Cohomology of torus manifold bundles
aa r X i v : . [ m a t h . K T ] S e p COHOMOLOGY OF TORUS MANIFOLD BUNDLES
JYOTI DASGUPTA, BIVAS KHAN, AND V. UMA
Abstract.
Let X be a 2 n -dimensional torus manifold with a locally standard T ∼ = (cid:0) S (cid:1) n action whose orbit space is a homology polytope. Smooth complete complex toric varieties andquasitoric manifolds are examples of torus manifolds. Consider a principal T -bundle p : E → B and let π : E ( X ) → B be the associated torus manifold bundle. We give a presentation of thesingular cohomology ring of E ( X ) as a H ∗ ( B )-algebra and the topological K -ring of E ( X ) asa K ∗ ( B )-algebra with generators and relations. These generalize the results in [17] and [19]when the base B = pt . These also extend the results in [20], obtained in the case of a smoothprojective toric variety, to any smooth complete toric variety. Introduction
A torus manifold is a 2 n -dimensional manifold acted upon effectively by an n -dimensional com-pact torus with non-empty fixed point set. Smooth complete complex toric varieties and quasitoricmanifolds are examples of torus manifolds. The notion of torus manifolds was introduced by A.Hattori and M. Masuda in [14]. In [17] M. Masuda and T. Panov studied relationships betweenthe cohomological properties of torus manifolds and the combinatorics of their orbit spaces. Thetopological K -ring of the torus manifolds with locally standard action and orbit space a homologypolotope was described by P. Sankaran in [19].Let p : E → B be a principal bundle with fibre and structure group the complex algebraic torus T ∼ = ( C ∗ ) n over a topological space B . For a smooth projective T -toric variety X , consider thetoric bundle π : E ( X ) → B , where E ( X ) = E × T X , π ([ e, x ]) = p ( e ). In [20], the authors describethe singular cohomology ring of E ( X ) as a H ∗ ( B )-algebra. Furthermore, when B is compactHausdorff, they describe the topological K -ring of E ( X ) as a K ∗ ( B )-algebra.In this paper we consider p : E → B to be a principal bundle with fibre and structure groupthe compact torus T ∼ = (cid:0) S (cid:1) n . We assume that B has the homotopy type of a finite CW complexso that H ∗ ( B ) and K ∗ ( B ) are finitely generated abelian groups. Without loss of generality, wefurther assume that B is compact and Hausdorff. Let X be a 2 n -dimensional torus manifold witha locally standard action of T such that the orbit space Q := X/T is a homology polytope. Wecall the associated bundle π : E ( X ) → B a torus manifold bundle, where E ( X ) = E × T X . InTheorem 3.3 we give a presentation of the singular cohomology ring of E ( X ) as a H ∗ ( B )-algebra.A presentation of the topological K -ring K ∗ ( E ( X )) as a K ∗ ( B )-algebra is obtained in Theorem4.7. As an application, we describe the cohomology ring and K -ring of toric bundles for a smoothcomplete toric variety in Corollary 5.2 extending the results in [20].The method of proof for Theorem 3.3 exploits the known presentation of the cohomology ring[17, Corollary 7 .
8] when the base B is a point. Applying the Leray-Hirsch theorem in cohomologywe first prove that H ∗ ( E ( X )) is a free module over H ∗ ( B ) of rank χ ( X ). Then we constructa surjective H ∗ ( B )-algebra homomorphism from R ( B, ( Q, Λ)) (see Definition 3.1) to H ∗ ( E ( X )).Here Λ denotes the characteristic map of the torus manifold (see Section 2). To verify that thisalgebra homomorphism is injective, we recall from [17] that the equivariant cohomology ring which Mathematics Subject Classification.
Primary 55N15, 57S25.
Key words and phrases. torus manifold bundles, cohomology, K -theory. is isomorphic to the face ring of Q , is a free H ∗ ( BT )-module of rank χ ( X ), where BT denotes theclassifying space of principal T -bundles. We then canonically extend the scalars of the face ring to H ∗ ( B ) and use that it is a finitely generated abelian group to conclude injectivity.Similarly, the method of proof for Theorem 4.7 exploits the known presentation of the topological K -ring [19, Theorem 5 .
3] when the base B is a point. Applying the Leray-Hirsch theorem in K -theory we first prove that K ∗ ( E ( X )) is a free module over K ∗ ( B ) of rank χ ( X ). Then we constructa surjective K ∗ ( B )-algebra homomorphism from R ( B, ( Q, Λ)) (see Definition 4.5) to K ∗ ( E ( X )).Let M := Hom( T, S ) denote the character lattice of T and RT := Z [ χ u : u ∈ M ] the ring offinite dimensional complex representations of T . In Proposition 4.4 we show that the K -theoreticface ring of Q denoted by K ( Q ) (see Definition 4.1) is a free RT -module of rank χ ( X ), usingmethods similar to [24] and [3] in the setting of smooth toric varieties. We then canonically extendthe scalars of K ( Q ) to K ∗ ( B ) and use that it is a finitely generated abelian group to concludeinjectivity. In the case of a smooth complete toric variety, the K -theoretic face ring is in factisomorphic to the algebraic T -equivariant K -ring [24, Theorem 6.4]. The authors believe that thetopological equivariant K -ring of any T -torus manifold is isomorphic to the K -theoretic face ringbut could not find it in literature. We prove this statement for a quasitoric manifold in a parallelwork [7].In Section 6 we consider a torus manifold X with a locally standard action of T such that X/T is not necessarily a homology polytope but only a face-acyclic nice manifold with corners (seeSection 2 for the definition). The equivariant cohomology ring as well as the ordinary cohomologyring of X have been described by Masuda and Panov in [17, Theorem 7.7, Corollary 7.8]. Let E ( X ) −→ B be the bundle with fiber X associated to the principal T -bundle over a topologicalspace B which is of the homotopy type of a finite CW complex. We generalize [17, Corollary7.8] to give a presentation H ∗ ( E ( X )) as a H ∗ ( B )-algebra in Theorem 6.1. Similar to Theorem3.3 we prove this by using the Leray-Hirsch theorem and the known presentation of H ∗ T ( X ) as a H ∗ ( BT )-algebra [17, Theorem 7.7]. We finally conjecture a similar presentation for K ∗ ( E ( X )) asa K ∗ ( B )-algebra. We note that difficulties arise in extending the result to this setting especiallybecause the cohomology ring H ∗ ( X ) is not generated in degree 2. Acknowledgements:
The authors are grateful to Prof. P. Sankaran for drawing our attention tothis problem and for his valuable comments on the initial versions of this manuscript. The first andthe second author thank the Council of Scientific and Industrial Research (CSIR) for their financialsupport. The authors wish to thank the unknown referee for a careful reading of the manuscriptand for very valuable comments and suggestions which led to improving the text. The final sectionhas been added taking into account the referee’s suggestions. The extension of Theorem 3.3 toTheorem 6.1 was also suggested by Prof. M. Masuda in a prior email correspondence. We aregrateful to him for this. 2.
