Definable C r vector bundles and bilinear spaces in an o-minimal structure and their homotopy theorems
aa r X i v : . [ m a t h . L O ] F e b DEFINABLE C r VECTOR BUNDLES AND BILINEAR SPACESIN AN O-MINIMAL STRUCTURE AND THEIR HOMOTOPYTHEOREMS
MASATO FUJITA
Abstract.
Consider an o-minimal structure on the real field. Let M be adefinable C r manifold, where r is a nonnegative integer. We first demonstratean equivalence of the category of definable C r vector bundles over M withthe category of finitely generated projective modules over the ring C r df ( M ).Here, the notation C r df ( M ) denotes the ring of definable C r functions on M .We also show an equivalence of the category of definable C r bilinear spacesover M with the category of bilinear spaces over the ring C r df ( M ). The maintheorems of this paper are homotopy theorems for definable C r vector bundlesand definable C r bilinear spaces over M . As an application, we show that theGrothendieck rings K ( C r df ( M )), K ( C ( M )) and the Witt ring W ( C r df ( M ))are all isomorphic. Introduction
O-minimal structure was initially a subject of mathematical logic, but it wasrealized that an o-minimal structure provides an excellent framework of geometry[vdD]. It is considered to be a generalization of semialgebraic geometry [BCR], andgeometric assertions on semialgebraic sets are generalized to the o-minimal case,such as triangulation and trivialization [vdD].In real algebraic geometry, a systematic study of algebraic vector bundles overreal algebraic varieties is given in a series of papers: [BBK, BK1, BK2, BK3,BK4, BK5, BK6, BK7, BK8, BK9, Ku], and a part of the results is summarized in[BCR, Chapter 12]. Semialgebraic vector bundles are also studied in [BCR, CKP].For instance, there is an equivalence of the category of finitely generated projectmodules over the ring of regular functions on an affine real algebraic variety withalgebraic vector bundles over the variety by [BCR, Proposition 12.1.12]. We alsofind a similar equivalence for semialgebraic objects in [BCR, Corollary 12.7.6]. Itis natural to generalize these studies to the o-minimal case. A study on definablefiber bundles was already done in [Ka2], but we cannot find an assertion on anequivalence of the category of definable C r vector bundles with the category offinitely generated projective modules over the ring of definable C r functions.We fix an o-minimal structure on the real field throughout this paper. Theterm ‘definable’ means ‘definable in the o-minimal structure’ in this paper. Let M be a definable C r manifold, where r is a nonnegative integer. Note that alldefinable C r manifolds are affine by [Ka1, Theorem 1.1] and [F, Theorem 1.3].We use this fact without explicitly stated in this paper. The notation C r df ( M )denotes the ring of definable C r functions on M . We investigate several categories Mathematics Subject Classification.
Primary 03C64; Secondary 57R22, 19A49.
Key words and phrases. o-minimal structure, vector bundle, Grothendieck ring, Witt ring. in this paper. The first category is VB r df ( M ); the objects are definable C r vectorbundles over M , and the arrows are definable C r M -morphisms between definable C r vector bundles. The second category is Proj ( R ), where R is a commutativering. The objects are finitely generated projective R -modules, and the arrows arehomomorphisms between R -modules.For any definable C r vector bundle ξ over a definable C r manifold M , Γ( ξ )denotes the set of all definable C r sections of ξ . It is a finitely generated projective C r df ( M )-module. The notation Γ( ϕ ) is the induced homomorphism from Γ( ξ ) toΓ( ξ ′ ) for any definable C r M -morphism ϕ : ξ → ξ ′ . Γ is a covariant functor from VB r df ( M ) to Proj ( C r df ( M )). The following is the first main theorem in this paper: Main Theorem . Let M be a definable C r manifold, where r is a nonnegativeinteger. Then, the functor Γ is an equivalence of the category VB r df ( M ) with thecategory Proj ( C r df ( M )).We define a definable C r bilinear space over a definable C r manifold in Section5. A bilinear space over a commutative ring is defined in [BCR, MH]. We showan equivalence of bilinear spaces. Let M be a definable C r manifold, where r is a nonnegative integer. The notation BS r df ( M ) is the category whose objectsare definable C r bilinear spaces over M and whose arrows are definable C r M -morphisms between definable C r bilinear spaces. We also consider the category BS ( R ), where R is a commutative ring. The objects are bilinear spaces over thering R and the arrows are morphisms of bilinear spaces over R . We also show thatthese categories are equivalent. Main Theorem . Let M be a definable C r manifold, where r is a nonnegativeinteger. The category BS r df ( M ) is equivalent to the category BS ( C r df ( M )).The homotopy theorem for vector bundles is well-known and, for instance, it isfound in [H]. We extend the theorem to the general o-minimal and C r case. Main Theorem C r vector bundles) . Considera definable C r manifold M , where r is a nonnegative integer. Let U be a definableopen subset of M × R containing M × [0 , C r vector bundleover U . Then, two definable C r vector bundles Ξ | M ×{ } and Ξ | M ×{ } are definably C r isomorphic. Here, the notation Ξ | M ×{ } denotes the restriction of the vectorbundle Ξ to M × { } .We can also prove a homotopy theorem for bilinear spaces. It is a generalizationof the homotopy theorem for semialgebraic C bilinear spaces in [BCR, Corollary15.1.9]. Main Theorem C r bilinear spaces) . Considera definable C r manifold M , where r is a nonnegative integer. Let U be a defin-able open subset of M × R containing M × [0 , , B ) be a definable C r bilinear space over U . Then, two definable C r bilinear spaces (Ξ , B ) | M ×{ } and(Ξ , B ) | M ×{ } are definably C r isometric. Here, the notation (Ξ , B ) | M ×{ } denotesthe restriction of the bilinear space (Ξ , B ) to M × { } .We show isomorphisms between Grothendieck rings and Witt rings as an appli-cation of the above theorems. The Grothendieck group of the ring is studied inK-theory [Miln, W]. It is defined as an abelian group generated by finitely gener-ated projective modules modulo some equivalence relation. The multiplication in EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 3 the abelian group is defined as the tensor product of two representative projectivemodules. The Witt ring [MH] of the ring of semialgebraic functions is isomorphicto its Grothendieck ring by [BCR, Theorem 15.1.2]. A similar assertion holds truefor the ring of definable C r functions. Main Theorem . Let M be a definable C r manifold, where r is a nonnegativeinteger. The Grothendieck ring K ( C r df ( M )) of the ring C r df ( M ) is isomorphic tothe Witt ring W ( C r df ( M )) of the same ring. The Grothendieck rings K ( C r df ( M ))and K ( C ( M )) are also isomorphic.This paper is organized as follows: In Section 2, we first introduce several basiclemmas used through the remaining sections. We show the equivalence of the cat-egory VB r df ( M ) with the category Proj ( C r df ( M )) in Section 3. An approximationtheorem for sections of definable C r vector bundles and its applications are intro-duced in Section 4. We demonstrate that the category BS r df ( M ) is equivalent tothe category BS ( C r df ( M )) in Section 5. The homotopy theorems for definable C r vector bundles and bilinear spaces are proved in Section 6. As an application ofthese results, we demonstrate that the Grothendieck ring of C r df ( M ) and the Wittring of C r df ( M ) are isomorphic in Section 7.We summarize basic notations used in this paper. The notation V denotes theclosure of a subset V of a topological space. The complement of the set V is denotedby V c in this paper. The transpose of a matrix X is denoted by t X . The notation[0 ,
1] denotes the closed interval { x ∈ R | ≤ x ≤ } in R .2. Preliminary
We provide several basic lemmas in this section. We first review the definitionof definable C r manifolds. Definition 2.1 (Definable manifolds) . Let r be a nonnegative integer. A definable C r manifold M of dimension n is a C r manifold having finite charts { ( U i , φ i ) } qi =1 such that, for any 1 ≤ i, j ≤ q , • U i is an open subset of M , • φ i : U i → V i is a homeomorphism, where V i is an definable open subset of R n , and, • φ i | U i ∩ U j ◦ ( φ j | φ j ( U i ∩ U j ) ) − : φ j ( U i ∩ U j ) → φ i ( U i ∩ U j ) is a definable C r diffeomorphism if U i ∩ U j = ∅ .Charts satisfying the above conditions are called definable C r charts of M . A C r map f : M → M ′ between definable C r manifolds is a definable C r map if, for anydefinable C r charts { ( U i , φ i : U i → V i ) } qi =1 of M and definable C r charts { ( U ′ i , φ ′ i : U ′ i → V ′ i ) } q ′ i =1 of M ′ , the map ( φ ′ j ) − ◦ f | U i ∩ f − ( U ′ j ) ◦ φ i : φ − i ( U i ∩ f − ( U ′ j )) → V ′ j is a definable C r map for any 1 ≤ i ≤ q and 1 ≤ j ≤ q ′ . An affine definable C r manifold M of dimension n is a definable subset of a Euclidean space R m suchthat there exist a finite definable open covering { U i } qi =1 of M in R m and a familyof definable C r diffeomorphisms onto definable open sets { φ i : U i → Ω i ⊂ R m } qi =1 with M ∩ U i = φ − i (( R n ×{ } ) ∩ Ω i ) for any 1 ≤ i ≤ q . We call the collection of pairs( U i , φ i ) qi =1 charts of the affine definable C r manifold. If a definable C r manifold M has a definable C r immersion ι : M → R m , we call it an affine definable C r manifoldidentifying it with its image. Note that all definable C r manifolds are affine by [Ka1, M. FUJITA
Theorem 1.1] and [F, Theorem 1.3]. We use this fact without explicitly stated inthis paper.
Lemma 2.2.
Let M be a definable C r manifold with ≤ r < ∞ . Given a definableclosed subset X of M , there exists a definable C r function f : M → R whose zeroset is X .Proof. We assume that M is a definable C r submanifold of a Euclidean space R m . Consider the closure X of X in R m . There exists a definable C r function F : R m → R with F − (0) = X by [vdDM, Theorem C.11]. The restriction of F to M satisfies the requirement. (cid:3) Lemma 2.3.
Let M be a definable C r manifold with ≤ r < ∞ . Let X and Y beclosed definable subsets of M with X ∩ Y = ∅ . Then, there exists a definable C r function f : M → [0 , with f − (0) = X and f − (1) = Y .Proof. There exist definable C r functions g, h : M → R with g − (0) = X and h − (0) = Y by Lemma 2.2. The function f : M → [0 ,
1] defined by f ( x ) = g ( x ) g ( x ) + h ( x ) satisfies the requirement. (cid:3) Lemma 2.4.
Let M be a definable C r manifold with ≤ r < ∞ . Let C and U beclosed and open definable subsets of M , respectively. Assume that C is containedin U . Then, there exists an open definable subset V of M with C ⊂ V ⊂ V ⊂ U .Proof. There is a definable continuous function h : M → [0 ,
1] with h − (0) = C and h − (1) = M \ U by Lemma 2.3. The set V = { x ∈ M ; h ( x ) < } satisfies therequirement. (cid:3) Lemma 2.5 (Fine definable open covering) . Let M be a definable C r manifoldwith ≤ r < ∞ . Let { U i } qi =1 be a finite definable open covering of M . For each ≤ i ≤ q , there exists a definable open subset V i of M satisfying the followingconditions: • the closure V i in M is contained in U i for each ≤ i ≤ q , and • the collection { V i } qi =1 is again a finite definable open covering of M .Proof. We inductively construct V i so that V i ⊂ U i and { V i } k − i =1 ∪{ U i } qi = k is a finitedefinable open covering of M . We fix a positive integer k with k ≤ q . Set C k = M \ ( S k − i =1 V i ∪ S qi = k +1 U i ). The set C k is a definable closed subset of M containedin U k . There exists a definable open subset V k of M with C k ⊂ V k ⊂ V k ⊂ U k by Lemma 2.4. It is obvious that { V i } ki =1 ∪ { U i } qi = k +1 is a finite definable opencovering of M . (cid:3) Lemma 2.6 (Partition of unity) . Let M be a definable C r manifold. Given afinite open definable covering { U i } qi =1 of M , there exist nonnegative definable C r functions λ i on M for all ≤ i ≤ q such that P qi =1 λ i = 1 and the closure of theset { x ∈ M | λ i ( x ) > } is contained in U i .Proof. Let { V i } qi =1 be a finite definable open covering given in Lemma 2.5. Thereexists a C r definable function f i on M with f − i (0) = M \ V i by Lemma 2.2. Set λ i = f i / P qj =1 f j . The definable C r functions λ i on M satisfy the requirements. (cid:3) The semialgebraic counterpart to the following lemma is found in
EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 5
Lemma 2.7.
Let X be a definable subset of R n and { V j } pj =1 be a finite definableopen covering of X × [0 , . Then, there exist a finite definable open covering { U i } qi =1 of X and finite definable C r functions ϕ i, < · · · < ϕ i,k < · · · < ϕ i,r i = 1 on U i such that, for any ≤ i ≤ q and ≤ k ≤ r i , the definable set { ( x, t ) ∈ U i × [0 , | ϕ i,k − ( x ) ≤ t ≤ ϕ i,k ( x ) } is contained in V j for some ≤ j ≤ p .Proof. Let π : R n × R → R n be the projection onto the first n coordinates. Applythe C r cell decomposition theorem [vdD, Chapter 7, Theorem 3.2 and Chapter 7,Exercise 3.3] to R n × R , then we have a definable C r cell decomposition { C i } si =1 partitioning X × [0 ,
1] and V , . . . , V p . Let { D i } qi =1 be the subfamily of cells in { π ( C i ) } si =1 contained in X . There exist definable C r functions ψ i, = −∞ <ψ i, < · · · < ψ i,l i < ψ i,l i +1 = ∞ on D i such that the cells contained in X × R areone of the following forms: { ( x, t ) ∈ D i × R | t = ψ i,k ( x ) } , { ( x, t ) ∈ D i × R | ψ i,k − ( x ) < t < ψ i,k ( x ) } .Since the cell decomposition partitions X × [0 , k and k ′ with ψ i,k = 0 and ψ i,k ′ = 1. Taking a subsequence, we may assume that 0 = ψ i, <ψ i, < · · · < ψ i,l i = 1 on D i . Set r i = 2 l i and define definable C r functions Ψ i,k on D i by Ψ i,k ( x ) = (cid:26) ψ i,k/ ( x ) if k is even, ψ i, ( k − / ( x )+ ψ i, ( k +1) / ( x )2 otherwise.We first show the following claim: Claim.
