Rational homotopy type of the space of immersions of a manifold in an euclidian space
aa r X i v : . [ m a t h . A T ] J un Rational homotopy type of the space ofimmersions of a manifold in an euclidian space
Abdoul Kader Yacouba Barma
Abstract
Let M be a simply-connected m dimensional manifold of finite type and k a positifinteger. In this paper we show that the rational Betti numbers of each component ofthe space of immersions of M in R m + k , have polynomial growth. As consequence, wededuce that, if M is a manifold with Euler characteristic χ ( M ) ≤ − , the Betti numbersof smooth embeddings, Emb (cid:0) M, R m + k (cid:1) , have exponential growth if k ≥ m + 1 .The main tool of this work is the construction of an explicit model of the space ofimmersions. Throughout the paper we fix a smooth differential manifold, M , of dimension m . We assumethat M is simply-connected and of finite type. An immersion of M in R d is a smooth map f from M to R d such that the differential of f at each x ∈ M is of rank m . The space ofimmersions of M in R d is the set Imm (cid:0) M, R d (cid:1) = { f : M R d | f is an immersion } , equipped with the weak C − topology.In this paper we determine the rational homotopy type of the path components of Imm (cid:0) M, R m + k (cid:1) .When k is odd and = 0 , that rational homotopy is easy to describe: Theorem 1.1 (Theorem 4.2) . Let M be a simply connected m -manifold of finite type. Andlet k = 2 s +1 with s = 0 . Suppose that p i ( τ M ) = 0 , ∀ i ≥ s +1 ,then each connected componentof Imm (cid:0) M, R m + k (cid:1) has the rational homotopy type of a product of Eilenberg-Maclane spaces,which depends only on the Betti numbers of M and on k . When the codimension, k , is even and ≥ , the rational homotopy type of the path com-ponents of Imm (cid:0) M, R m + k (cid:1) is more complicated to describe. In that case we obtain thefollowing results. 1 heorem 1.2 (Theorem 4.3) . Let M be a simply connected m -manifold of finite type and k an even integer. Then: • If k ≥ m + 1 , Imm (cid:0) M, R m + k (cid:1) is connected and its rational homotopy type is of theform Imm (cid:0) M, R m + k (cid:1) ≃ Q M ap (cid:0)
M, S k (cid:1) × K, where M ap (cid:0)
M, S k (cid:1) is the mapping space of maps from M into S k and K is a productof Eilenberg-Maclane spaces which depend only on the Betti numbers of M . • If ≤ k ≤ m , Suppose that p i ( τ M ) = 0 , ∀ i ≥ s then, the connected component of f in Imm (cid:0) M, R m + k (cid:1) has the rational homotopy type of a product Imm (cid:0) M, R m + k , f (cid:1) ≃ Q M ap (cid:16)
M, S k , ˜ f (cid:17) × K, where Imm (cid:0) M, R m + k , f (cid:1) is the component of Imm (cid:0) M, R m + k (cid:1) which contains f , K is the product of Eilenberg-Maclane spaces which depends only on the Betti numbersof M , M ap (cid:16)
M, S k , ˜ f (cid:17) is the path component of the mapping space of M in S k containingthe map ˜ f induced by the immersion f . As the special case of this theorem we have.
Theorem 1.3 (Corollary 4.4) . If H k ( M, Q ) = 0 , then all the path connected of Imm (cid:0) M, R m + k (cid:1) have the same rational homotopy type, and we can take ˜ f to be the constant map. As a consequence of Theorem 4.2 and Theorem 4.3 we have the following corollary
Theorem 1.4 (Corollary 4.5) . If M is simply connected manifold of dimension m andk an integer such that k ≥ . Then the rational Betti numbers of each component of Imm (cid:0) M, R m + k (cid:1) have polynomial growth. From this result we deduce the following.