Notation and Preliminaries
We recall some notation and preliminaries from [17] and [19].2.1.
Torus manifolds.
Let T ∼ = (cid:0) S (cid:1) n denote the compact n -dimensional torus. A 2 n -dimensionalclosed connected orientable smooth manifold X with an effective smooth action of T such that thefixed point set X T is non-empty, is called a torus manifold . Since X is compact it follows that X T is finite (see [18, Section 3.4], [5, Section 7.4]). A codimension-two connected submanifold iscalled a characteristic submanifold of X if it is pointwise fixed by a circle subgroup of T . Since X is compact, there are finitely many characteristic submanifolds, which we denote by V , . . . , V d . Itcan be shown that each V i is orientable. We say that X is omnioriented, if an orientation is fixedfor X and for every characteristic submanifold V i . We fix an omniorientation of X . OHOMOLOGY OF TORUS MANIFOLD BUNDLES 3
The T -action on the torus manifold X is said to be locally standard if it has a covering by T -invariant open sets U such that U is weakly equivariantly diffeomorphic to an open subset U ′ ⊂ C n invariant under the standard T -action on C n . The latter means that there is an automorphism θ : T → T and a diffeomorphism g : U → U ′ such that g ( ty ) = θ ( t ) g ( y ) for all t ∈ T , y ∈ U .Let Q := X/T be the orbit space and let Υ : X → Q be the projection map. If X is locallystandard, then Q becomes a nice manifold with corners (see [17, Section 5.1 p. 724] [5, Definition7.1.3]). We denote by Q i the image of V i under Υ for i = 1 , . . . , d ; these are the facets or thecodimension one faces of Q . A codimension- k preface is defined to be a non-empty intersectionof k facets for k = 1 , . . . , n . The connected components of prefaces are called faces . We regard Q itself as a face of codimension zero. We say that Q is face-acyclic if all its faces are acyclici.e. ˜ H i ( F ) = 0 , for every i , for each face F of Q . We say that Q is a homology polytope , if Q isface-acyclic and all its prefaces are faces. This is equivalent to saying that Q is acyclic and all itsprefaces are acyclic (in particular, connected). In this case the intersection of r facets Q i , . . . , Q i r is a codimension r face F of Q . Equivalently non-empty intersections of characteristic submanifoldsare connected submanifolds of X . Unless otherwise specified, we shall assume henceforth that X is a locally standard torus manifold with Q a homology polytope. Note that, H ∗ ( X ) is generatedin degree two if and only if X is locally standard and Q is homology polytope (see [17, Theorem8 . V i , there is a primitive element v i ∈ Hom ( S , T ) ∼ = Z n determined up to sign, whose image is the circle subgroup fixing V i pointwise. The sign of v i isdetermined by the omniorientation. Define the characteristic map Λ : { Q , . . . , Q d } → Hom( S , T ),such that Λ( Q i ) = v i . The local standardness of X implies that the characteristic map Λ satisfiesthe following smooth condition: if Q i ∩ · · · ∩ Q i k is non-empty, then Λ( Q i ) , . . . , Λ( Q i k ) is a partof a basis for the integral lattice Hom( S , T ) ∼ = Z n . Moreover, under our assumption of localstandardness and Q being a homology polytope, the manifold X is determined up to equivariantdiffeomorphisms by the pair ( Q, Λ) (see [17, Lemma 4 . Example 2.1. (1) Let T ∼ = ( C ∗ ) n be the algebraic torus, M = Hom ( T , C ∗ ) ∼ = Hom ( T, S )be the character lattice, and let N = Hom ( M, Z ) be the dual lattice. Consider the smoothcomplete T -toric variety X = X (∆) corresponding to a fan ∆ in N R := N ⊗ Z R ∼ = R n underthe action of the torus T . The orbit space of X under the action of the compact torus T ( ⊂ T ) is the manifold with corners X ≥ , which is formed by gluing ( U σ ) ≥ = Hom sg ( σ ∨ ∩ M, R ≥ )(see [10, Section 4 . ρ ∈ ∆(1), let v ρ ∈ Hom ( S , T ) = N be the primitive raygenerator of ρ . The characteristic submanifolds are given by the divisors D ρ for ρ ∈ ∆(1),these are fixed by the circle subgroups Image( v ρ ). In this case the characteristic map Λ isgiven by sending ( D ρ ) ≥ to v ρ . Since H ∗ ( X ) is generated in degree two by [6, Theorem10 . X ≥ is a homology polytope.(2) Another class of examples are quasitoric manifolds introduced by Davis andJanuszkiewicz in [9]. By definition a quasitoric manifold is locally standard under the T -action and the orbit space is a simple convex polytope and hence a homology polytope. Remark 2.2.
In [22], the author has constructed smooth complete toric varieties of complexdimension ≥ .
1] and [23, Proposition 2 . Lemma 2.3.