For any 1 ≤ i ≤ q and 1 ≤ k ≤ r i , there exists a positive integer j ( i, k )such that the definable set { ( x, t ) ∈ D i × [0 , | Ψ i,k − ( x ) ≤ t ≤ Ψ i,k ( x ) } is contained in V j ( i,k ) .We demonstrate the above claim. One of k − k is an even number. Weassume that k − { V j } pj =1 is an open covering and the decomposition partitions V , . . . V p ,there exists 1 ≤ j ( i, k ) ≤ p such that the set { ( x, t ) ∈ D i × [0 , | t = Ψ i,k − ( x ) = ψ i, ( k − / ( x ) } is contained in V j ( i,k ) . The set { ( x, t ) ∈ D i × R | ψ i, ( k − / ( x ) < t <ψ i, ( k +1) / ( x ) } is contained in V j ( i,k ) because V j ( i,k ) is open and the decompositionpartitions V j ( i,k ) . Hence, the definable set { ( x, t ) ∈ D i × [0 , | Ψ i,k − ( x ) ≤ t ≤ Ψ i,k ( x ) } is contained in V j ( i,k ) . We have demonstrated the claim.Let π l : R n → R l be the projection onto the first l coordinates. We inductivelydefine definable open subsets W i,l of R l and definable C r maps η i,l : W i,l → π l ( D i )as follows: When l = 1, the definable set π ( D i ) is a single point set { a } or an openinterval I . Set W i, = R and η i, ( x ) = a if π ( D i ) is a single point set { a } . Set W i, = I and η i, ( x ) = x if π ( D i ) is an open interval I . When l >
1, the definableset π l ( D i ) is one of the following forms: { ( x, t ) ∈ π l − ( D i ) × R | t = f ( x ) } , { ( x, t ) ∈ π l − ( D i ) × R | f ( x ) < t < f ( x ) } . M. FUJITA
Here, f, f and f are definable C r functions on π l − ( D i ). Set W i,l = W i,l − × R and η i,l ( x, t ) = ( η i,l − ( x ) , f ( η i,l − ( x ))) in the former case. Set W i,l = { ( x, t ) ∈ W i,l − × R | f ( η i,l − ( x )) < t < f ( η i,l − ( x )) } and η i,l ( x, t ) = ( η i,l − ( x ) , t ) in thelatter case.Set W i = W i,n and η i = η i,n . It is obvious that D i is contained in W i and therestriction of η i to D i is the identity map. For any 1 ≤ i ≤ q and 1 ≤ k ≤ r i , wedefine a definable set X i,k as follows: X i,k = { x ∈ W i | ( x, t ) ∈ V j ( i,k ) for all t ∈ R with Ψ i,k − ( η i ( x )) ≤ t ≤ Ψ i,k ( η i ( x )) } .It is obvious that D i is contained in X i,k by the above claim. We show that X i,k is an open set. Let x ∈ X i,k be fixed. Consider the closed definable subset Y i,k ( x ) = { t ∈ R | Ψ i,k − ( η i ( x )) ≤ t ≤ Ψ i,k ( η i ( x )) } of R . We also set Z i,k ( x ) = { ( x, t ) ∈ X i,k × R | Ψ i,k − ( η i ( x )) ≤ t ≤ Ψ i,k ( η i ( x )) } .The definable continuous function ρ on the closed definable set Y i,k ( x ) is definedas the distance between the point ( x, t ) and the closed set V cj ( i,k ) . Since Y i,k ( x ) iscompact, the function ρ takes the minimum m . The minimum m is positive becausethe intersection of Z i,k ( x ) with V cj ( i,k ) is empty. Take y ∈ W i sufficiently close to x satisfying the following conditions: • k y − x k n < m , where k · k n denotes the Euclidean norm in R n , • Ψ i,k − ( η i ( y )) < Ψ i,k ( η i ( x )), • Ψ i,k ( η i ( y )) > Ψ i,k − ( η i ( x )), • | Ψ i,k − ( η i ( y )) − Ψ i,k − ( η i ( x )) | < m and • | Ψ i,k ( η i ( y )) − Ψ i,k ( η i ( x )) | < m .We lead a contradiction assuming that y X i,k . There exists a real number t with Ψ i,k − ( η i ( y )) ≤ t ≤ Ψ i,k ( η i ( y )) and ( y, t ) V j ( i,k ) by the assumption. If t ∈ Y i,k ( x ), the distance between the point ( y, t ) and Z i,k ( x ) is k y − x k n and lessthan m . It is a contradiction to the assumption that ( y, t ) V j ( i,k ) . In the othercase, we have | t − Ψ i,k − ( η i ( x )) | ≤ | Ψ i,k − ( η i ( y )) − Ψ i,k − ( η i ( x )) | < m | t − Ψ i,k ( η i ( x )) | ≤ | Ψ i,k ( η i ( y )) − Ψ i,k ( η i ( x )) | < m y, t ) and Z i,k ( x ) is less than q(cid:0) m (cid:1) + (cid:0) m (cid:1) . Itcontradicts to the definition of m . We have demonstrated that X i,k is open.Set U i = T r i k =1 X i,k and ϕ i,k = Ψ i,k ◦ η i | U i for 1 ≤ i ≤ q . The set U i is adefinable open set and ϕ i,k is a definable C r function on U i . The set { ( x, t ) ∈ U i × [0 , | ϕ i,k − ( x ) ≤ t ≤ ϕ i,k ( x ) } is contained in V j ( i,k ) by the definition of X i,k .Since X = S qi =1 D i and D i ⊂ U i , we have X ⊂ S qi =1 U i . (cid:3) Equivalence of definable C r vector bundles with projectivemodules over the ring of definable C r functions We define definable C r vector bundles. The definition is lengthy, but it is almostthe same as the definition of semialgebraic vector bundles given in [BCR, Definition12.7.1]. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 7
Definition 3.1 (Definable C r vector bundle) . Let ξ = ( E, p, M ) be an R -vectorbundle of rank d over a definable C r manifold M . A family of local trivializations( U i , ϕ i : U i × R d → p − ( U i )) i ∈ I of the vector bundle ξ is a definable C r atlas of ξ if ( U i ) i ∈ I is a finite open definable covering of M and, for all pairs ( i, j ) ∈ I × I ,the maps ϕ − i ◦ ϕ j | ( U i ∩ U j ) × R d are definable C r maps. Two definable C r atlases areequivalent if their union is still a definable C r atlas. A definable C r vector bundle isa vector bundle equipped with an equivalence class of definable C r atlases. By abuseof notation, we denote a definable C r vector bundle by ξ = ( E, p, M ) specifyingthe total space, the base space and the map between them, but without specifyingthe definable C r atlas defining its structure. The trivial bundle of rank d over M is the triple ( M × R d , p, M ), where p is the natural projection from M × R d onto M . The notation ǫ dM denotes the trivial bundle of rank d over M . A section s ofthe vector bundle ξ is a definable C r section if the maps ϕ − i ◦ s | U i : U i → U i × R d are definable C r maps for all i ∈ I .Let ξ = ( E, p, M ) and ξ ′ = ( E ′ , p ′ , M ′ ) be two definable C r vector bundles.Let ( U i , φ i ) i ∈ I and ( U ′ j , φ ′ j ) j ∈ J be definable C r atlases of ξ and ξ ′ , respectively. Amorphism between vector bundles ψ = ( u, f ) : ξ → ξ ′ is a definable C r morphism if( φ ′ j ) − ◦ u ◦ φ i | ( U i ∩ U ′ j ) × R d are definable C r maps for all pairs ( i, j ) ∈ I × J , and themap f : M → M ′ between base spaces are also definable and of class C r . Here, themap u is a map between total spaces. When two definable C r manifolds M and M ′ are identical and the map between base spaces is the identity map on M , we call it a definable C r M -morphism . In this case, we use the same symbol for the morphismand the map between total spaces by abuse of notation. Two definable C r vectorbundles ξ and ξ ′ over a definable C r manifold M are definably C r isomorphic ifthere exist two definable C r M -morphism ψ : ξ → ξ ′ and ψ ′ : ξ ′ → ξ such that thecompositions of two morphisms ψ ◦ ψ ′ and ψ ′ ◦ ψ are the identity morphisms. Wecall ψ and ψ ′ definable C r isomorphisms . Remark . The notation GL( d, R ) denotes the general linear group of degree d with entries in R . Let ξ = ( E, p, M ) be a definable C r vector bundle of rank d overa definable C r manifold M . Let ( U i , φ i : U i × R d → p − ( U i )) qi =1 be a definable C r atlas of ξ . The map φ i | ( U i ∩ U j ) × R d ◦ (cid:0) φ j | ( U i ∩ U j ) × R d (cid:1) − : ( U i ∩ U j ) × R d → ( U i ∩ U j ) × R d induces a collection of definable C r maps { g ij : U i ∩ U j → GL( d, R ) } satisfying g ii = id and g ij ( x ) g jk ( x ) = g ik ( x ) for any x ∈ U i ∩ U j ∩ U k . The function g ij : U i ∩ U j → GL( d, R ) is called a transition function of ξ .On the other hand, we can construct a definable C r vector bundle of rank d inthe same way as the case of general topological vector bundles [H, Section 4.2] ifa finite definable open covering { U i } qi =1 and a family of definable C r map { g ij : U i ∩ U j → GL( d, R ) } are given and satisfy the above conditions.The proof of the following proposition is straightforward and we omit it. Proposition 3.3.
Let r be a nonnegative integer. Let M and N be definable C r manifolds.(i) If ξ is a definable C r vector bundle over M , and f : N → M is a definable C r map, then f ∗ ( ξ ) is a definable C r vector bundle over N .(ii) If ξ and η are definable C r vector bundles over M , the Whitney sum ξ ⊕ η ,the tensor product ξ ⊗ η , the dual ξ ∨ and Hom( ξ, η ) are definable C r vectorbundles over M . M. FUJITA
The following two lemmas are important and used several times in this paper.
Lemma 3.4.
Given a definable C r vector bundle ξ = ( E, p, M ) of rank d , thereexist a finite number of definable C r sections s , . . . , s m of ξ such that, for any x ∈ M , the vectors s ( x ) , . . . , s m ( x ) generate the fiber ξ x = p − ( x ) .Proof. Let ( U i , ϕ i : U i × R d → p − ( U i )) qi =1 be a definable C r atlas of ξ and { λ i } qi =1 be a definable C r partition of unity subordinate to the covering { U i } qi =1 given inLemma 2.6. For any 1 ≤ i ≤ q and 1 ≤ j ≤ d , define a definable C r section s i,j by s i,j ( x ) = ϕ i ( x, λ i ( x ) e j ) for any x ∈ U i and s i,j ( x ) = ϕ k ( x,
0) for any x ∈ U k \ U i for all k = i , where e j is the j -th vector of the canonical basis of R d . The sections s i,j obviously generate the vector space p − ( x ) for any x ∈ M . (cid:3) Lemma 3.5.
Let ξ = ( E, p, M ) be a definable C r vector bundle of rank d . Considera finite number of definable C r sections s , . . . , s m of ξ such that, for any x ∈ M ,the vectors s ( x ) , . . . , s m ( x ) generate the fiber ξ x = p − ( x ) . Then, for any definable C r section s of ξ , there exist definable C r functions c , . . . , c m on M with s = P mj =1 c j s j .Proof. Let ( U i , ϕ i : U i × R d → p − ( U i )) qi =1 be a definable C r atlas of ξ . Fix1 ≤ i ≤ q . For any subset J ⊂ { , . . . , m } of cardinality d , set U i,J = { x ∈ U i | the fiber p − ( x ) is generated by { s j ( x ) | j ∈ J }} .The collection { U i,J } is a finite definable open covering of M . The fiber ξ x = p − ( x )is generated by { s j ( x ) } j ∈ J for any x ∈ U i,J . Taking a finer finite definable opencovering, we may assume that, ( U i , ϕ i : U i × R d → p − ( U i )) qi =1 is a definable C r atlas and, for any 1 ≤ i ≤ q , there exists a subset J ( i ) ⊂ { , . . . , m } suchthat the fiber ξ x is generated by { s j ( x ) } j ∈ J ( i ) for any x ∈ U i . Let j i ( k ) be the k -th smallest element of the subset J ( i ) ⊂ { , . . . , m } . The map τ i : ǫ dU i → ξ | U i defined by τ i ( x, ( a , . . . , a d )) = P dk =1 a k s j i ( k ) ( x ) is a definable C r isomorphism. LetΠ i : U i × R d → R d be the natural projection. Let π k : R d → R be the projectiononto the k -th coordinate for any 1 ≤ k ≤ d . We have s ( x ) = d X k =1 π k (Π i ( τ − i ( s ( x )))) s j i ( k ) ( x )for any x ∈ U i . Define definable C r functions b i,j on U i as follows: b i,j = (cid:26) π k (Π i ( τ − i ( s ( x )))) if j = j i ( k ) for some 1 ≤ k ≤ d ,0 otherwise.For any 1 ≤ i ≤ q , the collection of definable functions { b ij } mj =1 satisfies the follow-ing equality: s | U i = m X j =1 b i,j s j | U i .Let { λ i } qi =1 be a definable C r partition of unity subordinate to the definableopen covering { U i } qi =1 given by Lemma 2.6. For any 1 ≤ i ≤ q and 1 ≤ k ≤ m , thefunction c i,k : M → R is the definable C r function defined by c i,k ( x ) = (cid:26) λ i ( x ) b i,k ( x ) if x ∈ U i ,0 elsewhere. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 9
The equality s = P qi =1 ( P mk =1 c i,k s k ) follows from the following calculation: q X i =1 m X k =1 c i,k ( x ) s k ( x ) ! = X ≤ i ≤ q,x ∈ U i λ i ( x ) m X k =1 b i,k ( x ) s k ( x ) ! = X ≤ i ≤ q,x ∈ U i λ i ( x ) s ( x ) = q X i =1 λ i ( x ) s ( x )= s ( x )Set c j = P qi =1 c i,j for any 1 ≤ j ≤ m . We have s = P mj =1 c j s j . (cid:3) The following lemma is similar to [BCR, Theorem 12.1.7, Corollary 12.7.5]. Theproof is also similar.