Theorem 1.5 (Remark 4.6) . Let M be a simply connected m-manifold of finite type withEuler characteristic χ ( M ) ≤ − , and k = 0 an integer. Then if k ≥ m +1 , the Betti numbersof the space of smooth embedding of M into R m + k , Emb (cid:0) M, R m + k (cid:1) , have exponential growth. In [6], M.W Hirsh proves that
Imm (cid:0) M, R m + k (cid:1) has the same homotopy type of the space ofsections of a bundle Framed m ( τ M ) (see 2.1 for the definition of Framed m ( τ M )) whose fiberis the Stiefel manifold V m (cid:0) R m + k (cid:1) of m-frames in R m + k . That is, Imm (cid:0) M, R m + k (cid:1) ≃ Γ (Framed m ( τ M )) . Framed m ( ξ ) (see Propositon 3.5 for its construction) from which we deduce the followingresult. Theorem 1.6 (Proposition 3.7) . Let ξ be a vector bundle of rank m and p i ( ξ ) , ≤ i ≤ m ,its Pontryagin classes. If p i ( τ M ) = 0 ∀ i ≥ k , then Framed m ( ξ ) is rationally trivial. Outline of the paper - In Section 2 we will construct the framed bundle associated to a vector bundle. Wewill start with the construction of the principal bundle associated to a vector bundle,afterward we will construct a framed bundle associated to a vector bundle and at theend we will prove that the framed framed bundle is a pullback of Borel bundle.- In Section 3 we will construct the model of framed bundle. We will start with a reviewof the construction of the model of bundle, later we recall the model of Borel bundlein general case. We will prove that under certains conditions this bundle is rationaltrivial.- In Section 4 we will prove Theorem 4.2, and Theorem 4.3 and we will prove the polyno-mial growth of the space of embeddings. We will start this section with a quick reviewof the Small Hirsch theorem.
Acknowlegment .The present work is a part of my thesis. I would like to thank my advisor, Pascal Lambrechts,for making it possible throughout his advices and encouragements. I also thank Yves Felixfor suggesting that Corollary 4.4 is true without a codimension condition.
In this section we study the framed bundle associated to a vector bundle. First we recall itsconstruction and give some of its property which allow us to identify it with a pullback ofcertain Borel bundle.
We start with some generality. 3 .1.1 GL ( m ) prinicipal bundle Let ξ = ( E, π, B ) be a real vector bundle of rank m . For b ∈ B , we denote by E b ξ the fiberof ξ over b : E b ξ := π − ( b ) . We associate to ξ the GL ( m ) -principal bundle P ξ = ( P m ξ, p, B ) . Where P m ξ is the set ofpairs ( b, ( v · · · , v m )) with b ∈ B and ( v , · · · , v m ) is an m − frames in E b . P m ξ is topologizeas a subspace of the Whitney sum E ⊕ · · · ⊕ E = mE . p denote the projection map from P m ξ to B , which send the frame ( b, ( v · · · , v m )) on b . Note that GL ( m ) acts freely on the rightof P m ξ . The action is defined as follow. Let g = { g ij } ∈ GL ( m ) and ( b, ( v · · · , v m )) ∈ P m ξ . ( b, ( v · · · , v m )) g = ( b, ( w , · · · , w m )) where w j = X i v i g ij . This make
P ξ = ( P m ξ, p, B ) into a principal bundle over B . Let W be a vector space of dimension m + k . We denote by V m W the space of m framesin W . V m W is a right GL ( m ) space. By consequence GL ( m ) acts on P m ξ × V m W . Set by P m ξ × GL ( m ) V m W the orbit of P m ξ × V m W under this action, and q : P m ξ × V m W −→ P m ξ × GL ( m ) V m W the corresponding projection. q determine a map ρ : P m ξ × GL ( m ) V m W −→ B via the commutative diagram P m ξ × V m W q / / pr (cid:15) (cid:15) P m ξ × GL ( m ) V m W ρ (cid:15) (cid:15) P m ξ p / / B Proposition 2.1. [5, Proposition 1, p 198] There is an unique topology on P m ξ × GL ( m ) V m W such that Framed m ( ξ ) = (cid:0) P m ξ × GL ( m ) V m W, ρ, B (cid:1) is a locally trivial bundle with fiber V m W . Definition 2.2.
Framed m ( ξ ) is called framed bundle with fiber V m W associated to ξ . P = ( EGL ( m ) , p , BGL ( m )) the universal GL ( m ) principal bundle, where EGL ( m ) is a contractile space and BGL ( m ) = EGL ( m ) GL ( m ) . By the theorem of classification ofprincipal bundle there exist a map f : B −→ BGL ( m ) such that P m ξ is the pullback of P via f . f is called the classifying map. Considerthe Borel bundle (cid:0) EGL ( m ) × GL ( m ) V m W, ˜ p , BGL ( m ) (cid:1) with fiber V m W associated to P =( EGL ( m ) , p , BGL ( m )) . Then we have the following result. Proposition 2.3.