Let X be a locally standard torus manifold with orbit space X/T = Q . For each i , ≤ i ≤ d , there exists a T -equivariant complex line bundle L i such that c ( L i ) = [ V i ] ∈ H ( X ) ,where [ V i ] denotes the cohomology class dual to V i and each L i admits an equivariant section s i : X → L i which vanishes precisely along V i . JYOTI DASGUPTA, BIVAS KHAN, AND V. UMA
Proof:
Set V = V i and recall that V is a closed T - invariant codimension 2 submanifold of X .Since T is a compact Lie group we can assume that X is endowed with a T -invariant Riemannianmetric (see [4, Chapter VI, Theorem 2 . ν denote the normal bundle to V in X . We have thedecomposition T ( V ) ⊕ ν = T ( X ) | V . Since V is T -invariant, T ( V ) and T ( X ) | V are T -equivariantvector bundles. Moreover, since the Riemannian metric is also T -invariant, ν = T ( V ) ⊥ ⊆ T ( X ) | V is naturally a T -equivariant real vector bundle. Furthermore, we see that ν is a canonically orientedreal 2-plane bundle since T ( V ) and T ( X ) | V are oriented by the choice of the omniorientation. Thus ν admits a reduction of structure group to SO (2 , R ) ∼ = S giving ν the structure of a complex linebundle. Since T is a connected Lie group and ν is T -equivariant, T preserves the orientation underthe linear action on the fibre. (Fixing an oriented basis for the R -vector space ν x , for every x ∈ V , t ψ t ∈ Hom( ν x , ν tx ) defines a continuous map from T to SO (2 , R ) ⊆ O (2 , R ).) This implies that T preserves the complex structure on the fibre, making ν a T -equivariant complex line bundle.Now (by [4, Chapter VI, Theorem 2 . V has a closed invariant tubular neighbourhood denotedby D which is equivariantly diffeomorphic to the disk bundle associated to the normal bundle ν .The restriction of the equivariant diffeomorphism to the zero section of ν is the inclusion of V in D ⊂ X . We denote by ̟ : D → V the projection map of the disk bundle. The complex line bundle ̟ ∗ ( ν ) admits an equivariant section s : D → ̟ ∗ ( ν ) which vanishes precisely along V . Considerthe trivial complex line bundle E := ( X \ int D ) × C on ( X \ int D ), with the canonical T -action on( X \ int D ) and the trivial T -action on the fibre C . Consider the equivariant bundle isomorphism η : E | ∂D → ̟ ∗ ( ν ) | ∂D given by ( x, λ ) λs ( x ), for all x ∈ ∂D . Now using clutching of bundles (see[16, Theorem 3 . E | ∂D along ̟ ∗ ( ν ) | ∂D using the equivariant identification η we get anequivariant line bundle, say L on X . Note that L admits an equivariant section ˜ s (which restrictsto s on D and x ( x,
1) on ( X \ int D )) that vanishes precisely along V . Hence c ( L ) = [ V ] andthis completes the proof. (cid:3) Remark 2.4.
Let p ′ : ET → BT be the universal principal T -bundle with the associated bundle π ′ : ET × T X → BT . For a T -equivariant line bundle q : L → X , we obtain the line bundle ET × T L on ET × T X with the projection [ e, l ] [ e, q ( l )]. If L has a T -invariant section s whichvanishes precisely along V ⊆ X , we obtain a section ˜ s of ET × T L , defined by [ e, x ] [ e, s ( x )].Thus ˜ s vanishes precisely along ET × T V ⊆ ET × T X . It follows that c T ( L ) = c ( ET × T L ) =[ ET × T V ] := [ V ] T . Remark 2.5.
Note that in Lemma 2.3 we do not assume that Q is a homology polytope or evenface-acyclic. It holds when Q is simply a nice manifold with corners. Remark 2.6.
Throughout this text by H ∗ ( ) we shall always mean cohomology ring with Z -coefficients unless specified otherwise.2.2. Cohomology ring and K -ring of torus manifolds. We now recall the presentation of thecohomology ring and K -ring of torus manifolds from [17] and [19]. Theorem 2.7. ([17, Corollary 7 . , [19, Proposition 5 . Let I be the ideal in Z [ x , . . . , x d ] gen-erated by the elements:(i) x i · · · x i r whenever V i ∩ · · · ∩ V i r = ∅ ,(ii) X ≤ i ≤ d h u, v i i x i where u ∈ Hom ( T, S ) .We have an isomorphism of Z -algebras Z [ x , . . . , x d ] I ∼ → H ∗ ( X ) which maps x i to c ( L i ) = [ V i ] ∈ H ( X ) for ≤ i ≤ d . Furthermore, by [17, Equation (5 . . , H ∗ ( X ) is a free abeliangroup of rank χ ( X ) = | X T | = m . Here m equals the number of vertices of Q . OHOMOLOGY OF TORUS MANIFOLD BUNDLES 5
Theorem 2.8. [19, Theorem 5 . Let J ′ be the ideal in Z [ x , . . . , x d ] generated by the followingelements: (i) x i · · · x i r , whenever V i ∩ · · · ∩ V i r = ∅ , (ii) Y { ≤ i ≤ d : h u,v i i > } (1 − x i ) h u,v i i − Y { ≤ j ≤ d : h u,v j i < } (1 − x j ) −h u,v j i for u ∈ Hom ( T, S ) .We have an isomorphism of Z -algebras Z [ x , . . . , x d ] J ′ ∼ → K ∗ ( X ) which maps x i to − [ L i ] , ≤ i ≤ d .Furthermore, K ∗ ( X ) is a free abelian group of rank equal to χ ( X ) = m ( see [19, Remark 3.2]) . Remark 2.9.
Let J the ideal in Z [ y ± , . . . , y ± d ] generated by the following elements:(i) Y ≤ j ≤ r (cid:0) − y i j (cid:1) , whenever V i ∩ · · · ∩ V i r = ∅ ,(ii) Y ≤ i ≤ d y h u,v i i i for u ∈ Hom(
T, S ).In Theorem 2.8, by making the transformation y i = 1 − x i , 1 ≤ i ≤ d we get the followingalternative presentation Z [ y ± , . . . , y ± d ] J for K ∗ ( X ) which sends y i to [ L i ], 1 ≤ i ≤ d (see [19,Remark 4.2]). 3. Cohomology ring of torus manifold bundles
Let p : E → B be a principal bundle with fibre and structure group the compact torus T over atopological space B . Then one has the associated fibre bundle π : E ( X ) → B with fibre the torusmanifold X , where E ( X ) := E × T X , and π ([ e, x ]) = p ( e ). For u ∈ Hom (
T, S ), let C u denote thecorresponding 1-dimensional T -representation. One has a T -line bundle ξ u on B whose total spaceis E × T C u . For the T -equivariant complex line bundle L i from Lemma 2.3, let E ( L i ) := E × T L i denote the associated line bundle on E ( X ). Definition 3.1.