Lemma 3.6.
Let ξ = ( E, p, M ) be a definable C r vector bundle of rank d . Then: (i) There exists an injective definable C r morphism from ξ into a trivial bundle ǫ nM . (ii) There exists another definable C r vector bundle ξ ′ over M such that the Whit-ney sum ξ ⊕ ξ ′ is definably C r isomorphic to a trivial bundle ǫ nM .Proof. We first show that the assertion (ii) follows from the assertion (i). Wefirst show that there exists a definable C r map f : M → Gr ( n, d ) such that ξ isdefinably C r isomorphic to f ∗ ( γ n,d ). Here, Gr ( n, d ) denotes the Grassmanian of d -dimensional subspaces of an n -dimensional vector space, and γ n,d denotes theuniversal vector bundle over Gr ( n, d ). Let ψ : ξ → ǫ nM be an injective definable C r morphism. Define f : M → Gr ( n, k ) by { x } × f ( x ) = ψ ( p − ( x )). It sufficesto prove that f is a definable C r map. Let ( U i , ϕ i : ǫ dU i → ξ | U i ) qi =1 be a definable C r atlas of ξ . Define η ij : U i → R d by ( x, η ij ( x )) = ψ ( ϕ i ( x, e j )) for any 1 ≤ i ≤ q and 1 ≤ j ≤ d , where e j is the j -th vector of the canonical basis of R d . It is adefinable C r map. Since f ( x ) is generated by η ij ( x ) for any x ∈ U i , the map f isalso a definable C r map.Let γ ⊥ n,d denote the vector bundle orthogonal to γ n,d . We have f ∗ ( γ n,d ) ⊕ f ∗ ( γ ⊥ n,k ) ≃ ǫ M because γ n,d ⊕ γ ⊥ n,d ≃ ǫ n Gr ( n,d ) . The vector bundle ξ ′ = f ∗ ( γ ⊥ n,d )is a definable C r vector bundle satisfying the required condition.We next show the assertion (i). Let s , . . . , s n be the sections of ξ given inLemma 3.4. The definable C r morphism ψ : ǫ nM → ξ defined by ψ ( x, a , . . . , a n ) = x, n X i =1 a i s i ( x ) ! is surjective. Taking the dual, we obtain an injective definable C r morphism ψ ∨ : ξ ∨ ֒ → ( ǫ nM ) ∨ ≃ ǫ nM . We have already shown that the condition (i) implies thecondition (ii). Hence, there exists a definable C r vector bundle ξ ′ over M suchthat the Whitney sum ξ ∨ ⊕ ξ ′ is isomorphic to a trivial bundle ǫ nM . We have ξ ⊕ ( ξ ′ ) ∨ ≃ ( ξ ∨ ) ∨ ⊕ ( ξ ′ ) ∨ ≃ ( ξ ∨ ⊕ ξ ′ ) ∨ ≃ ( ǫ nM ) ∨ ≃ ǫ nM . In particular, there existsan injective definable C r morphism from ξ into a trivial bundle ǫ nM . (cid:3) We begin to prove Main Theorem 1.1. We first review the notations introducedin Section 1.
Notation . Let M be a definable C r manifold, where r is a nonnegative integer.The notation C r df ( M ) denotes the ring of definable C r functions on M . The objectsof the category VB r df ( M ) are definable C r vector bundles over M and its arrowsare definable C r M -morphisms between them. Given a commutative ring R , thecategory Proj ( R ) is the category whose objects are finitely generated projective R -modules and whose arrows are homomorphisms between projective modules.For any definable C r vector bundle ξ over a definable C r manifold M , the no-tation Γ( ξ ) denotes the set of all definable C r sections of ξ . The notation Γ( ϕ ) isthe induced homomorphism from Γ( ξ ) to Γ( ξ ′ ) for any definable C r M -morphism ϕ : ξ → ξ ′ between definable C r vector bundles over M .We next show that Γ is a covariant functor from VB r df ( M ) to Proj ( C r df ( M )). Proposition 3.8.
Let M be a definable C r manifold, where r is a nonnegativeinteger. The map Γ is a covariant functor from the category VB r df ( M ) to thecategory Proj ( C r df ( M )) .Proof. The only non-trivial assertion is that the set of all definable C r sectionsΓ( ξ ) is a finitely generated projective C r df ( M )-module for any definable C r vectorbundle ξ .We first show that Γ( ξ ) is a finitely generated C r df ( M )-module. There are de-finable C r sections s , . . . , s n of ξ which generate the fibers ξ x for all x ∈ M byLemma 3.4. The C r df ( M )-module Γ( ξ ) is generated by s , . . . , s n by Lemma 3.5.We next show that Γ( ξ ) is a projective module. We have a definable C r vectorbundle ξ ′ such that ξ ⊕ ξ ′ is definably C r isomorphic to a trivial bundle ǫ nM byLemma 3.6. We have Γ( ξ ) ⊕ Γ( ξ ′ ) ≃ Γ( ǫ nM ) = ( C r df ( M )) n .Since Γ( ξ ) is a direct summand of a free module of finite rank, it is a finitelygenerated projective module. (cid:3) We finally show Main Theorem 1.1.
Theorem 3.9.
Let M be a definable C r manifold, where r is a nonnegative integer.Then, the functor Γ is an equivalence of the category VB r df ( M ) with the category Proj ( C r df ( M )) .Proof. In order to show that Γ is an equivalence of the categories, we have only toshow that the functor Γ is faithful and full, furthermore; for any finitely generatedprojective C r df ( M )-module P , there is a definable C r vector bundle ξ = ( E, p, M )such that Γ( ξ ) is isomorphic to P by [Mac, Chapter IV, Section 4, Theorem 1].We first show that the functor Γ is faithful. Let ξ = ( E, p, M ) and ξ ′ =( E ′ , p ′ , M ) be two definable C r vector bundles. Let u : ξ → ξ ′ and u ′ : ξ → ξ ′ be definable C r M -morphisms with Γ( u ) = Γ( u ′ ). Fix an arbitrary element v ∈ E .There exists a definable C r section s with s ( p ( v )) = v by Lemma 3.4. We have u ( v ) = u ( s ( p ( v ))) = (Γ( u )( s ))( p ( v )) = (Γ( u ′ )( s ))( p ( v )) = u ′ ( s ( p ( v ))) = u ′ ( v ). Wehave shown that u = u ′ .We next show that the functor Γ is full. Let ξ = ( E, p, M ) and ξ ′ = ( E ′ , p ′ , M )be two definable C r vector bundles. Let Φ : Γ( ξ ) → Γ( ξ ′ ) be a homomorphism. Weconstruct a definable C r M -morphism u : ξ → ξ ′ with Γ( u ) = Φ. Fix an arbitraryelement v ∈ E . Set x = p ( v ). There exists a definable C r section s with s ( x ) = v EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 11 by Lemma 3.4. We define u ( v ) as u ( v ) = Φ( s )( x ).We first show that it is a well-defined map. Let m x be the maximal ideal of C r df ( M )of definable C r functions on M vanishing at the point x . Let s ′ be another sectionwith s ′ ( x ) = v . We have s − s ′ ∈ m x Γ( ξ ). We get Φ( s ) − Φ( s ′ ) ∈ m x Γ( ξ ′ ) becauseΦ is a homomorphism. It means that Φ( s )( x ) = Φ( s ′ )( x ), and u is well-defined.We have only to show that u is a definable C r M -morphism because the equalityΓ( u ) = Φ is obviously satisfied.Let ( U i , ϕ i : U i × R d → p − ( U i )) qi =1 and ( U ′ i , ϕ ′ j : U ′ j × R d ′ → ( p ′ ) − ( U ′ j )) q ′ j =1 be definable C r atlases of ξ and ξ ′ , respectively. Let s , . . . s m be sections of ξ generating the fibers ξ x = p − ( x ) for all x ∈ M given by Lemma 3.4. Fix 1 ≤ i ≤ q and 1 ≤ j ≤ q ′ with U i ∩ U ′ j = ∅ . We have only to show that the restriction of u to U i ∩ U ′ j is a definable C r U i ∩ U ′ j -morphism. For any subset J ⊂ { , . . . , m } ofcardinality d , set U i,J = { x ∈ U i | the fiber ξ x is generated by { s j | j ∈ J }} .Taking a finer finite definable open covering and arranging the order of the sections,we may assume that the fiber ξ x is generated by { s ( x ) , . . . , s d ( x ) } without loss ofgenerality. Let u ′ : ( U i ∩ U ′ j ) × R d → ( U i ∩ U ′ j ) × R d ′ be the definable C r U i ∩ U ′ j -morpshism defined by u ′ ( x, v ) = ( ϕ ′ j ) − ◦ u ◦ ϕ i ( x, v ). Let π i : U i × R d → R d be thenatural projection. Consider the map τ : ( U i ∩ U ′ j ) × R d → ( U i ∩ U ′ j ) × R d given by τ ( x, ( a , . . . , a d )) = x, d X k =1 a k π i ( ϕ − i ( s k ( x ))) ! for any x ∈ U i ∩ U ′ j and ( a , . . . , a d ) ∈ R d . It is obviously a definable C r U i ∩ U ′ j -isomorphism. Consider the following commutative diagram:( U i ∩ U ′ j ) × R d ( U i ∩ U ′ j ) × R d ( U i ∩ U ′ j ) × R d ′ p − ( U i ∩ U ′ J ) ( p ′ ) − ( U i ∩ U ′ j ) ❍❍❍❍❍❍❥ u ′ ◦ τ ❄ τ ✲ u ′ ❄ ϕ i | ( Ui ∩ U ′ j ) × R d ❄ ϕ ′ j | ( Ui ∩ U ′ j ) × R d ′ ✲ u | ( Ui ∩ U ′ j ) × R d Here, the map u ′ ◦ τ is a map given by u ′ ◦ τ ( x, ( a , . . . , a d )) = x, d X k =1 a k π i (( ϕ ′ j ) − (Φ( s k )( x ))) ! .It is obviously a definable C r map. The restriction of u to U i ∩ U ′ j is also a definable C r U i ∩ U ′ j -morphism. Hence, the map u is a definable C r M -morphism.We finally demonstrate that, for any finitely generated projective C r df ( M )-module P , there exists a definable C r vector bundle ξ = ( E, p, M ) such that Γ( ξ ) is isomor-phic to P . We may assume that M is connected without loss of generality. Since P is a direct summand of a free module by [Ei, Proposition A3.1], we may assumethat P ⊂ ( C r df ( M )) n . Let s , . . . , s m ∈ ( C r df ( M )) n be generators of P . Set E = { ( x, v ) ∈ M × R n | v = σ ( x ) for some σ ∈ P } ,and p : E → M be the natural projection.We demonstrate that ξ = ( E, p.M ) is a definable C r vector bundle. Let m x denote the set of definable C r functions on M vanishing at x for any x ∈ M . The C r df ( M ) m x -module P m x is free by [Ei, Exercise 4.11]. For all subsets I ⊂ { , . . . , m } ,we define the subsets U I of M by { x ∈ M | { s i } i ∈ I is a basis of P m x } .The set U I is open. In fact, if x ∈ U I , there exist a j ∈ C r df ( M ) \ m x and b j,k ∈ C r df ( M ) for any j I and k ∈ I with a j ( x ) s j ( x ) = P k ∈ I b j,k ( x ) s k ( x ). Set V = { y ∈ M | a j ( y ) = 0 for any j I } , then V is an open neighbourhood of x . The set { s i } i ∈ I is a basis of the localization P m y for any y ∈ V . Hence, V is contained in U I .It means that U I is open. Since M is connected, the rank of P m x is constant, say d . Let A I be the n × d -matrix whose column vectors are { s i } i ∈ I . It is obvious that x ∈ U I if and only if at least one of d × d -minor determinants of A I is not zero at x . Hence, U I is a definable set. Let P be the collection of all subsets of { , . . . , m } with U I = ∅ . It is obvious that the family of definable open sets { U I } I ∈P covers M . We have shown that { U I } I ∈P is a finite definable open covering of M .Fix an arbitrary subset I ∈ P . Let j I ( k ) be the k -th smallest element of thesubset I ⊂ { , . . . , m } . Define ϕ I : U I × R d → p − ( U I ) by ϕ I ( x, ( a , . . . , a d )) = d X k =1 a k s j I ( k ) ( x ).It is obviously definable and of class C r , and for any x ∈ U I , the restriction ϕ I | { x }× R d of ϕ I to { x } × R d is an isomorphism between vector spaces. Let J ⊂{ , . . . , m } be another subset with U J = ∅ and U I ∩ U J = ∅ . We want to showthat (cid:0) ϕ J | ( U I ∩ U J ) × R d (cid:1) − ◦ ϕ I | ( U I ∩ U J ) × R d is a definable C r diffeomorphism. It isdefinable because ϕ I and ϕ J are definable. It is also obvious that it is a bijective.We have only to show that it is a C r diffeomorphism. Let x ∈ U I ∩ U J be an arbi-trary point. Since both { s k | k ∈ I } and { s k | k ∈ J } are bases of P m x , there exists g ∈ GL ( d, C r df ( M ) m x ) with (cid:0) ϕ J | ( U I ∩ U J ) × R d (cid:1) − ◦ ϕ I | ( U I ∩ U J ) × R d ( y, v ) = ( y, g ( y ) v ) forany y ∈ M sufficiently close to x . It shows that (cid:0) ϕ J | ( U I ∩ U J ) × R d (cid:1) − ◦ ϕ I | ( U I ∩ U J ) × R d is a C r diffeomorphism. We have shown that ( U I , ϕ I ) I ∈P is a definable C r -atlas of ξ . It remains to show that P = Γ( ξ ). The inclusion P ⊂ Γ( ξ ) is obvious becauseΓ( ξ ) contains the generators s , . . . , s m of P . We demonstrate the opposite inclu-sion. Let s ∈ Γ( ξ ) be fixed. The fiber p − ( x ) is generated by s , . . . , s m by thedefinition of the total space E . Hence, there exist definable C r function c , . . . , c m on M with s = P mi =1 c i s i by Lemma 3.5. We have shown s ∈ P because s , . . . , s m are generators of P . (cid:3) Proposition 3.10.