Framed m ( ξ ) is the pullback of (cid:0) EGL ( m ) × GL ( m ) V m W, ˜ p , BGL ( m ) (cid:1) via f .Proof. Since
P ξ = ( P m ξ, p, B ) is the pullback of P = ( EGL ( m ) , p , BGL ( m )) via g . Thismeans that, if T is a space and p : T −→ EGL ( m ) , p : T −→ B are maps such that p ◦ p = g ◦ p then there is a unique map: Φ : T −→ P m ξ such that the following diagram is commutative T p & & p (cid:29) (cid:29) Φ ! ! P m ξ F / / p (cid:15) (cid:15) EGL ( m ) p (cid:15) (cid:15) B f / / BGL ( m ) Therefore, if T ′ is a space, p ′ : T −→ EGL ( m ) × GL ( m ) V m W and p ′ : T −→ B two maps such that ˜ p ◦ p ′ = g ◦ p ′ there is a unique map Φ ′ : T ′ −→ P m ξ × GL ( m ) V m W T ′ p ′ + + p ′ $ $ Φ ′ ' ' P m ξ × GL( m ) V m W ′ F / / ρ (cid:15) (cid:15) EGL ( m ) × GL( m ) V m W ˜ p (cid:15) (cid:15) B f / / BGL ( m ) with F ′ ([ X, v ]) = [ F ( x, λ ) , v ] ,ρ ([ X, v ]) = p ( X )˜ p ([ e, v ]) = p ( e ) . This proves that
Framed m ( ξ ) is the pullback of (cid:0) EGL ( m ) × GL ( m ) V m W, ˜ p , BGL ( m ) (cid:1) via f . In this section we construct the rational model of de framed bundle associated to a vectorbundle construct in Section 2. We start with some notions of rational homotopy theory. Aferwe give model of the Borel bundle constructed by K. Matsuo in [9], and give this constructionin a particular case. And finally we will use this model of Borel bundle to construct the modelof a framed bundle.
In this section we recall some classics notions of rational homotopy theory.
In all this paper we wil use the standard tools of rational homotopy theory, following thenotation and terminologie of [3]. Recall that A pl is the Sullivan- de Rham contravariantfunctor and that for a simply connected space of finite type , A pl ( X ) is a CGDA(commutativedifferential graded algebra). Any CGDA weakly equivalent to A pl ( X ) is called a CGDAmodel of X and it completely encodes the rational homotopy of X .6 .1.2 Model of bundle Let ξ = ( E, p, B ) be a bundle of fiber F . Assume that F, B are simply connected CWcomplexes of finite type. If ( A, d A ) is a CGDA model of B . In [3, Theorem 15.3] it’s provedthat we can form a commutative diagramme A pl ( B ) A pl ( p ) / / A pl ( E ) / / A pl ( F )( A, d A ) m O O (cid:31) (cid:127) / / ( A ⊗ Λ V, d ) φ / / O O (cid:0) Λ V, ¯ d (cid:1) ¯ φ O O where φ , ¯ φ and m are quasi isomorphism. ( d | A = d A dV ∈ A ⊗ Λ V. Definition 3.1.
The inclusion ( A, d A ) ֒ → ( A ⊗ Λ V, d ) is called the model of ξ = ( E, p, B ) . We will said that its rationally trivial if ( A ⊗ Λ V, d ) ≃ ( A, d A ) ⊗ (Λ V, d ) . Remark 3.2. If ξ Q = ( E Q , p Q , B Q ) is the rationalization of ξ . If ξ is rationally trivial, then ξ Q is trivial. And we have the following result proved in [8]
Proposition 3.3.
Let ξ = ( E, p, B ) be a bundle of fibre F with F and B simply connected.Set ξ Q = ( E Q , p Q , B Q ) the rationalization of ξ and ξ ( Q ) = (cid:0) E ( Q ) , p ( Q ) , B ( Q ) (cid:1) its fiberwiserationalization. If ξ Q is trivial then ξ ( Q ) is trivial.Proof. It comes from the unicity of p ( Q ) : E ( Q ) −→ B up to homotopy pvoved by LIerena [7,Proposition 6.1], and the fact that it’s the pullback of p Q : E Q −→ B Q along B −→ B Q G is a simply connected Lie group.Let P = ( P, p, B ) be a G -principal bundle. It is pull back from the universal bundle ( EG, p, BG ) via a classifying map φ : B −→ BG , and then the associated bundle ( P × G F, p, B ) is the pullback of ( EG × G F, p, BG ) via φ .Suppose that F is a homogenous space, that is , F = HK , where H is a compact Lie group7imply connected and K a simply connected closed subgroup of H . If there is a morphism ofLie group µ : G −→ H then, HK admets the action of G defined by ghK = ( µ ( g ) h ) K . Since G, H, K are simply connected
BG, BH , and BK are simply connected and by the result ofBorel [2, Theorem 19.1] their rational cohomology are finitely generated polynomial algebras Λ V G , Λ V H and Λ V K . In particular thier minimals models are given by: (Λ V G , , (Λ V H , and (Λ V K , . Consider Bµ : BG −→ BH and Bν : BK −→ BH the morphisms inducedby µ and the inclusion of ν of K in H . K. Matsuo prove in [9] that the model of : EG × µG HK −→ BG is of the form (Λ V G , ֒ → (Λ V G ⊗ Λ V K ⊗ Λ sV H , d ) where ( sV H ) i = V i +1 H and ( d | V G = d | V K = 0 dsv = ( Bµ ) ∗ ( v ) − ( Bν ) ∗ ( v ) for sv ∈ sV H . Let φ ∗ : (Λ V G , −→ ( A, d A ) be the Sullivan representative of φ . Then combining and [4,Theorem 270] we obtain the following result. Proposition 3.4.