Let R ( B, ( Q, Λ)) denote the ring H ∗ ( B )[ x , . . . , x d ] I where the ideal I is generatedby the following elements:(i) x i · · · x i r , whenever Q i ∩ · · · ∩ Q i r = ∅ ,(ii) d X i =1 h u, v i i x i − c ( ξ u ) for u ∈ Hom(
T, S ).Recall that the face ring Z [ Q ] of the homology polytope Q is defined to be Z [ x , . . . , x d ] I wherethe ideal I is generated by elements of the form x i · · · x i r , whenever Q i ∩ · · · ∩ Q i r = ∅ . (3.1)For h ( x , . . . , x d ) ∈ Z [ x , . . . , x d ] ⊆ H ∗ ( B )[ x , . . . , x d ] we shall denote by ¯ h ( x , . . . , x d ) its classin Z [ Q ] and by ¯¯ h ( x , . . . , x d ) its class in R ( B, ( Q, Λ)).We have a canonical H ∗ ( BT )-algebra structure on Z [ Q ] given by the ring homomorphism H ∗ ( BT ) → Z [ Q ] which maps u to d X i =1 h u, v i i ¯ x i . A canonical H ∗ ( B )-module structure on H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ] is obtained by extending scalars to H ∗ ( B ) via the homomorphism H ∗ ( BT ) → H ∗ ( B ) thatsends u to c ( ξ u ). Lemma 3.2.
We have an isomorphism R ( B, ( Q, Λ)) ∼ = H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ] of H ∗ ( B ) -modules.In particular, R ( B, ( Q, Λ)) is a free H ∗ ( B ) -module of rank m . JYOTI DASGUPTA, BIVAS KHAN, AND V. UMA
Proof:
Define α : H ∗ ( B )[ x , . . . , x d ] → H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ] by sending x i ⊗ ¯ x i and b b ⊗ b ∈ H ∗ ( B ). Clearly the generators of I listed in ( i ) of Definition 3.1 map to zero under α . Now α ( d X i =1 h u, v i i x i − c ( ξ u )) = 1 ⊗ d X i =1 h u, v i i ¯ x i − c ( ξ u ) ⊗ ⊗ u · − u · ⊗ H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ]. Hence α induces a well defined H ∗ ( B )-module homomor-phism ¯ α : R ( B, ( Q, Λ)) → H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ].We define β : H ∗ ( B ) ⊗ Z Z [ Q ] → R ( B, ( Q, Λ)) by b ⊗ ¯ h ( x , . . . , x d ) b ¯¯ h ( x , . . . , x d ), for b ∈ H ∗ ( B )and ¯ h ( x , . . . , x d ) ∈ Z [ Q ]. Clearly this is well defined. Now β (1 ⊗ u · − u · ⊗
1) = d X i =1 h u, v i i ¯¯ x i − c ( ξ u )which is zero in R ( B, ( Q, Λ)). Hence β induces a map ¯ β : H ∗ ( B ) ⊗ H ∗ ( BT ) Z [ Q ] → R ( B, ( Q, Λ)).Noting that ¯ α and ¯ β are inverses of each other proves the first assertion. Now Z [ Q ] is free H ∗ ( BT )-module of rank m by [17, Theorem 7 .
7, Lemma 2 . (cid:3) The following is the main theorem of this section.
Theorem 3.3.
Let B have the homotopy type of a finite CW complex. The map Φ : R ( B, ( Q, Λ)) → H ∗ ( E ( X )) which sends x i to c ( E ( L i )) is an isomorphism of H ∗ ( B ) -algebras. Proof:
Suppose that Q i ∩ · · · ∩ Q i r = ∅ , which implies V i ∩ · · · ∩ V i r = ∅ . So by Lemma 2.3, thebundle L i ⊕ · · · ⊕ L i r has a nowhere vanishing T -equivariant section. Hence by Remark 2.4, thebundle E ( L i ) ⊕ · · · ⊕ E ( L i r ) admits a nowhere vanishing section. This shows that c ( E ( L i ) · · · c ( E ( L i r )) = 0 (3.2)in H r ( E ( X )). Hence the elements listed in (i) of Definition 3.1 map to zero under Φ.Let L u be the trivial line bundle X × C u on X . Consider the associated line bundle ξ ′ u := ET × T C u on BT . Note that ET × T L u is isomorphic to the pullback π ′∗ ( ξ ′ u ) where π ′ is as inRemark 2.4. By the naturality of Chern classes c T ( L u ) := c ( ET × T L u ) = π ′∗ ( c ( ξ ′ u )) . (3.3)By [17, Proposition 3 . π ′∗ ( c ( ξ ′ u )) = d X i =1 h u, v i i [ V i ] T (3.4)and by Remark 2.4, c T d Y i =1 L h u,v i i i ! = d X i =1 h u, v i i [ V i ] T (3.5)in H T ( X ). Now, (3.3), (3.4) and (3.5) together imply c T ( d Y i =1 L h u,v i i i ) = c T ( L u ). This in turn impliesthat L u ∼ = d Y i =1 L h u,v i i i as T -equivariant line bundles by [11, Theorem C . π ∗ ( ξ u ) ∼ = E ( L u ) ∼ = d Y i =1 E ( L i ) h u,v i i . (3.6) OHOMOLOGY OF TORUS MANIFOLD BUNDLES 7
Taking first Chern classes on both sides of (3.6) we get d X i =1 h u, v i i c ( E ( L i )) = c ( π ∗ ( ξ u )) . (3.7)This implies that the generators of I listed in (ii) of Definition 3.1 map to zero under Φ, hence itis a well-defined ring homomorphism.By Theorem 2.7, there exist p i ( x , . . . , x d ) ∈ Z [ x , . . . , x d ], 1 ≤ i ≤ m such that p i := p i ( c ( L ) , . . . , c ( L d )) : 1 ≤ i ≤ m form a Z -basis of H ∗ ( X ). Consider P i := p i ( c ( E ( L )) , . . . , c ( E ( L d ))) : 1 ≤ i ≤ m in H ∗ ( E ( X )). Since E ( L i ) | X = L i , it follows that P j | X = p j . Since H k ( X ) is free for all k ,by the Leray-Hirsch theorem, (see [12, Theorem 4 D. H ∗ ( E ( X )) is a free H ∗ ( B )-module with P , . . . , P m as a basis. Moreover, since Φ( x i ) = c ( E ( L i )), each P i has a preimage under Φ. Henceby Lemma 3.2, Φ is a surjective H ∗ ( B )-module map between two free H ∗ ( B )-modules of thesame rank. Furthermore, since H ∗ ( B ) is a finitely generated abelian group, it follows that Φ is asurjective map from a finitely generated abelian group to itself, and hence an isomorphism. (Moregenerally, a surjective morphism from a finitely generated module over a Noetherian commutativering to itself is an isomorphism (see [2, Chapter 6, Exercise 1 . ( i )])). (cid:3) K -ring of torus manifold bundles Definition 4.1.