Let M be a definable C r manifold, where r is a nonnegativeinteger. Then, we have Γ( ξ ⊕ ξ ) = Γ( ξ ) ⊕ Γ( ξ ) and Γ( ξ ⊗ ξ ) = Γ( ξ ) ⊗ Γ( ξ ) for any definable C r vector bundles ξ and ξ over M .Proof. We omit the proof. (cid:3)
EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 13 Approximation of sections of definable C r vector bundles This short section is devoted for demonstrating that any definable continuoussection of a definable C r vector bundle can be approximated by a definable C r section. We introduce two lemmas prior to the proof of the above assertion. Lemma 4.1.
Let C be a definable closed subset of R n . Let U ⊂ C × R q be adefinable open neighbourhood of C ×{ } contained in C × B q (0; 1) . Then, there existsa positive definable continuous function ρ : C → R with { ( x, v ) ∈ C × R q | k v k q <ρ ( x ) } ⊂ U . Here, B q (0; r ) denotes the open ball in R q centred at the origin of radius r , and k v k q denotes the Euclidean norm of a vector v in R q .Proof. Since U is an open neighbourhood of C × { } , for any x ∈ C , there existpositive real numbers δ x and s x such that the set { y ∈ C | k y − x k < δ x } × B q (0; s x )is contained in U . Set S = { s ∈ R | ∃ x ∈ C ⊂ R n , k x k n = s } . For any s ∈ S ,the notation C s denotes the definable closed set { x ∈ C | k x k n = s } . Since thedefinable closed set C s is compact, there exists a positive real number t such that C s × B q (0; t ) is contained in U . The function φ : S → R given by φ ( s ) = sup { t ∈ R | C s × B q (0; t ) ⊂ U } is a well-defined definable function, and positive for any point in S .There exist finite points s , · · · , s p ∈ S such that the restriction of φ to S \{ s , · · · , s p } is continuous by [vdD, Chapter 3, Theorem 1.2]. We may assume that S \{ s , · · · , s p } is open by enlarging the discrete set { s , · · · , s p } if necessary. Let T be a connected component of S \{ s , · · · , s p } such that s i is contained in the closureof T . The notation graph( φ | T ) denotes the graph of the restriction of φ to T . Weshow that ( s i ,
0) is not contained in the closure graph( φ | T ). Assume the contrary.There exists a convergent sequence { t m } ⊂ T with t m → s i and φ ( t m ) → m → ∞ . It means that, for any positive real number ε >
0, there exist a positiveinteger m ε and x ε ∈ C t mε with { x ε } × B (0; ε ) U . There exists a real number R with t m ≤ R for any positive integer m because { t m } is a convergent sequence. Set D = { x ∈ C | k x k n ≤ R } . We have x ε ∈ D for any ε >
0. Let { ε m } be a sequenceof positive real numbers converging to 0. Since D is compact, we may assume thatthe sequence { x ε m } converges to a point x ′ ∈ C by taking a subsequence of { ε m } if necessary. Since U is an open neighbourhood of C × { } , there exist positivereal numbers δ x ′ and s x ′ such that the set { y ∈ C | k y − x ′ k < δ x ′ } × B q (0; s x ′ ) iscontained U . We have { x ε m } × B q (0; s x ′ ) ⊂ U for sufficiently large m . On the otherhand, for sufficiently large m , we have ε m < s x ′ and we get { x ε m } × B q (0; ε m ) U by the assumption. It is a contradiction. We have shown that ( s i , graph( φ | T ).There exist positive real numbers y , . . . , y p and η , . . . , η p such that the restric-tion of φ to the intersection S ∩ ( s i − η i , s i + η i ) is larger than y i for all 1 ≤ i ≤ p because ( s i , graph( φ | T ). Set V = S \ { s , · · · , s p } and V i = ( s i − η i , s i + η i )for 1 ≤ i ≤ p . The collection of definable open sets { V i } pi =0 is a definable opencovering of the neighborhood V = S pi =0 V i of S . Let { τ i : V → R } pi =0 be a definablecontinuous partition of unity subordinate to the open covering { V i } pi =0 given byLemma 2.6. Define a definable continuous function g : V → R by g ( x ) = φ ( x ) / g i : V i → R by g i ( x ) = y i / i = 1 , . . . , p . Then, the function ρ : C → R defined by ρ ( x ) = P qi =0 τ i ( x ) g i ( x ) isa well-defined definable continuous function with 0 < ρ ( x ) < φ ( x ) for any x ∈ C .The definable continuous function ρ satisfies the desired properties. (cid:3) The following lemma is [Es, Theorem 1.1].
Lemma 4.2.
Let M and N be definable C r manifolds. Assume that N is a definable C r submanifold of R n . Let f : M → N be a definable continuous map and ε : M → R be a positive definable continuous function on M . There exists a definable C r function g : M → R such that k f ( x ) − g ( x ) k n < ε ( x ) for any x ∈ M . We show an approximation theorem for definable C r sections using the abovelemmas. Theorem 4.3.
Let ξ = ( E, p, M ) be a definable C r vector bundle of rank d over adefinable C r manifold M , where r is a nonnegative integer. Let σ : M → E be adefinable continuous section of ξ . For any definable open neighbourhood U of σ ( M ) in E , there exists a definable C r section s : M → E with s ( M ) ⊂ U .Proof. We may assume that M is a definable C r submanifold of a Euclidean space R n which is simultaneously closed in R n . In fact, M is a definable C r submanioldof a Euclidean space R n because M is affine. Take a definable C r function H on R n with H − (0) = M \ M using [vdDM, Theorem C.11]. The image of M underthe definable immersion ι : M → R n +1 given by ι ( x ) = ( x, /H ( x )) is a definableclosed subset of R n +1 .Let s , . . . , s m be definable C r sections of ξ given in Lemma 3.4. There existdefinable continuous functions α , . . . , α m on M with σ ( x ) = P mi =1 α i ( x ) s i ( x ) forany x ∈ M by Lemma 3.5. Define τ : M × R m → E by τ ( x, ( c , . . . , c m )) = P mi =1 ( α i ( x ) + c i ) s i ( x ). It is a definable continuous map. The definable open set V = τ − ( U ) is a definable open neighbourhood of M × { } . Taking an intersectionof V with M × B m (0; 1), we may assume that V is contained in M × B m (0; 1).There exists a positive definable continuous function ρ on M such that { ( x, v ) ∈ M × R m | k v k m < ρ ( x ) } ⊂ V by Lemma 4.1. Take a definable C r approximation β i of α i with | β i ( x ) − α i ( x ) | < ρ ( x ) / √ m for any 1 ≤ i ≤ m using Lemma 4.2, thenthe definable C r section s : M → E given by s ( x ) = P mi =1 β i ( x ) s i ( x ) satisfies therequirement. (cid:3) Corollary 4.4.
Let r be a nonnegative integer. If two definable C r vector bundlesover a definable C r manifold are definably C isomorphic, they are definably C r isomorphic.Proof. Let ξ and ξ be definable C r vector bundles over a definable C r manifold M which are definably C isomorphic. Remember that Hom( ξ , ξ ) is a definable C r vector bundle over M by Proposition 3.3. SetIso( ξ , ξ ) = { ( φ, x ) | φ is an isomorphism between ( ξ ) x and ( ξ ) x } .It is a definable open subset of the total space of the vector bundle Hom( ξ , ξ ).A definable C isomorphism between ξ and ξ corresponds to a definable contin-uous section of Hom( ξ , ξ ) whose image is contained in Iso( ξ , ξ ). There exists adefinable C r section of Hom( ξ , ξ ) contained in Iso( ξ , ξ ) by Theorem 4.3. Thissection corresponds to a definable C r isomorphism between ξ and ξ . (cid:3) We give another important corollary of Lemma 4.2.
Theorem 4.5.
Let r be a nonnegative integer. Any definable C vector bundleover a definable C r manifold is definably C isomorphic to a definable C r vectorbundle. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 15
Proof.
Let ξ be a definable C vector bundle of rank d over a definable C r manifold M . There exist a trivial bundle ǫ nM over M and a definable C vector subbundle ξ ⊥ of ǫ nM such that ξ is a definable C r vector subbundle of ǫ nM and ξ ⊕ ξ ⊥ = ǫ nM by Lemma 3.6. Let ϕ : M → Gr ( n, d ) be the definable C map naturally inducedfrom the vector bundle ξ . Let ˜ ϕ : M → Gr ( n, d ) be a definable C r approximationof ϕ given by Lemma 4.2. If we take ˜ ϕ sufficiently close to ϕ , we may assume that ϕ ( x ) ⊥ ∩ ˜ ϕ ( x ) = { } for any x ∈ M . Let π : ǫ nM = ξ ⊕ ξ ⊥ → ξ be the orthogonalprojection. Let ˜ ξ be the definable C r vector bundle on M corresponding to themap ˜ ϕ . The restriction π | ˜ ξ of π to ˜ ξ gives a definable C isomorphism between thevector bundles ξ and ˜ ξ . (cid:3) Equivalence of definable C r bilinear forms over vector bundleswith bilinear forms over projective modules The purpose of this section is to demonstrate an equivalence of the category ofdefinable C r bilinear spaces over a definable C r manifold M with the category ofbilinear spaces over C r df ( M ). We first define definable C r bilinear spaces. Definition 5.1 (Definable C r bilinear spaces) . Let r be a nonnegative integer.Consider a definable C r manifold M and a definable C r vector bundle ξ = ( E, p, M )over M . A definable C r function s : E ⊕ E = { ( u, v ) ∈ E × E | p ( u ) = p ( v ) } → R is called definable C r bilinear form over the definable C r vector bundle ξ if therestriction of s to the set { ( u, v ) ∈ E × E | p ( u ) = p ( v ) = x } is a nondegeneratesymmetric bilinear form for any x ∈ M . We call the bilinear form s positive definite if the inequality s ( v, v ) > v ∈ E with v = 0 as a vector inthe fiber p − ( p ( v )). We define a negative definite bilinear form in the same way. A definable C r bilinear space ( ξ, s ) over M is a pair of definable C r vector bundle ξ over M and a definable C r bilinear form s defined over it.Let M ′ be a definable C r manifold. Let ( ξ = ( E, p, M ) , s ) and ( ξ ′ = ( E ′ , p ′ , M ′ ) , s ′ )be two definable C r bilinear spaces over M and M ′ , respectively. Let ϕ = ( u, f ) bedefinable C r morphism from the vector bundle ξ to the vector bundle ξ ′ . It meansthat u : E → E ′ and f : M → M ′ are definable C r maps with p ′ ◦ u = f ◦ p such that u | p − ( x ) is a linear map for any x ∈ M . It is a definable C r morphism from the bilin-ear space ( ξ, s ) to the bilinear space ( ξ ′ , s ′ ) if the equality s ( v, w ) = s ′ ( u ( v ) , u ( w ))holds true for any ( v, w ) ∈ E ⊕ E . We define a definable C r M -morphism from( ξ, s ) to ( ξ ′ , s ′ ) in the same way when the base spaces M and M ′ are identical.Two definable C r bilinear space ( ξ = ( E , p , M ) , s ) and ( ξ , s ) over a de-finable C r manifold are definably C r isometric if there exists a definable C r M -isomorphism ( u, id) : ξ → ξ ′ between definable C r vector bundles such that s ( v, w ) = s ′ ( u ( v ) , u ( w )) for any ( v, w ) ∈ E ⊕ E .Let ǫ dM be the trivial bundle of rank d over a definable C r manifold M . Adefinable C r bilinear space ( ǫ dM , b ) is a trivial bilinear space of type ( r + , r − ) if d = r + + r − and b (( x, ( v , . . . , v d )) , ( x, ( w , . . . , w d ))) = P r + i =1 v i w i − P r − i =1 v i + r + w i + r + .The notation r + h i ⊥ r − h− i denotes this bilinear form. Proposition 5.2.
Let r be a nonnegative integer. Any definable C r vector bun-dle over a definable C r manifold has a positive definite definable C r bilinear formdefined over it.Proof. Let ξ be a definable C r bilinear space over a definable C r manifold M . Wemay assume that ξ be a subbundle of a trivial bundle ǫ nM by Lemma 3.6. There exists a positive definite bilinear form over the trivial bundle ǫ nM . Its restriction to ξ is a desired positive definite bilinear form over ξ . (cid:3) The notation SGL( d, R ) denotes the set of all d × d symmetric invertible matrices. Proposition 5.3.
