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama,Meguro-ku, Tokyo 152-8551, Japan.
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