The K -theoretic face ring of the homology polytope Q is defined to be K ( Q ) := Z [ y ± , . . . , y ± d ] J where J is the ideal generated by elements of the form(1 − y i ) · · · (1 − y i r ) , whenever Q i ∩ · · · ∩ Q i r = ∅ . (4.1)We show that K ( Q ) is a free RT -module in Proposition 4.4. We first set up the notation. Recallthat v i ∈ Hom ( S , T ) determines the circle subgroup of T fixing V i for i = 1 , . . . , d . Let V denotethe set of vertices of Q and let a ∈ V . Write a = Q i ∩ · · · ∩ Q i n as an intersection of facets. Thenby [17, Proposition 3 . v i , . . . , v i n form a basis of Hom ( S , T ). We set RT a := Z [ χ ± u i , . . . , χ ± u in ]where u i , . . . , u i n denotes the dual basis of v i , . . . , v i n . For any b ∈ V , denote by a ∨ b theminimal face of Q containing both a and b . If a ∨ b = Q , set RT a ∨ b = Z and the projectionmap RT a → RT a ∨ b to be the augmentation map. Otherwise when a ∨ b is a proper face, write a ∨ b = Q i ∩ · · · ∩ Q i l and set RT a ∨ b := Z [ M h v i , . . . , v i l i ⊥ ] = Z [ χ ± u i , . . . , χ ± u il ] . Then we have the canonical projection map RT a → RT a ∨ b given by χ u ij χ u ij for j = 1 , . . . , l and χ u ij j = l + 1 , . . . , n .The following lemma is analogous to [24, Theorem 6 .
4] in the setting of torus manifolds. Weprove it along similar lines.
Lemma 4.2.
There is an inclusion of rings ¯ φ : K ( Q ) ֒ → Y a ∈V RT a . JYOTI DASGUPTA, BIVAS KHAN, AND V. UMA
The image consists of elements of the form ( r a ) ∈ Y a ∈V RT a , where for any two distinct a, b ∈ V ,the restriction of r a and r b to RT a ∨ b coincide. Proof:
Define the map φ : Z [ y ± , . . . , y ± d ] → Y a ∈V RT a given by y i r i := ( r ia ) where r ia = (cid:26) a / ∈ Q i χ u i if a ∈ Q i Set W = { ( r a ) ∈ Y a ∈V RT a : r a | a ∨ b = r b | a ∨ b , for all a = b ∈ V} . Note that W is a subring of Y a ∈V RT a .Let a, b ∈ V be distinct. If a ∨ b = Q , then there is nothing to prove. Otherwise write a ∨ b = Q i ∩ · · · ∩ Q i l , where a = Q i ∩ · · · ∩ Q i n and b = Q i ∩ · · · ∩ Q i l ∩ Q j l +1 ∩ · · · ∩ Q j n . Nowconsider the following cases:(1) a, b / ∈ Q i : Then r ia = 1 = r ib , hence r a | a ∨ b = r b | a ∨ b .(2) a / ∈ Q i and b ∈ Q i : Then r ia = 1 and r ib = χ u i . Note that under the restriction map RT b → RT a ∨ b , χ u i
1, since u i ∈ h v i , . . . , v i l i ⊥ . Hence we are done in this case.(3) a, b ∈ Q i : Then r ia | a ∨ b = χ u i = r ib | a ∨ b and under the respective projection they map tothe same image since i ∈ { i , . . . , i l } .This proves that r i ∈ W for 1 ≤ i ≤ d . We show that elements of W can be written as Laurentpolynomials in r i ’s. Set V = { a , . . . , a m } and let α = ( α a i ) ∈ W . Let a = Q i ∩ · · · ∩ Q i n , then α a ∈ RT a = Z [ χ ± u i , . . . , χ ± u in ] andhence we can find a Laurent polynomial p ( y i , . . . , y i n ) such that p ( r i , . . . , r i n ) a = α a . Let α := α − p ( r i , . . . , r i n ). Then we see that α a = 0.Now let a = Q i ∩· · ·∩ Q i l ∩ Q j l +1 ∩· · ·∩ Q j n such that a ∨ a = Q i ∩· · ·∩ Q i l . Similarly as abovethere is a Laurent polynomial p ( y i , . . . , y i l , y j l +1 , . . . , y j n ) such that p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a = α a . Note that p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a = p ( χ u i , . . . , χ u il , , . . . , RT a ∨ a remains unchanged, i.e. p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a = p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a | a ∨ a . (4.2)Since α ∈ W , α a | a ∨ a = α a | a ∨ a = 0. Moreover, p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) ∈ W implies p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a | a ∨ a = p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a | a ∨ a = α a | a ∨ a = 0 . Now, (4.2) implies p ( r i , . . . , r i l , r j l +1 , . . . , r j n ) a = 0 . Letting α := α − p ( r i , . . . , r i l , r j l +1 , . . . , r j n ), we have α a = 0 = α a . Repeating this processfor a , . . . , a m , where a k = Q k ∩ · · · ∩ Q k n for k = 1 , . . . , m , we get that α m a = α m a = · · · = α m am = 0, for α m = α − P mk =1 p k ( r k , . . . , r k n ). Thus α m = 0, so that α is in the image of φ .Since α ∈ W was arbitrary, φ is surjective. It remains to show that ker( φ ) = J .For a ∈ V , consider the map φ a : Z [ y ± , . . . , y ± d ] → RT a which sends y i r ia for 1 ≤ i ≤ d . Wesee that ker( φ a ) = J a := h y j − a / ∈ Q j i and clearly ker( φ ) = ∩ a ∈V J a . Then ∩ a ∈V J a = J followsfrom [24, Lemma 6 . φ : K ( Q ) ∼ → W ֒ → Y a ∈V RT a as required. (cid:3) OHOMOLOGY OF TORUS MANIFOLD BUNDLES 9
Note that one has a monomorphism of rings RT ι → K ( Q ) defined by χ u Y ≤ i ≤ d y h u,v i i i , u ∈ Hom(
T, S ) = M , which gives an RT -algebra structure on K ( Q ).Moreover, for every a k = Q k ∩ · · · ∩ Q k n , in V , we have the isomorphism ζ k : Z [ M ] = RT → RT a k which maps χ u n Y j =1 χ h u,v kj i u kj for 1 ≤ k ≤ m . Thus m Y k =1 ζ k identifies ( RT ) m with Y a ∈V RT a = m Y k =1 RT a k . Now, ( RT ) m has a canonical RT -algebra structure via the diagonalembedding δ . Hence ζ = ( m Y k =1 ζ k ) ◦ δ : RT −→ m Y k =1 RT a k which maps χ u ( n Y j =1 χ h u,v kj i u kj ) givesthe canonical RT -algebra structure on Y a ∈V RT a . Corollary 4.3.