Let r be a nonnegative integer. Let ξ be a definable C r vectorbundle of rank d over a definable C r manifold M with a definable C r atlas ( U i , φ i : U i × R d → p − ( U i )) qi =1 . The definable C r map g ij : U i ∩ U j → GL( d, R ) is atransition map defined in Remark 3.2 for any ≤ i, j ≤ q .There exists a one-to-one correspondence of definable C r bilinear forms over ξ with families of definable C r maps { s i : U i → SGL( d, R ) } qi =1 satisfying the equality t g ji ( x ) s j ( x ) g ji ( x ) = s i ( x ) for any x ∈ U i ∩ U j .Proof. When a bilinear form s over ξ = ( E, p, M ) is given, we define definable C r maps s i : U i → SGL( d, R ) by t vs i ( x ) w = s ( φ i ( x, v ) , φ i ( x, w )) for any x ∈ U i and v, w ∈ R d . These maps satisfy the requirement.On the other hand, if a family of definable C r maps { s i : U i → SGL( d, R ) } qi =1 isgiven. Let π i : U i × R d → R d be the projection. Define s : E ⊕ E → R by s ( v, w ) = t (cid:0) π i ( φ − i ( v )) (cid:1) s i ( p ( v )) π i ( φ − i ( w )) for any v, w ∈ p − ( U i ) with p ( v ) = p ( w ). It iseasy to check that it is a well-defined definable C r bilinear form over ξ . (cid:3) Definition 5.4 (Orthogonal sum and Tensor product) . Let ( ξ = ( E , p , M ) , s )and ( ξ = ( E , p , M ) , s ) be definable C r bilinear spaces over the same definable C r manifold M . Their orthogonal sum is the bilinear space ( ξ ⊕ ξ , s ⊥ s ) whosevector bundle is the Whitney sum of ξ and ξ , and the bilinear form s ⊥ s isdefined as follows:( s ⊥ s )( x ⊕ x , y ⊕ y ) = s ( x , y ) + s ( x , y ).The tensor product of two bilinear spaces ( ξ , s ) and ( ξ , s ) is the bilinear spacewhose vector bundle is the tensor product of ξ and ξ , and the bilinear form s ⊗ s is defined as follows:( s ⊗ s )( x ⊗ x , y ⊗ y ) = s ( x , y ) s ( x , y ). Proposition 5.5.
Let r be a nonnegative integer. Let ( ξ , s ) and ( ξ , s ) be defin-able C r bilinear spaces over the same definable C r manifold M . Their orthogonalsum ( ξ , s ) ⊥ ( ξ , s ) and tensor product ( ξ , s ) ⊗ ( ξ , s ) are definable C r bilinearspaces over M .Proof. Obvious. (cid:3)
Definition 5.6.
Let ( ξ = ( E, p, N ) , s ) be a definable C r bilinear spaces over adefinable C r manifold N . Consider a definable C r map f : M → N betweendefinable C r manifolds. The definable C r bilinear space f ∗ ( ξ, s ) induced by f isa definable C r bilinear space whose vector bundle is f ∗ ξ and whose definable C r bilinear form is f ∗ s defined by f ∗ s (( x, v ) , ( x, w )) = s ( v, w )for x ∈ M and v, w ∈ E with p ( v ) = p ( w ) = f ( x ).We next review the definition of bilinear spaces over a commutative ring. Thefollowing definitions are found in [BCR, Section 15.1] and [MH]. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 17
Definition 5.7 (Bilinear spaces over a commutative ring) . Let R be a commutativering. Let P be a finitely generated projective module over R and b : P × P → R bea symmetric bilinear form. The bilinear form b is nondegenerate if the linear map h b : P → P ∨ from P into its dual, induced by b , is an isomorphism. A bilinearspace over R is a pair ( P, b ), where P is a finitely generated projective R -moduleand b : P × P → R is a nondegenerate symmetric bilinear form. A morphism from a bilinear space ( P, b ) to another bilinear space ( P ′ , b ′ ) is a homomorphism ϕ : P → P ′ between R -modules with b ′ ( ϕ ( v ) , ϕ ( w )) = b ( v, w ) for any v, w ∈ P .An isometry of two bilinear spaces ( P, b ) and ( P ′ , b ′ ) over R is an isomorphism ϕ : P → P ′ between R -modules with b ′ ( ϕ ( v ) , ϕ ( w )) = b ( v, w ) for any v, w ∈ P .Let ( P , b ) and ( P , b ) be bilinear spaces over the same commutative ring R .Their orthogonal sum is the bilinear space ( P ⊕ P , b ⊥ b ) whose projectivemodule is the direct sum of P and P , and the bilinear form b ⊥ b is defined asfollows: ( b ⊥ b )( x ⊕ x , y ⊕ y ) = b ( x , y ) + b ( x , y ).The tensor product of ( P , b ) and ( P , b ) is the bilinear space whose projectivemodule is the tensor product of P and P , and the bilinear form b ⊗ b is definedas follows: ( b ⊗ b )( x ⊗ x , y ⊗ y ) = b ( x , y ) b ( x , y ).It is obvious that P ⊕ P and P ⊗ P are finitely generated projective R -modulesbecause an R -module is projective if and only if it is a direct summand of a freemodule [Ei, Proposition A3.1]. Notation . Let M be a definable C r manifold, where r is a nonnegative integer.The notation BS r df ( M ) denotes the category whose objects are definable C r bilinearspaces over M and whose arrows are definable C r M -morphisms between them. Let R be a commutative ring. The notation BS ( R ) denotes the category whose objectsare bilinear spaces over R and whose arrows are morphisms between bilinear spacesover R .For any definable C r bilinear space ( ξ, s ) over M , we define a bilinear space( P, b ) = Γ b ( ξ, s ) over the commutative ring C r df ( M ) as follows: The finitely gener-ated projective module P is Γ( ξ ) and the bilinear form b : Γ( ξ ) × Γ( ξ ) → C r df ( M )over it is given by b ( σ, σ ′ )( x ) = s ( σ ( x ) , σ ′ ( x ))for any σ, σ ′ ∈ Γ( ξ ) and x ∈ M . We denote the bilinear form b as Γ b ( s ) by abuse ofnotation. A definable C r M -morphism between definable C r bilinear spaces over M is simultaneously a definable C r M -morphisms between definable C r vectorbundles. The notation Γ b ( u ) denotes the homomorphism Γ( u ) between finitelygenerated projective C r df ( M )-modules for any definable C r M -morphism u betweendefinable C r bilinear forms.We show that the above map Γ b is a covariant functor from the category BS r df ( M )to the category BS ( C r df ( M )). Proposition 5.9.
Let M be a definable C r manifold, where r is a nonnegativeinteger. The map Γ b is a covariant functor from the category BS r df ( M ) to thecategory BS ( C r df ( M )) .Proof. It is easy to check that the map b defined in Notation 5.8 is a bilinear formover Γ( ξ ). We show that Γ b ( u ) is a morphism between bilinear spaces for any definable C r M -morphism u from a definable C r bilinear space ( ξ, s ) over M toa definable C r bilinear space ( ξ ′ , s ′ ) over M . Set ( P, b ) = Γ b ( ξ, s ) and ( P ′ , b ′ ) =Γ b ( ξ ′ , s ′ ). We have only to show that b ′ (Γ b ( u )( σ ) , Γ b ( u )( σ ′ )) = b ( σ, σ ′ ) for any σ, σ ′ ∈ Γ( ξ ). Let x ∈ M be fixed. We have b ′ (Γ b ( u )( σ ) , Γ b ( u )( σ ′ ))( x ) = s ′ ( u ◦ σ ( x ) , u ◦ σ ′ ( x )) = s ( σ ( x ) , σ ′ ( x )) = b ( σ, σ ′ )( x ).We have finished the proof. (cid:3) Finally, we prove Main Theorem 1.2 introduced in Section 1.
Theorem 5.10.
Let M be a definable C r manifold, where r is a nonnegative in-teger. Then, the functor Γ b is an equivalence of the category BS r df ( M ) with thecategory BS ( C r df ( M )) .Proof. In the same way as the proof of Theorem 3.9, we have to show that thefunctor Γ b is faithful and full, furthermore; for any bilinear space ( P, b ) over the ring C r df ( M ), there is a definable C r bilinear space ( ξ, s ) such that Γ b ( ξ, s ) is isotropicto ( P, b ) by [Mac, Chapter IV, Section 4, Theorem 1]. The functor Γ b is obviouslyfaithful because the functor Γ is faithful.We next show that the functor Γ b is full. Consider an arbitrary morphism ϕ : Γ b ( ξ, s ) → Γ b ( ξ ′ , s ′ ) of the category BS ( C r df ( M )). Since the functor Γ is full byTheorem 3.9, there exists a definable C r morphism u : ξ → ξ ′ with Γ( u ) = ϕ . Let ξ = ( E, p, M ). We want to show that this u is also a morphism between ( ξ, s ) and( ξ ′ , s ′ ) in the category BS r df ( M ). We have only to show s ( v, w ) = s ′ ( u ( v ) , u ( w ))for any v, w ∈ E with p ( v ) = p ( w ). Set x = p ( v ) = p ( w ). There exist definable C r sections σ v , σ w of ξ with σ v ( x ) = v and σ w ( x ) = w by Lemma 3.4. The followingcalculation indicates that u is a morphism between definable C r bilinear spaces( ξ, s ) and ( ξ ′ , s ′ ). s ′ ( u ( v ) , u ( w )) = s ′ ( u ( σ v ( x )) , u ( σ w ( x ))) = (Γ b ( s ′ )(Γ( u )( σ v ) , Γ( u )( σ w )))( x )= (Γ b ( s ′ )( ϕ ( σ v ) , ϕ ( σ w )))( x ) = (Γ b ( s )( σ v , σ w ))( x )= s ( σ v ( x ) , σ w ( x )) = s ( v, w ).We finally construct a definable C r bilinear space ( ξ, s ) such that Γ b ( ξ, s ) isisotropic to ( P, b ) for a given bilinear space (
P, b ) over the ring C r df ( M ). We havealready shown that Γ gives an equivalence of the category VB r df ( M ) with the cat-egory Proj ( C r df ( M )) in Theorem 3.9. There exists a definable C r vector bundle ξ = ( E, p, M ) of rank d such that Γ( ξ ) is isomorphic to the module P . We haveonly to construct a definable C r bilinear form s over ξ such that Γ b ( ξ, s ) is isotropicto ( P, b ).Let ψ : Γ( ξ ) → P be an isomorphism between C r df ( M )-modules. For any v, w ∈ E with p ( v ) = p ( w ) = x , there exists definable C r sections σ v , σ w ∈ Γ( ξ ) with σ v ( x ) = v and σ w ( x ) = w by Lemma 3.4. We set s ( u, v ) = ( b ( ψ ( σ v ) , ψ ( σ w )))( x ).We want to demonstrate that s : E ⊕ E → R is a definable C r bilinear form over ξ . We first show that s is a well-defined map. Let σ ′ v and σ ′ w be definable C r sections of ξ with σ ′ v ( x ) = v and σ ′ w ( x ) = w . The notation m x denotes the setof definable C r functions vanishing at the point x ∈ M . It is a maximal ideal of C r df ( M ). Since σ v − σ ′ v ∈ m x Γ( ξ ), we have b ( ψ ( σ v ) , ψ ( σ w )) − b ( ψ ( σ ′ v ) , ψ ( σ w )) = b ( ψ ( σ v − σ ′ v ) , ψ ( σ w )) ∈ m x . We get b ( ψ ( σ v ) , ψ ( σ w ))( x ) = b ( ψ ( σ ′ v ) , ψ ( σ w ))( x ). We EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 19 can show that b ( ψ ( σ ′ v ) , ψ ( σ w ))( x ) = b ( ψ ( σ ′ v ) , ψ ( σ ′ w ))( x ) in the same way. We haveshown that b ( ψ ( σ v ) , ψ ( σ w ))( x ) = b ( ψ ( σ ′ v ) , ψ ( σ ′ w ))( x ).It means that s is a well-defined map.We next show that s is a definable C r bilinear form over ξ . Let ( U i , φ i : U i × R d → p − ( U i )) qi =1 be a definable C r atlas of ξ . We can get definable C r sections σ , . . . , σ m which generate p − ( x ) for any x ∈ M by Lemma 3.4. Fix 1 ≤ i ≤ q .We may assume that σ ( x ) , . . . σ d ( x ) generate p − ( x ) for any x ∈ U i in the sameway as the proof of Theorem 3.9. Let τ be the definable C r isomorphism from U i × R d × R d onto p − ( U i ) ⊕ p − ( U i ) defined by τ ( x, ( a , . . . , a d ) , ( b , . . . , b d )) =( P dj =1 a j σ j ( x ) , P dj =1 b j σ j ( x )). The composition s | p − ( U i ) ⊕ p − ( U i ) ◦ τ is given by s | p − ( U i ) ⊕ p − ( U i ) ◦ τ ( x, ( a , . . . , a d ) , ( b , . . . , b d )) = d X j,k =1 a j b k b ( ψ ( σ j ) , ψ ( σ k ))( x ).It is a definable C r bilinear form over U i × R d . Hence, the restriction of s to p − ( U i ) ⊕ p − ( U i ) is a definable C r bilinear form over the restriction ξ | U i of ξ to U i . It means that s itself is a definable C r bilinear form over ξ .We finally show that Γ b ( ξ, s ) is isotropic to ( P, b ). We have Γ b ( s )( σ , σ )( x ) = s ( σ ( x ) , σ ( x )) = ( b ( ψ ( σ ) , ψ ( σ )))( x ) for any σ , σ ∈ Γ( ξ ) and x ∈ M . Hence, wehave Γ b ( s )( σ , σ ) = b ( ψ ( σ ) , ψ ( σ )). We have shown that Γ b ( ξ, s ) is isotropic to( P, b ) via the isotropy ψ . (cid:3) Proposition 5.11.
Let M be a definable C r manifold, where r is a nonnega-tive integer. Then, we have Γ b (( ξ , s ) ⊥ ( ξ , s )) = Γ b ( ξ , s ) ⊥ Γ b ( ξ , s ) and Γ b (( ξ , s ) ⊗ ( ξ , s )) = Γ b ( ξ , s ) ⊗ Γ b ( ξ , s ) for any definable C r bilinear spaces ( ξ , s ) and ( ξ , s ) over M .Proof. We omit the proof. (cid:3) Homotopy theorems for definable C r vector bundles and bilinearspaces In this section, we show the homotopy theorems introduced in Section 1. Wefirst show the homotopy theorem for definable C r vector bundles. It can be shownfollowing the standard argument in [H] using the following three lemmas: Lemma 6.1.