The inclusion of rings ¯ φ in Lemma 4.2 is a monomorphism of RT -algebras. Proof:
The proof follows readily since it can be seen that ¯ φ ◦ ι = ζ . (cid:3) Proposition 4.4. K ( Q ) is a free RT -module of rank χ ( X ) . Proof:
We see that K ( Q ) is isomorphic to a localization of Z [ Q ] by a similar argument asin the proof of [3, Theorem 2 . Z [ Q ] ∼ = Z [ y , . . . , y d ] J ∩ Z [ y , . . . , y d ] which sends x i to y i −
1. This remains an isomorphism if we localize at therespective multiplicative systems S I = { ( x i + 1) k } k ∈ N and S J = { y ki } k ∈ N : S − I Z [ Q ] ∼ = S − J Z [ y , . . . , y d ] J ∩ Z [ y , . . . , y d ] = K ( Q )Now Z [ Q ] is Cohen-Macaulay by [17, Lemma 8 . K ( Q ) is also Cohen-Macaulay. Note that K ( Q ) is a finite RT -module since it is a submodule of a Noetherian module Q a ∈V RT a ≃ RT m byLemma 4.2. Hence ι : RT ⊆ K ( Q ) is an integral extension. Since RT is an integrally closed domainand K ( Q ) is a torsion free RT -module, by the Going Down Theorem ([15, Corollary 2 . . M of K ( Q ) which contracts to the maximal ideal m of RT , ht m = ht M . Thenby [3, Lemma 2 . K ( Q ) is a projective RT -module. Moreover, since RT is a Laurent polynimialring, K ( Q ) is in fact a free RT -module. Now note that the presentation of K ∗ ( X ) in [19, Theorem5.3] and Remark 2.9, implies that K ∗ ( X ) ∼ = Z [ y ± , . . . , y ± d ] J ∼ = Z ⊗ RT K ( Q )where the extension of scalars to Z is via the augmentation homomorphism RT ǫ → Z . On the otherhand it is also known that K ∗ ( X ) is a free abelian group of rank χ ( X ). Hence the propositionfollows. (cid:3) Definition 4.5.
Let R ( B, ( Q, Λ)) := K ∗ ( B )[ y ± , . . . , y ± d ] J where the ideal J is generated by thefollowing elements:(i) (1 − y i ) · · · (1 − y i r ), whenever Q i ∩ · · · ∩ Q i r = ∅ ,(ii) Y ≤ i ≤ d y h u,v i i i − [ ξ u ] for u ∈ Hom(
T, S ). Consider the ring K ∗ ( B ) ⊗ RT K ( Q ) obtained from the RT -algebra K ( Q ) by extending scalars to K ∗ ( B ) via the homomorphism RT → K ∗ ( B ) which maps χ u [ ξ u ]. In particular, by Proposition4.4, K ∗ ( B ) ⊗ RT K ( Q ) is a free K ∗ ( B )-module of rank χ ( X ) = m . Lemma 4.6.
We have an isomorphism R ( B, ( Q, Λ)) ∼ = K ∗ ( B ) ⊗ RT K ( Q ) as K ∗ ( B ) -modules. Inparticular, R ( B, ( Q, Λ)) is a free K ∗ ( B ) -module of rank χ ( X ) . The proof is similar to the proof of Lemma 3.2.
Theorem 4.7.
Let B has the homotopy type of a finite CW complex. Then we have an isomor-phism Ψ : R ( B, ( Q, Λ)) ∼ → K ∗ ( E ( X )) of K ∗ ( B ) -modules, which maps y i [ E ( L i )] . Suppose that Q i ∩ · · · ∩ Q i r = ∅ . Recall from the proof of the Theorem 3.3, the bundle E ( L i ) ⊕· · · ⊕ E ( L i r ) admits a nowhere vanishing section. Then applying γ r -operation, we obtain γ r ([ L i ⊕· · ·⊕ L i r ] − r ) = ( − r c r ( L i ⊕· · ·⊕ L i r ) = 0. Also note that γ r ([ L i ⊕ · · · ⊕ L i r ] − r ) = Y ≤ j ≤ r (cid:0) [ L i j ] − (cid:1) .This shows that the elements listed in (i) of Definition 4.5 map to zero under Ψ.Note that, we have π ∗ ( ξ u ) ∼ = E ( L u ) ∼ = d Y i =1 E ( L i ) h u,v i i from the proof of Theorem 3.3. Thisimplies that the generators of J listed in (ii) of Definition 4.5 map to zero under Ψ.The surjectivity of Ψ follows from the same argument as in the proof of Theorem 3.3, using aversion of the Leray-Hirsch theorem in the setting of K-theory (see [13, Theorem 2 . (cid:3) Some applications
As an illustration of the above results, we derive both the cohomology and K -ring of E ( X ),where X = X (∆) is a smooth complete toric variety (see Example 2.1). Definition 5.1.
For a smooth complete fan ∆ we define the following rings.(1) Let R ( H ∗ ( B ) , ∆) denote the ring H ∗ ( B )[ X , . . . , X d ] I where the ideal I is generated bythe following elements:(i) X i · · · X i r , whenever ρ i , . . . , ρ i r do not generate a cone in ∆,(ii) d X i =1 h u, v i i X i − c ( ξ u ) for u ∈ Hom(
T, S ).(2) Let R ( K ∗ ( B ) , ∆) denote the ring K ∗ ( B )[ Y ± , . . . , Y ± d ] J where the ideal J is generatedby the following elements:(i) (1 − Y i ) · · · (1 − Y i r ), whenever ρ i , . . . , ρ i r do not generate a cone in ∆,(ii) Y ≤ i ≤ d Y h u,v i i i − [ ξ u ] for u ∈ Hom(
T, S ). Corollary 5.2.
Let X = X (∆) be a smooth complete T ∼ = ( C ∗ ) n -toric variety. Let p : E → B bea principal T -bundle, where B has the homotopy type of a finite CW complex. (1) The cohomology ring of E ( X ) is isomorphic as an H ∗ ( B ) -algebra to R ( H ∗ ( B ) , ∆) underthe isomorphism Φ : R ( H ∗ ( B ) , ∆) → H ∗ ( E ( X )) which sends X i to c ( E ( L i )) . (2) The topological K -ring of E ( X ) is isomorphic as a K ∗ ( B ) -algebra to R ( K ∗ ( B ) , ∆) underthe isomorphism Ψ : R ( K ∗ ( B ) , ∆) → K ∗ ( E ( X )) which sends Y i to [ E ( L i )] . OHOMOLOGY OF TORUS MANIFOLD BUNDLES 11
Proof:
We consider X as a torus manifold with locally standard T ∼ = ( S ) n action and orbitspace the homology polytope X ≥ (see Example 2.1). We then have the principal T -bundle p ′ : E → E/T since T is an admissible subgroup of T (i.e. T → T /T is a principal T -bundle). Notethat E × T X and E × T X are homotopy equivalent, since T = T × ( R ≥ ) n and ( R ≥ ) n is contractible.Similarly B and E/T are homotopy equivalent. The assertions (1) and (2) of the corollary nowfollow by applying Theorem 3.3 and Theorem 4.7 respectively for the space E × T X associated tothe principal T -bundle p ′ : E → E/T . Note that in the proof of assertion (2) above, Proposition4.4 is immediate from [24, Theorem 6 .
9] because the ring K ( Q ) in Proposition 4.4 is the algebraic T -equivariant K -ring of X . (cid:3) Remark 5.3.
Let X be a torus manifold with locally standard action and orbit space a homologypolytope Q whose nerve is a shellable simplicial complex (see [23]), e.g. quasitoric manifolds. ThenTheorem 3.3 (respectively, Theorem 4.7) can be proved for for B any topological space (respectively, B compact Hausdorff topological space) using [20, Lemma 2 .
1, Lemma 2 . Torus manifold bundles when
X/T is not a homology polytope
In the preceeding sections we considered torus manifolds X defined in Section 2.1 with theadditional assumption that X/T = Q is a homology polytope. This ensured that the cohomologyring H ∗ ( X ) was generated by the degree 2 classes corresponding to the fundamental classes [ V i ] ofthe characteristic submanifolds (see Theorem 2.7).In this section we shall consider a torus manifold X with a locally standard action of T as definedin Section 2.1, with the exception that X/T = Q is not assumed to be a homology polytope butonly face acyclic. In particular, we do not assume that the prefaces are connected. Since Q isface acyclic the cohomology ring of X satisfies the property that H odd ( X ) = 0 (see [5, Theorem7.4.46]), which in particular also implies by the universal coefficient theorem that H ∗ ( X ) is torsionfree and hence free of rank χ ( X ). Moreover, [17, Corollary 7.8] gives an explicit presentation ofthe ring H ∗ ( X ).Let p : E −→ B be a principal T -bundle and let E ( X ) := E × T X be the associated torusmanifold bundle. Let B be a topological space having the homotopy type of a finite CW complex.We then have the following theorem which gives a presentation of H ∗ ( E ( X )) as a H ∗ ( B )-algebra. Theorem 6.1.
Let I be the ideal in the ring R := H ∗ ( B )[ x F : F a face of Q ] generated by thefollowing relations:(i) x G x H − x G ∨ H X E ∈ G ∩ H x E ; (ii) d X i =1 h u, v i i x Q i − c ( ξ u ) for u ∈ Hom ( T, S ) where Q i are the facets of Q , v i = Λ( Q i ) isthe primitive vector in Hom ( S , T ) ≃ Z n which determines the circle subgroup of T fixing thecharacteristic submanifold V i for ≤ i ≤ d , and ξ u = E × T C u is the line bundle on B associatedto the character u ∈ M . Since X is omnioriented v i is well defined. (See Section 2 and Section 3) .The map Φ : R → H ∗ ( E ( X )) which sends x F to [ E ( V F )] defines an isomorphism of H ∗ ( B ) -algebras from R / I → H ∗ ( E ( X )) . Here V F denotes the connected T -stable submanifold Υ − ( F ) of X corresponding to a face F of Q and [ E ( V F )] denotes the P oincar ´ e dual of E ( V F ) := E × T V F in H ∗ ( E ( X )) . Proof:
By [17, Corollary 7.8] it follows that H ∗ ( X ) is a free Z -module of rank χ ( X ) and thatthere exists p , . . . , p m polynomials in Z [ x F : F a face of Q ] such that p i ([ V F ]) for 1 ≤ i ≤ m form a Z -module basis of H ∗ ( X ). Since E ( V F ) | X = V F for each face F of Q , by the Leray-Hirschtheorem P i := p i ([ E ( V F )]) for 1 ≤ i ≤ m form a basis of H ∗ ( E ( X )) as an H ∗ ( B )-module. Recall from (3.6) that we have the isomorphism of line bundles d Y i =1 ( E ( L i )) h u,v i i ≃ π ∗ ( ξ u ) over E ( X ) (see Lemma 2.3, Remark 2.5 for the definition of the line bundles L i on X ). Since c ( E ( L i )) =[ E ( V i )] we see that the relation ( ii ) holds in H ∗ ( E ( X )) by (3.7).Consider the classifying map f : B −→ BT of the principal T -bundle E −→ B . Thus we have themap e f : E ( X ) −→ ET × T X over f since E ( X ) is the pull back of ET × T X under f . This inducesthe canonical maps of cohomology rings e f ∗ : H ∗ T ( X ) −→ H ∗ ( E ( X )) over f ∗ : H ∗ ( BT ) −→ H ∗ ( B )giving a commuting square H ∗ T ( X ) e f ∗ −→ H ∗ ( E ( X )) ↑ π ′∗ ↑ π ∗ H ∗ ( BT ) f ∗ −→ H ∗ ( B )(see Remark 2.4).Furthermore, the submanifold ET × T V F of ET × T X pulls back to the submanifold E ( V F ) of E ( X ) under f . Thus the class τ F := [ ET × T V F ] ∈ H ∗ T ( X ) maps to the class τ ′ F := [ E ( V F )] in