Let M be a definable C r manifold, where r is a nonnegative integer.Let U and V be definable open subsets of M with U ⊂ V . The maps π and π arethe projections of M × [0 , onto the first and the second components, respectively.Then, there exists a definable C r map ϕ : M × [0 , → M × [0 , satisfying thefollowing conditions: • The restriction of ϕ to V c × [0 , is the identity map. • The restriction of the composition π ◦ ϕ to U × [0 , is constantly one. • The equality π ◦ ϕ ( x, t ) = x is satisfied for any x ∈ M .Proof. Since U and V c are closed definable sets, there exists definable C r functions f, g : M → R with f − (0) = U and g − (0) = V c by Lemma 2.2. The map ϕ : M × [0 , → M × [0 ,
1] defined by ϕ ( x, t ) = (cid:18) x, f f + g t + g f + g (cid:19) satisfies the requirements. (cid:3) The proofs of the following lemmas are almost the same as [H, Section 3.4].
Lemma 6.2.
Let U be a definable open subset of a definable C r manifold, and W be a definable open subset of U × R containing U × [0 , . Let ϕ , ϕ , ϕ : U → [0 , be definable C r functions with ϕ < ϕ < ϕ on U . We consider a definable C r vector bundle ξ = ( E, p, W ) over W . Define definable sets B and B as follows: B = { ( x, t ) ∈ U × [0 , | ϕ ( x ) ≤ t ≤ ϕ ( x ) } and B = { ( x, t ) ∈ U × [0 , | ϕ ( x ) ≤ t ≤ ϕ ( x ) } .For i = 1 , , let U i be a definable open subset of W such that B i ⊂ U i and therestriction ξ | U i of ξ to U i is definably C r isomorphic to a trivial bundle over U i .Then, there exists a definable open subset W ′ of W such that B ∪ B ⊂ W ′ andthe restriction ξ | W ′ of ξ to W ′ is definably C r isomorphic to a trivial bundle over W ′ .Proof. Set W ′ = { ( x, t ) ∈ U | t ≤ ϕ ( x ) } ∪ { ( x, t ) ∈ U | t ≥ ϕ ( x ) } . Wehave only to show that ξ | W ′ is definably C isomorphic to a trivial bundle byCorollary 4.4. Let ξ | U i = ( E i , p i , U i ) and u i : U i × R d → E i be a definable C r isomorphism. Set B = B ∩ B and let v i be the restriction of u i to B × R d .Then, h = v ◦ v − : B × R d → B × R d is a definable C r isomorphism. Thereexists a definable C r map g : U → GL( d, R ) with h ( x, v ) = ( x, g ( x ) v ). Define adefinable continuous map u : W ′ × R d → p − ( W ′ ) by u (( x, t ) , v ) = (cid:26) u (( x, t ) , v ) if t ≤ ϕ ( x ) and u (( x, t ) , g ( x ) v ) if t > ϕ ( x ).The map u is a definable continuous isomorphism between the restriction ξ | W ′ of ξ to W ′ and a trivial vector bundle. (cid:3) Lemma 6.3.
Let M be a definable C r manifold and W be a definable open subset of M × R containing M × [0 , . We consider a definable C r vector bundle ξ = ( E, p, W ) over W . There exist a finite definable open covering { U i } qi =1 of M and a definableopen subset V i of U i × R containing U i × [0 , such that the restriction ξ | V i of ξ to V i is definably C r isomorphic to a trivial bundle.Proof. Let ( W i , ψ i ) pi =1 be a definable C r atlas of ξ . By definition, the restriction ξ | W i is definably C r isomorphic to a trivial bundle for any 1 ≤ i ≤ p . By Lemma2.7, there exists a finite definable open covering { U i } qi =1 of M and definable C r functions 0 = ϕ i, < · · · < ϕ i,k < · · · < ϕ i,r i = 1 on U i such that, for any 1 ≤ i ≤ q and 1 ≤ k ≤ r i , the definable set B i,k = { ( x, t ) ∈ U i × [0 , | ϕ i,k − ( x ) ≤ t ≤ ϕ i,k ( x ) } is contained in W j for some 1 ≤ j ≤ p . In particular, for any 1 ≤ i ≤ q and1 ≤ k ≤ r i , there exists a definable open subset U i,k of U ∩ ( U i × R ) such that B i,k ⊂ U i,k and the restriction ξ | U i,k is definably C r isomorphic to a trivial bundle.Apply Lemma 6.2, then, for any 1 ≤ i ≤ q , there exists a definable open subset V i of U i × R such that it contains U i × [0 ,
1] and ξ | V i is definably C r isomorphic to atrivial bundle. (cid:3) We are now ready to show the homotopy theorem for definable C r vector bundles. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 21
Theorem 6.4 (Homotopy theorem for definable C r vector bundles) . Consider adefinable C r manifold M , where r is a nonnegative integer. Let U be a definableopen subset of M × R containing M × [0 , . Define the definable C r map r : U → U by r ( x, t ) = ( x, . Let ξ = ( E, p, U ) be a definable C r vector bundle over U . Then,shrinking U if necessary, two definable C r vector bundles ξ and r ∗ ( ξ ) are definably C r isomorphic.Proof. We construct a definable C r isomorphism ˜ u : ξ → r ∗ ( ξ ). There exist a finitedefinable open covering { U i } qi =1 and a definable open subset V i of U i × R containing U i × [0 ,
1] such that the restriction ξ | V i of ξ to V i is definably C r isomorphic to atrivial bundle by Lemma 6.3. Let h i : V i × R d → p − ( V i ) be a definable C r isomorphism for any 1 ≤ i ≤ q . Shrinking U if necessary, we may assume that U = S qi =1 V i . There exist definable open sets U ′ i such that U ′ i ⊂ U i and { U ′ i } qi =1 isagain a finite definable open covering of M by Lemma 2.5. There exists a definable C r map r i : M × [0 , → M × [0 ,
1] such that the restriction of r i to U ci × [0 ,
1] is theidentity map, the restriction of the composition π ◦ r i to U ′ i × [0 ,
1] is constantlyone, and π ◦ r i ( x, t ) = x for any x ∈ M by Lemma 6.1. Here, π : M × R → M and π : M × R → R are the projections. We may assume that r i is a definable C r map defined on U , shrinking U if necessary. We have r = r q ◦ · · · ◦ r ◦ r .Set s = id and s i = r i ◦ · · · ◦ r ◦ r for i >
0. Set s ∗ i ξ = ( s ∗ i E, p i , U ). Considerthe map h i,j : V j × R d → s ∗ i E | V j given by h i,j (( x, t ) , w ) = (( x, t ) , h j ( s i ( x, t ) , w ))for any ( x, t ) ∈ V j and w ∈ R d . It is a definable C r diffeomorphism by the definitionof the induced vector bundle. Let u i : s ∗ i − ξ → s ∗ i − ξ be the definable C r morphismsuch that it is defined by u i ( h i − ,i (( x, t ) , v )) = h i − ,i ( r i ( x, t ) , v ) for any (( x, t ) , v ) ∈ V i × R d and u i is the identity map off the set p − i − ( V i ). Consider the definable C r U -morphism ˜ u i : s ∗ i − ξ → s ∗ i ξ given by ˜ u i ( v ) = ( p i − ( v ) , u i ( v )). We show that ˜ u i is a definable C r U -isomorphism. It is clear that ˜ u i is a bijective definable C r U -morphism. We have only to show that ˜ u i is a C r diffeomorphism. Let ( x, t ) ∈ U befixed. There exists 1 ≤ j ≤ q with ( x, t ) ∈ V j . We have the following commutativediagram: V j × R d p − i − ( V j ) { (( x, t ) , v ) ∈ V j × s ∗ i − E | r i ( x, t ) = p i − ( v ) } p − i ( V j ) ❄ id Vj × ( u i ◦ h i − ,j ) ✲ h i − ,j ❄ ˜ u i | p − i − Vj ) ✲ ≃ The definable C r map id V j × ( u i ◦ h i − ,j ) is a C r diffeomorphism. Hence, ˜ u i is alsoa C r diffeomorphism.Set ˜ u = ˜ u q ◦ · · · ◦ ˜ u ◦ ˜ u . The U -morphism ˜ u is a definable C r U -isomorphismbetween ξ and r ∗ ( ξ ). (cid:3) The following corollary is Main Theorem 1.3.
Corollary 6.5.
Consider a definable C r manifold M , where r is a nonnegativeinteger. Let U be a definable open subset of M × R containing M × [0 , . Let Ξ be a definable C r vector bundle over U . Then, two definable C r vector bundles Ξ | M ×{ } and Ξ | M ×{ } are definably C r isomorphic.Proof. Immediate from Theorem 6.4. (cid:3)
Definition 6.6.
Let f, g : M → N be definable C r maps between definable C r manifolds. We call them definably C r homotopic if there exists a definable C r map H : M × [0 , → N with H ( x,
0) = f ( x ) and H ( x,
1) = g ( x ) for any x ∈ M . Corollary 6.7.
Consider definable C r manifolds M and N , where r is a nonneg-ative integer. Let f, g : M → N be definably C r homotopic definable C r maps. Let ξ be a definable C r vector bundle over N . Then, the induced definable C r vectorbundles f ∗ ξ and g ∗ ξ are definably C r isomorphic.Proof. Immediate from Corollary 6.5. (cid:3)
Corollary 6.8.
Any definable C r vector bundle over R n is definably C r isomorphicto a trivial bundle.Proof. Consider the definable C r function H : R n × [0 , → R n given by H ( x , . . . , x n , t ) = ( tx , . . . , tx n ).It means that the identity map on R n and the constant map given by c ( x , . . . , x n ) =(0 , . . . ,
0) are definably C r homotopic. Any definable C r vector bundle ξ is definably C r isomorphic to c ∗ ξ , which is a trivial bundle, by Corollary 6.7. (cid:3) We begin to prove the homotopy theorem for definable C r bilinear spaces. Itscounterpart for semialgebraic vector bundles is proved in [BCR, Chapter 15]. Wefirst show a theorem on Nash vector bundles in order to prove the homotopy theoremfor definable C r bilinear spaces. The proof is the same as that of [BCR, Theorem15.1.6], which is a theorem on semialgebraic C vector bundles. We give a prooffor readers’ convenience. Nash vector bundles are defined in [BCR, Section 12.7].We use the real spectrum Spec r ( N ( M )) of the ring N ( M ) of Nash functions on anaffine Nash manifold M in the proof of the following theorem. Real spectrum isdefined and its features are investigated in [BCR, Chapter 7]. Theorem 6.9.
Let M be an affine Nash manifold and ξ = ( E, p, M ) be a Nashvector bundle of rank d . Let ( U i , φ i : U i × R d → p − ( U i )) i =1 ,...,q be a Nash atlasof ξ . Let s be a Nash bilinear form over ξ . The families of Nash maps { s i : U i → SGL( d, R ) } i =1 ,...,q are induced from the Nash bilinear form s defined in thesame way as Proposition 5.3. Then, there exist a finite semialgebraic open covering { V j } ri =1 of M , Nash maps g j : V j → GL( d, R ) and nonnegative integers ≤ r + ,j ≤ d for any ≤ j ≤ q such that V j is included in some U i and t g j ( x ) s i ( x ) g j ( x ) = (cid:18) I r + ,j OO − I r − ,j (cid:19) for any x ∈ V j , where r − ,j = d − r + ,j and I m is the m × m identity matrix.Proof. We may assume that ξ is a trivial bundle over M without loss of generality.Consider an arbitrary prime cone α ∈ Spec r ( N ( M )). Let k ( α ) be the real closure of N ( M ) / supp( α ) under the ordering induced from the prime cone α . A nondegener-ate symmetric bilinear form s α : k ( α ) d × k ( α ) d → k ( α ) is induced from the bilinearform s by [BCR, Theorem 8.4.4, Theorem 8.5.2]. We can find h α ∈ GL( d, k ( α )) EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 23 and 0 ≤ r + ,α ≤ d with t h α s α h α = (cid:18) I r + ,α OO − I r − ,α (cid:19) by Sylvester’s law of inertia,where r − ,α = d − r + ,α . We identify Spec r ( N ( M )) with f M by [BCR, Proposi-tion 8.8.1]. There exist an open semialgebraic subset W α of M and a Nash map h : W α → GL( d, R ) with α ∈ g W α and h ( α ) = h α by [BCR, Proposition 8.8.3].The definitions of f M , g W α and h ( α ) are found in [BCR, Proposition 7.2.2, Theorem7.2.3]. Let w i : W α → R d be the i -th columns of h for all 1 ≤ i ≤ d . Apply theGram-Schmidt orthogonalization process, and we define u i and v i as follows: v = | s α ( w , w ) | − / w , v = | s α ( u , u ) | − / u , where u = w − s α ( w , v ) s α ( v , v ) v , · · · v d = | s α ( u d , u d ) | − / u d , where u d = w d − d − X i =1 s α ( w d , v i ) s α ( v i , v i ) v i .Consider the semialgebraic open set V α := { x ∈ W α | s α ( w ( x ) , w ( x )) = 0 , s α ( u ( x ) , u ( x )) = 0 , . . . , s α ( u d ( x ) , u d ( x )) = 0 } .The maps u i and v i are Nash maps on V α for all 1 ≤ i ≤ d . We have s α ( w i ( α ) , w j ( α )) = ± δ ij for all 1 ≤ i, j ≤ d by the definition of w i , where δ ij denotes the Kroneckerdelta. We can inductively show that s α ( u i ( α ) , u i ( α )) = ± k ( α ). Hence, thepoint α is contained in the set f V α . Define g : V α → GL( d, R ) so that the i -th rowof g ( x ) is v i ( x ) for any 1 ≤ i ≤ d . We have t g ( x ) s ( x ) g ( x ) = (cid:18) I r + ,α OO − I r − ,α (cid:19) forany x ∈ V α .The family { f V α } α ∈ Spec r ( N ( M )) is an open covering of Spec r ( N ( M )). SinceSpec r ( N ( M )) is compact by [BCR, Corollary 7.1.13], a finite subfamily { f V j } rj =1 of { f V α } α ∈ Spec r ( N ( M )) covers Spec r ( N ( M )). The family { V j } rj =1 covers M by [BCR,Proposition 7.2.2]. By the definition of V j , there exists a Nash map g j : V j → GL( d, R ) with t g j ( x ) s ( x ) g j ( x ) = (cid:18) I r + ,j OO − I r − ,j (cid:19) for any x ∈ V j and some0 ≤ r + ,j ≤ d , where r − ,j = d − r + ,j . (cid:3) Theorem 6.9 has several useful corollaries.
Corollary 6.10.
There exist a semialgebraic open covering { U i } pi =1 of SGL( d, R ) and Nash maps g i : U i → GL( d, R ) and ≤ r + ,i ≤ d such that t g i ( s ) sg i ( s ) = (cid:18) I r + ,i OO − I r − ,i (cid:19) for any s ∈ U i and ≤ i ≤ p , where r − ,i = d − r + ,i .Proof. Consider the trivial bundle ǫ d SGL( d, R ) over SGL( d, R ) of rank d . Let π :SGL( d, R ) × R d → SGL( d, R ) and π : SGL( d, R ) × R d → R d be the naturalprojections. Define a Nash bilinear form b : ǫ d SGL( d, R ) ⊕ ǫ d SGL( d, R ) → R by b ( v, w ) = t π ( v ) Sπ ( w ) for any v, w ∈ SGL( d, R ) × R d with S = π ( v ) = π ( w ). The corollaryis obtained by applying Theorem 6.9 to ǫ d SGL( d, R ) and b . (cid:3) Corollary 6.11.
Consider a definable C r manifold M , where r is a nonnegativeinteger. Let ( ξ, s ) be a definable C r bilinear space over M of rank d . There exista finite definable open covering { V i } qi =1 of M and ≤ r + ,i ≤ d such that therestriction of the bilinear space ( ξ, s ) to V i is definably C r isotropic to a trivialbilinear space of type ( r + ,i , r − ,i ) for any ≤ i ≤ q . Here, r + ,i = d − r − ,i .Proof. We may assume that ξ is a trivial bundle over M without loss of generality.Let b : M → SGL( d, R ) be the definable C r map induced from the bilinear form s .Let { U i } pi =1 be a finite semialgebraic open covering of SGL( d, R ) given in Corollary6.10. Set V i = b − ( U i ). The restriction of ξ to V i is obviously definably C r isotropicto the trivial bilinear space of type ( r + ,i , r − ,i ) for some 0 ≤ r + ,i ≤ d and r − ,i = d − r + ,i . (cid:3) Lemma 6.12.
Consider a definable C r manifold M , where r is a nonnegativeinteger. Let U and V be definable open subsets of M with M = U ∪ V . Let ( ξ, s ) bea definable C r bilinear space over M of rank d . We have a definable C r isometry ϕ : ( ǫ r + + r − U , r + h i ⊥ r − h− i ) → ( ξ, s ) | U , a positive definite definable bilinearspace ( ν + , s | ν + ) of rank r + over V and a negative definite definable bilinear space ( ν − , s | ν − ) of rank r − over V with ν + ⊕ ν − = ξ | V . Then, there exists a decomposition ξ = ξ + ⊕ ξ − by definable C r vector subbundles ξ + and ξ − of ξ over M such thatthe restrictions s | ξ + and s | ξ − are positive and negative definite, respectively.Proof. Set d = r + + r − . Let σ : U ∩ V → Gr ( d, r + ) be the definable C r map givenby ϕ − (( ν + ) x ) = { x } × σ ( x ). Here, Gr ( d, r + ) denotes the Grassmanian of r + -dimensional subspaces of an d -dimensional vector space. Note that the restrictionof the bilinear form r + h i ⊥ r − h− i to the vector space σ ( x ) is positive definitefor any x ∈ U ∩ V . Let M ( r − , r + ) be the set of all real-valued r − × r + matrices.Set V = { V ∈ Gr ( d, r + ) | V ∩ ( { } × R r − ) = { }} . Consider the biregularisomorphism ψ : M ( r − , r + ) → V given by ψ ( θ ) = { ( v, θv ) ∈ R d | v ∈ R r + } andset Ω = { θ ∈ M ( r − , r + ) | k θv k r − < k v k r + for any 0 = v ∈ R r + } . Remember that k · k r + and k · k r − denote the Euclidean norm in R r + and R r − , respectively. Thesemialgebraic set Ω is convex. We can immediately show that r + h i ⊥ r − h− i ispositive definite on the vector space ψ ( θ ) if and only if θ ∈ Ω. Let λ and µ be adefinable C r partition of unity on M subordinate to the cover M = U ∪ V givenby Lemma 2.6. Define the subbundle ξ + of ξ as follows:( ξ + ) x = ϕ ( { x } × ψ ( µ ( x ) ψ − ( σ ( x ))) if x ∈ U ∩ V , ϕ ( { x } × ( R r + × { } )) if x ∈ U \ U ∩ V and( ν + ) x if x ∈ V \ U ∩ V .Since θ is convex, we have µ ( x ) ψ − ( σ ( x )) ∈ Ω for any x ∈ U ∩ V ; hence, therestriction s | ξ + of s to ξ + is positive definite. We can construct ξ − in the sameway. (cid:3) Corollary 6.13.
Consider a definable C r manifold M , where r is a nonnegativeinteger. Let ( ξ, s ) be a definable C r bilinear space over M of rank d . Then, thereexists a decomposition ξ = ξ + ⊕ ξ − by definable C r vector subbundles ξ + and ξ − of ξ over M such that the restrictions s | ξ + and s | ξ − of the bilinear form s to ξ + and ξ − are positive and negative definite, respectively. The subbundles ξ + and ξ − areuniquely determined up to definable C r isomorphism. EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 25
Proof.
By Corollary 6.11, there exist a finite definable open covering { V i } qi =1 of M and 0 ≤ r i ≤ d such that the restriction of the bilinear space ( ξ, s ) to V i isdefinably C r isotropic to a trivial bilinear space of type ( r i , s i ) for any 1 ≤ i ≤ q .Here, s i = d − r i . The existence of decomposition follows from Lemma 6.12 andinduction.We next show the uniqueness. Let ξ = ξ ′ + ⊕ ξ ′− be another decomposition. Let i : ξ ′ + → ξ be the natural inclusion and π : ξ = ξ + ⊕ ξ − → ξ + be the naturalprojection. The composition π ◦ i : ξ ′ + → ξ + is a definable C r isomorphism. Hence, ξ + is unique up to definable C r isomorphism. (cid:3) We return to the proof of the homotopy theorem for definable C r bilinear spaces.The proof is similar to that of Theorem 6.4. Lemma 6.14.
Let U be a definable open set, and W be a definable open subset of U × R containing U × [0 , . Let ϕ , ϕ , ϕ : U → [0 , be definable C r functions with ϕ < ϕ < ϕ on U . We consider a definable C r bilinear space ( ξ = ( E, p, W ) , s ) over W . Define definable sets B and B as follows: B = { ( x, t ) ∈ U × [0 , | ϕ ( x ) ≤ t ≤ ϕ ( x ) } and B = { ( x, t ) ∈ U × [0 , | ϕ ( x ) ≤ t ≤ ϕ ( x ) } .For i = 1 , , let U i be a definable open subset of W such that B i ⊂ U i and therestriction ( ξ, s ) | U i of ( ξ, s ) to U i is definably C r isotropic to a trivial bilinear spaceover U i of type ( r + , r − ) . Then, there exists a definable open subset W ′ of W suchthat B ∪ B ⊂ W ′ and the restriction ( ξ, s ) | W ′ of ( ξ, s ) to W ′ is definably C r isotropic to a trivial bilinear space over W ′ of type ( r + , r − ) .Proof. We may assume that U is connected without loss of generality. Let ξ | U i =( E i , p i , U i ), and let u i : U i × R d → E i be a definable C r isometry for i = 1 , W ′ , B and u in the same way as Lemma 6.2. We showed that the map u is a definable continuous isomorphism between the restriction ξ | W ′ of ξ to W ′ and a trivial bundle in Lemma 6.2. Set J ( r + , r − ) = (cid:18) I r + OO − I r − (cid:19) . We have s ( u i (( x, t ) , v ) , u i (( x, t ) , w )) = t vJ ( r + , r − ) w for any ( x, t ) ∈ U i and v, w ∈ R d . Wewant to show that the map u is a definable C isometry between ( ξ, s ) | W ′ and( ǫ dW ′ , r + h i ⊥ r − h− i ), where d = r + + r − . We have only to show that s ( u (( x, t ) , v ) , u (( x, t ) , w )) = t vJ ( r + , r − ) w for any ( x, t ) ∈ W ′ and v, w ∈ R d . When t ≤ ϕ ( x ), it is obvious because u = u . Since u (( x, t ) , v ) = u (( x, t ) , g ( x ) v ) on B , we have u (( x, ϕ ( x )) , v ) = u (( x, ϕ ( x )) , g ( x ) v ). We get t vJ ( r + , r − ) w = s ( u (( x, ϕ ( x )) , v ) , u (( x, ϕ ( x )) , w ))= s ( u (( x, ϕ ( x )) , g ( x ) v ) , u (( x, ϕ ( x )) , g ( x ) w ))= t v t g ( x ) J ( r + , r − ) g ( x ) w .We have shown that t vJ ( r + , r − ) w = t v t g ( x ) J ( r + , r − ) g ( x ) w for any x ∈ U . When t > ϕ ( x ), we finally obtain s ( u (( x, t ) , v ) , u (( x, t ) , w )) = s ( u (( x, t ) , g ( x ) v ) , u (( x, t ) , g ( x ) w ))= t v t g ( x ) J ( r + , r − ) g ( x ) w = t vJ ( r + , r − ) w . There exists a semialgebraic open subset U of SGL( d, R ) and a Nash map ˜ g : U →
GL( d, R ) such that J ( r + , r − ) ∈ U and t ˜ g ( S ) S ˜ g ( S ) = J ( r + , r − ) for any S ∈ U by Corollary 6.10. Take a definable C r isomorphism ˜ u : ξ | W ′ → ǫ dW ′ sufficientlyclose to u as in the proof of Corollary 4.4. Define a definable C r bilinear form ˜ s over the trivial bundle ǫ dW ′ by ˜ s ((( x, t ) , v ) , (( x, t ) , w )) = s (˜ u (( x, t ) , v ) , ˜ u (( x, t ) , w ))for any ( x, t ) ∈ W ′ and v, w ∈ R d . Let ˜ S : W ′ → SGL( d, R ) be the definable C r map naturally induced from the bilinear form ˜ s . Since ˜ u is sufficiently close to u ,the nondegenerate symmetric matrix ˜ S ( x, t ) is sufficiently close to J ( r + , r − ) for any( x, t ) ∈ W ′ . In particular, we may assume that ˜ S ( x, t ) ∈ U for any ( x, t ) ∈ W ′ .Define a definable C r isomorphism ψ : W ′ × R d → W ′ × R d by ψ (( x, t ) , v ) =(( x, t ) , ˜ g ( ˜ S ( x, t )) v ) for any ( x, t ) ∈ W ′ and v ∈ R d . The definable C r isomorphism˜ u ◦ ψ − is a definable C r isotropy between ( ξ, s ) | W ′ and a trivial bilinear space over W ′ of type ( r + , r − ). (cid:3) Lemma 6.15.
Let M be a definable C r manifold and W be a definable open subsetof M × R containing M × [0 , . We consider a definable C r bilinear space ( ξ, s ) over W . Then, there exist a finite definable open covering { U i } qi =1 of M and adefinable open subset V i of U i × R containing U i × [0 , such that the restriction ( ξ, s ) | V i of ( ξ, s ) to V i is definably C r isotropic to a trivial bilinear space.Proof. We can prove the lemma in the same way as Lemma 6.3 using Corollary6.11 and Lemma 6.14 in place of Lemma 6.2. (cid:3)
We are now ready to show the homotopy theorem for definable C r bilinear spaces. Theorem 6.16 (Homotopy theorem for definable C r bilinear spaces) . Consider adefinable C r manifold M , where r is a nonnegative integer. Let U be a definableopen subset of M × R containing M × [0 , . Define the definable C r map r : U → U by r ( x, t ) = ( x, . Let ξ = ( E, p, U ) be a definable C r vector bundle over U and s bea definable C r bilinear form over ξ . Then, shrinking U if necessary, two definable C r bilinear spaces ( ξ, s ) and r ∗ ( ξ, s ) are definably C r isometric.Proof. We may assume that M is connected without loss of generality. Thereexist a finite definable open covering { U i } qi =1 and a definable open subset V i of U i × R containing U i × [0 ,
1] such that the restriction ( ξ, s ) | V i of ( ξ, s ) to V i isdefinably C r isometric to a trivial bundle of type ( r + , r − ) by Lemma 6.15. Let h i : V i × R d → p − ( V i ) be a definable C r isometry for any 1 ≤ i ≤ q . Shrinking U if necessary, we may assume that U = S qi =1 V i .We define r i , s i , u i and ˜ u i in the same way as the proof of Theorem 6.4. Set s ∗ i ξ = ( s ∗ i E, p i , U ). The notation s ∗ i s denotes the induced bilinear form over s ∗ i ξ . Wehave shown that ˜ u i is a definable C r isomorphism between the definable C r vectorbundles s ∗ i − ξ and s ∗ i ξ in the proof of Theorem 6.4. We want to demonstrate that˜ u i is a definable C r isometry between the definable C r bilinear spaces s ∗ i − ( ξ, s ) and s ∗ i ( ξ, s ). For that purpose, we have only to show that s ∗ i s (˜ u i ( v ) , ˜ u i ( w )) = s ∗ i − ( v, w )for any v, w ∈ s ∗ i − E with p i − ( v ) = p i − ( w ). Identifying s ∗ i E with the space { (( x, t ) , v ) ∈ U × s ∗ i − E | p i − ( v ) = r i ( x, t ) } , we have s ∗ i s ((( x, t ) , v ) , (( x, t ) , w )) = s ∗ i − s ( v, w ) for any v, w ∈ s ∗ i − E with p i − ( v ) = p i − ( w ) = r i ( x, t ). We easily get s ∗ i s (˜ u i ( v ) , ˜ u i ( w )) = s ∗ i s (( r i ( x, t ) , v ) , ( r i ( x, t ) , w )) = s ∗ i − ( v, w ). We have shown that˜ u i is a definable C r isometry.Set ˜ u = ˜ u q ◦ · · · ◦ ˜ u ◦ ˜ u . The U -morphism ˜ u is a definable C r U -isometrybetween ( ξ, s ) and r ∗ ( ξ, s ). (cid:3) EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 27
The following corollary is Main Theorem 1.4.