thecohomology ring H ∗ ( E ( X )). This in particular implies that the element τ G τ H − τ G ∨ H X E ∈ G ∩ H τ E maps to τ ′ G τ ′ H − τ ′ G ∨ H X E ∈ G ∩ H τ ′ E in H ∗ ( E ( X )). However, by [17, Theorem 7.7], τ G τ H − τ G ∨ H X E ∈ G ∩ H τ E =0 in H ∗ T ( X ). Hence the relation ( i ) holds in H ∗ ( E ( X )). Thus Φ induces a well defined mapfrom R / I −→ H ∗ ( E ( X )). On the other hand [17, Theorem 7.7, Corollary 7.8] imply that asan H ∗ ( BT )-algebra, the ring H ∗ T ( X ) has the presentation R ′ / I ′ , where R ′ := H ∗ ( BT )[ x F : F a face of Q ] and I ′ is the ideal in R ′ generated by the relations ( i ) above and the relations( ii ) ′ d X i =1 h u, v i i x Q i − u for u ∈ Hom ( T, S ) = H ( BT ). This further implies that R / I is isomorphicto the ring H ∗ T ( X ) ⊗ H ∗ ( BT ) H ∗ ( B ) which is a free H ∗ ( B )-module of rank χ ( X ) as in Lemma 3.2above. Here H ∗ ( B ) is a H ∗ ( BT )-module by the map f ∗ which sends u ∈ Hom ( T, S ) to the class c ( ξ u ) ∈ H ∗ ( B ).Since H ∗ ( B ) is a finitely generated abelian group the proof follows by the arguments similar tothe proof of Theorem 3.3. (cid:3) Example 6.2. (see [17, Example 3.2, Example 5.8] and [5, Example 7 . . X = S be the4-sphere identified with the following subset { ( z , z , y ) ∈ C × R : | z | + | z | + | y | = 1 } . Define a T = S × S -action on X given by ( t , t ) · ( z , z , y ) = ( t z , t z , y ). The T action on X is locally standard with X/T homeomorphic to Q = { ( x , x , y ) ∈ R : x + x + y = 1 , x ≥ , x ≥ } . It has 2 characteristic submanifolds z = 0 and z = 0. The intersection of the two characteristicsubmanifolds is disconnected and it is the union of the two T -fixed points (0 , ,
1) and (0 , , − T which fixes { z = 0 } is given by { ( t,
1) : t ∈ S } which corresponds to e ∈ Hom ( S , T ) ∼ = Z . Similarly the circle subgroup fixing { z = 0 } is given by { (1 , t ) : t ∈ S } which corresponds to e ∈ Hom ( S , T ) ∼ = Z . Here e , e are the standard basis of Z . Here theorbit space Q is a 2-ball with two 0-faces denoted by a and b respectively and two 1-faces denotedby G and H respectively. Thus the orbit space is not a homology polytope, but is a face-acyclicmanifold with corners. OHOMOLOGY OF TORUS MANIFOLD BUNDLES 13
Let B = CP and E −→ B denote the principal T -bundle associated to the direct sum of theline bundles O ⊕ O (1) where O denotes the trivial line bundle and O (1) denotes the tautologicalline bundle on CP . Consider the associated S bundle E ( S ) over B . By Theorem 6.1, H ∗ ( E ( S ))has the presentation R / I where R = H ∗ ( CP )[ x G , x H , x a , x b ] with x a and x b are of degree 2 and x G and x H are of degree 4 and I is the ideal in R generated by the following two relations ( i ) x G · x H − x a − x b ; x a x b ( ii ) x G − c ( O ) = x G − x G ; x H − c ( O (1)).6.1. K -ring of a torus manifold bundle. Since H odd ( X ) = 0 the Atiyah Hirzebruch spectralsequence with E p,q = H p ( X ; K q ( pt )) collapses at the E term and converges to K p + q ( X ) (see [1,p. 208 ]). Moreover, since H ∗ ( X ) is free abelian of rank χ ( X ) by [1, p. 209] we have K r ( X ) = 0when r is odd and K r ( X ) ≃ Z m when r is even. Here m = χ ( X ) is also equal to the number ofvertices of Q . In particular, K ( X ) is free abelian of rank m .Let E −→ B be a principal T -bundle and E ( X ) := E × T X the associated bundle over a base B having the homotopy type of a finite CW complex.Let S := K ∗ ( B )[ x F : F a face of Q ] and J denote the ideal in S defined by the followingrelations:( i ) x G x H − x G ∨ H X E ∈ G ∩ H x E ;( ii ) Y i : h u,v i i > (1 − x Q i ) h u,v i i − [ ξ u ] Y i : h u,v i i < (1 − x Q i ) −h u,v i i for u ∈ Hom ( T, S ).We have the following conjecture on K ∗ ( E ( X )) as a K ∗ ( B )-algebra. When B = pt this shallgive a presentation of the K -ring of X which will generalize Sankaran’s result stated in Theorem2.8. For arbitrary B this shall generalize our Theorem 4.7 proved above. Conjecture 6.3.
The ring K ∗ ( E ( X )) is a free K ∗ ( B ) module of rank m = χ ( X ) and is isomorphicto S / J . Remark 6.4.
The difficulty in this case is because the cohomology is not generated in degree 2(see [17, Example 4.10]), we cannot find canonical complex line bundles whose classes generate the K -ring as in [19, Section 3].On the other hand it may be useful to define the analogue of the K -theoretic face ring K ′ ( Q )when Q is a nice manifold with corners so that when Q is a homology polytope it agrees with K ( Q )(see Definition 4.1). One can then check whether K ′ ( Q ) has the structure of a free RT -module ofrank m generalizing the Proposition 4.4 above.Consider the fibration E ( X ) −→ B where X is as above. When B is a path connected, finite-dimensional CW-complex then by [8, Theorem 9 . E p,q = H p ( B, K q ( X )) ⇒ K p + q ( E ( X )). Since K q ( X ) = 0 for q odd this spectral sequencecollapses at the E term. We wonder if this gives enough information to deduce the structure of K ∗ ( E ( X )) as a K ∗ ( B )-module. References [1] ATIYAH, M. F. — HIRZEBRUCH, F. — ADAMS, J. F. — SHEPHERD, G. C.:
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Department of Mathematics, Indian Institute of Technology-Madras, Chennai, India
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