Corollary 6.17.
Consider a definable C r manifold M , where r is a nonnegativeinteger. Let U be a definable open subset of M × R containing M × [0 , . Let (Ξ , B ) be a definable C r bilinear space over U . Then, two definable C r bilinear spaces (Ξ , B ) | M ×{ } and (Ξ , B ) | M ×{ } are definably C r isometric. Here, the notation (Ξ , B ) | M ×{ } denotes the restriction of the bilinear space (Ξ , B ) to M × { } .Proof. Immediate from Theorem 6.16. (cid:3)
Corollary 6.18.
Consider definable C r manifolds M and N , where r is a non-negative integer. Let f, g : M → N be definably C r homotopic definable C r maps.Let ( ξ, s ) be a definable C r bilinear space over N . Then, the induced definable C r bilinear spaces f ∗ ( ξ, s ) and g ∗ ( ξ, s ) are definably C r isometric.Proof. Immediate from Corollary 6.17. (cid:3)
Corollary 6.19.
Let ξ be a definable C r vector bundle over a definable C r manifold,where r is a nonnegative integer. Consider two positive definable C r bilinear forms s and s ′ over ξ . Two definable C r bilinear spaces ( ξ, s ) and ( ξ, s ′ ) are definably C r isometric.Proof. Let ξ = ( E, p, M ). Consider the definable C r vector bundle η = ( E × R , q, M × R ), where q : E × R → M × R is the definable C r map given by q ( v, t ) =( p ( v ) , t ) for any v ∈ E and t ∈ R . We next consider the definable C r bilinear form σ over η given by σ ( v, t ) = (1 − t ) s ( v ) + ts ′ ( v ). The bilinear form σ is positive definitefor any 0 ≤ t ≤
1. There exists a definable open subset U of M × R containing M × [0 ,
1] such that the restriction σ | q − ( U ) of σ to q − ( U ) is positive definite. Twodefinable C r bilinear space ( ξ, s ) and ( ξ, s ′ ) are definably C r isometric by applyingCorollary 6.17 to the bilinear space ( η, σ ) | U . (cid:3) We show counterparts of Corollary 4.4 and Theorem 4.5 for definable C r bilinearspaces. Theorem 6.20.
Let r be a nonnegative integer. Consider two definable C r bilinearspaces over a definable C r manifold which are definably C isometric. They aredefinably C r isometric.Proof. Let ( ξ, s ) and ( ξ ′ , s ′ ) be two definable C r bilinear spaces. Let ϕ : ( ξ, s ) → ( ξ ′ , s ′ ) be a definable C isometry between them. We demonstrate that ( ξ, s ) and( ξ ′ , s ′ ) are definably C r isometric. We may assume that ( ξ, s ) is the Whitney sumof a positive definite definable C r bilinear space ( ξ + , s + ) and a negative definitedefinable C r bilinear space ( ξ − , s − ) by Corollary 6.13. We may also assume that( ξ ′ , s ′ ) is the Whitney sum of a positive definite definable C r bilinear space ( ξ ′ + , s ′ + )and a negative definite definable C r bilinear space ( ξ ′− , s ′− ) in the same way. Let ι : ξ + → ξ be the natural inclusion and π ′ : ξ ′ → ξ ′ + be the natural projection. Thecomposition π ′ ◦ ϕ ◦ ι is a definable C isometry between ( ξ + , s + ) and ( ξ ′ + , s ′ + ). Wecan show that ( ξ − , s − ) and ( ξ ′− , s ′− ) are definably C isometric in the same way.Hence, we may assume that ( ξ, s ) and ( ξ ′ , s ′ ) are positive definite.Since two vector bundles ξ and ξ ′ are definably C isomorphic, they are definably C r isomorphic by Corollary 4.4. Let ψ : ξ → ξ ′ be definable C r isomorphism. Thepositive definite definable C r bilinear form s over ξ naturally induces a positivedefinite definable C r bilinear form ˜ s over ξ ′ via the isomorphism ψ . Two definable bilinear spaces ( ξ ′ , s ′ ) and ( ξ ′ , ˜ s ) are definably C r isometric by Corollary 6.19. Wehave finished the proof. (cid:3) Theorem 6.21.
Let r be a nonnegative integer. Any definable C bilinear spacesover a definable C r manifold is definably C isometric to a definable C r bilinearspace over the same manifold.Proof. The definable C bilinear space ( ξ, s ) over a definable C r manifold is theWhitney sum of a positive definite definable C bilinear space and a negative def-inite definable C bilinear space by Corollary 6.13. We may assume that the C bilinear space is positive definite without loss of generality. There exists a definable C r vector bundle ˜ ξ definably C isomorphic to ξ by Theorem 4.5. Let ϕ : ˜ ξ → ξ be a definable C isomorphism. The positive definite definable C bilinear form s over ξ naturally induces a positive definite definable C bilinear form s ′ over ˜ ξ viathe isomorphism ϕ . Especially, two definable C bilinear spaces ( ξ, s ) and ( ˜ ξ, s ′ )are definably C isometric.On the other hand, there exists a definable C r bilinear form ˜ s over ˜ ξ by Proposi-tion 5.2. Two definable bilinear spaces ( ˜ ξ, s ′ ) and ( ˜ ξ, ˜ s ) are definably C isometricby Corollary 6.19. We have finished the proof. (cid:3) Grothendieck rings and Witt rings are isomorphic
The goal of this section is Main Theorem 1.5. We first review the definitions ofthe Grothendieck ring and the Witt ring over a commutative ring.
Definition 7.1 (Grothendieck ring over a commutative ring) . Let R be a com-mutative ring. The notation Proj( R ) denotes the set of all isomorphism class offinitely generated projective R -modules. We define an equivalence relation ∼ onProj( R ) × Proj( R ). Let ( P , P ) , ( P ′ , P ′ ) ∈ Proj( R ) × Proj( R ). They are equiv-alent, namely, ( P , P ) ∼ ( P ′ , P ′ ) if there exist a nonnegative integer r and anisomorphism of R -modules P ⊕ P ′ ⊕ R r ≃ P ′ ⊕ P ⊕ R r . It is easy to check thatthe relation ∼ is an equivalence relation using the fact that a projective module isa direct summand of a free module.The set K ( R ) = Proj( R ) × Proj( R ) / ∼ is the Grothendieck ring of R . Thenotation [( P , P )] denotes the equivalence class of ( P , P ). The sum of [( P , P )]and [( P ′ , P ′ )] is given by [( P ⊕ P ′ , P ⊕ P ′ )]. The inverse of [( P , P )] is givenby [( P , P )] and denoted by − [( P , P )]. The notation [ P ] denotes the element[( P, K ( R ) is of the form [ P ] − [ P ]. The product in K ( R ) isdefined using the tensor product of modules.For instance, the Grothendieck rings K ( C r df ( R n )) are isomorphic to Z by Corol-lary 6.8 for all nonnegative integers n and r . Definition 7.2 (Witt ring over a commutative ring) . Let R be a commutative ringsuch that 2 is invertible in R . The bilinear space H ( P ) given by ( P ⊕ P ∨ , β ) forsome finitely generated projective module P is called a hyperbolic space , where P ∨ is the dual of P and β is the bilinear form given by β ( x ⊕ ϕ, y ⊕ ψ ) = ψ ( x ) + ϕ ( y ).Two bilinear spaces ( P, b ) and ( P ′ , b ′ ) over the ring R are Witt equivalent if thereare two finitely generated projective modules Q and Q ′ such that ( P, b ) ⊥ H ( Q )and ( P ′ , b ′ ) ⊥ H ( Q ′ ) are isometric. The notation W ( R ) denotes the set of all Wittequivalence classes of bilinear spaces over R . The sum of two elements [( P, b )] and[( P ′ , b ′ )] of W ( R ) is defined as [( P, b ) ⊥ ( P ′ , b ′ )]. The product of them are defined EFINABLE VECTOR BUNDLES AND BILINEAR SPACES 29 as [(
P, b ) ⊗ A ( P ′ , b ′ )]. They are well-defined by [MH, Theorem I.7.3]. The additiveinverse of [( P, b )] is [( P, − b )] by [BCR, Theorem 15.1.4].The following theorem is Main Theorem 1.5. The proof is similar to [BCR,Theorem 15.1.2]. Theorem 7.3.
Let M be a definable C r manifold, where r is a nonnegative inte-ger. The Grothendieck ring K ( C r df ( M )) of the ring C r df ( M ) is isomorphic to theWitt ring W ( C r df ( M )) of the same ring. The Grothendieck rings K ( C r df ( M )) and K ( C df ( M )) are also isomorphic.Proof. Set A = C r df ( M ). We first define a ring homomorphism ∆ : K ( A ) → W ( A ).Since K ( A ) is generated by [ P ], where P is a finitely generated projective A -module, we have only to define ∆([ P ]). There exists a definable C r vector bundle ξ over M such that Γ( ξ ) is isomorphic to P by Theorem 3.9. There exists apositive definite definable C r bilinear form σ over ξ by Proposition 5.2. A bilinearform b over P is the bilinear form induced from the bilinear form σ over ξ . Theequivalent class [( P, b )] in W ( A ) does not depend on the choice of a positive definitedefinable C r bilinear form σ on ξ by Corollary 6.19. We set ∆([ P ]) = [( P, b )]. When[ P ] = [ P ′ ] in K ( A ), there exists a nonnegative integer s and an isomorphism ϕ : P ′ ⊕ A s ≃ P ⊕ A s . Let ˜ b be a positive definite bilinear form over P ⊕ A s and ϕ ∗ ˜ b be a positive definable bilinear form over P ′ ⊕ A s induced by ˜ b via ϕ .Two bilinear space ( P ⊕ A s , b ) and ( P ′ ⊕ A s , ϕ ∗ ˜ b ) are isometric. Let b ′ be anotherpositive definite bilinear form over P ′ . We get [( P ′ , b ′ )] = [( P ′ , ϕ ∗ ˜ b | P ′ )] = [( P ′ ⊕ A s , ϕ ∗ ˜ b )] − [( A s , ϕ ∗ ˜ b | A s )] = [( P ⊕ A s , ˜ b )] − [( A s , ˜ b | A s )] = [( P, ˜ b | P )] = [( P, b )] byCorollary 6.19. Therefore, the map ∆ is well-defined. Note that b ⊥ b ′ and b ⊗ A b ′ are positive definite when b and b ′ are positive definite. We have shown that ∆ isa ring homomorphism.We next define a map ∇ : W ( A ) → K ( A ). Let ( P, b ) be a bilinear space over A . There exists a definable C r bilinear space ( ξ, s ) over M such that Γ b ( ξ, s ) isisometric to ( P, b ) by Theorem 5.10. There exists a unique decomposition ( ξ, s ) =( ξ + , s + ) ⊥ ( ξ − , s − ) up to isomorphism by Corollary 6.13 such that ( ξ + , s + ) and( ξ − , s − ) are positive and negative definite definable C r bilinear spaces over M ,respectively. We set ∇ ([( P, b )]) = [Γ( ξ + )] − [Γ( ξ − )]. It is obvious that ∇ ([( P, b ) ⊥ ( P ′ , b ′ )]) = ∇ ([( P, b )]) + ∇ ([( P ′ , b ′ )]). We show that ∇ is well-defined. Let ( P ′ , b ′ )be another bilinear space over A with [( P, b )] = [( P ′ , b ′ )] in W ( A ). There existfinitely generated projective A -modules Q and Q ′ such that ( P, b ) ⊥ H ( Q ) and( P ′ , b ′ ) ⊥ H ( Q ′ ) are isometric. Let ˜ b be a positive definite bilinear form over Q .It exists by Theorem 3.9, Proposition 5.2 and Theorem 5.10. The orthogonal sum( Q, ˜ b ) ⊥ ( Q, − ˜ b ) is isomorphic to H ( Q ) by the proof of [BCR, Theorem 15.1.4]. Wehave ∇ ([( P, b )]) = ∇ ([( P, b )]) + ∇ ([( Q, ˜ b )]) + ∇ ([( Q, − ˜ b )]) = ∇ ([( P, b ) ⊥ H ( Q )]) = ∇ ([( P ′ , b ′ ) ⊥ H ( Q ′ )]) = ∇ ([( P ′ , b ′ )]). It shows that ∇ is well-defined.We can easily show that ∇ ◦ ∆ is the identity map. We next show that ∆ ◦ ∇ is the identity map. Let ( P, b ) be a bilinear space over A . We define a definable C r bilinear spaces ( ξ, s ), ( ξ + , s + ) and ( ξ − , s − ) over M in the same way as above.We have ∆ ◦ ∇ ([( P, b )]) = ∆([Γ( ξ + )]) − ∆([Γ( ξ − )]) = [Γ b ( ξ + , s ′ + )] − [Γ b ( ξ − , s ′− )] forsome positive definite definable C r bilinear forms s ′ + and s ′− by Theorem 5.10. Wehave [Γ b ( ξ + , s ′ + )] = [Γ b ( ξ + , s + )] and [Γ b ( ξ − , s ′− )] = [Γ b ( ξ − , − s − )] by Corollary 6.19.We get ∆ ◦ ∇ ([( P, b )]) = [Γ b ( ξ + , s ′ + )] − [Γ b ( ξ − , s ′− )] = [Γ b ( ξ + , s + )] + [Γ b ( ξ − , s − )] =[Γ b ( ξ + , s + ) ⊥ Γ b ( ξ − , s − )] = [Γ b ( ξ + ⊕ ξ − , s + ⊥ s − )] = [Γ b ( ξ, s )] = [( P, b )]. 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