The model of P × µG HK −→ B is of the form ( A, d A ) ֒ → ( A ⊗ Λ V K ⊗ Λ sV H , D ) where D | A = d A D | V K = 0 Dsv = φ ∗ (( Bµ ) ∗ ( v )) − ( Bν ) ∗ ( v ) . In this section we will construct a model of framed bundle associated to the vector bundle ξ = ( E, p, B ) . We suppose that B is simply connected. In this case, the GL ( m ) structureon its associated principal bundle P ξ can be restreinte to SO ( m ) . We suppose also that W = R m + k with k ≥ , in this case we have an SO ( m ) equivariant homeomorphism between SO ( m + k ) SO ( k ) and V m (cid:0) R m + k (cid:1) . This homeomorphism induce an isomorphism of bundle over B between Framed m ( ξ ) = (cid:0) P m ξ × SO ( m ) V m (cid:0) R m + k (cid:1) , ρ, B (cid:1) and (cid:16) P m ξ × SO ( m ) SO ( m + k ) SO ( k ) , ˜ ρ, B (cid:17) .We will use the following notation for certains cohomology classes of BSO ( m ) , BSO ( k ) and BSO ( m + k ) : 8 p i ∈ H i ( BSO ( m ) , Q ) ; i : 1 · · · , [ m − ] is the universal Pontryagin classes, e m ∈ H m ( BSO ( m ) , Q ) , for m even is Euler classes, • b i ∈ H i ( BSO ( k ) , Q ) ; i : 1 · · · , [ k − ] is the universal Pontryagin classes, e k ∈ H k ( BSO ( k ) , Q ) , for k even is Euler classes, • c i ∈ H i ( BSO ( m + k ) , Q ) ; i : 1 · · · , [ k − ] is the universal Pontryagin classes, e m + k ∈ H m + k ( BSO ( m + k ) , Q ) , for m + k even is Euler classes. Proposition 3.5.
Let ξ := ( E, π, B ) be a vectoriel bundle with fibre R m and ( A, d A ) ≃ −→ A pl ( B ) a CGDA model of B. If p i ( ξ ) ∈ A ∩ kerd A are the Pontrjagin classes of ξ , then the model of ˜ ρ : P m ξ × SO ( m ) SO ( m + k ) SO ( k ) −→ B is of the form : • if m = 2 l + 1 , k = 2 s + 1( A, d A ) −→ ( A ⊗ Λ( x s +1 , · · · , x l + s , ¯ e m + k − ) , D ) Da = d A a if a ∈ ADx i = p i ( ξ ) if i ≤ m , Dx i = 0 if i > m D ¯ e m + k − = 0 , • if m = 2 l , k = 2 s + 1 ( A, d A ) −→ ( A ⊗ Λ( x s +1 , · · · , x l + s ) , D ) ( Da = d A a if a ∈ ADx i = p i ( ξ ) if i ≤ m , Dx i = 0 if i > m • if m = 2 l + 1 , k = 2 s ( A, d A ) −→ ( A ⊗ Λ( x s , · · · , x l + s , e k ) , D ) Da = d A a if a ∈ ADx s = e k + p s ( ξ ) Dx i = p i ( ξ ) if i ≤ m, Dx i = 0 if i > mDe k = 0 if m = 2 l , k = 2 s ( A, d A ) −→ ( A ⊗ Λ( x s , · · · , x l + s − , ¯ e m + k − , e k ) , D ) Da = d A a if a ∈ ADx s = e k + p s ( ξ ) Dx i = p i ( ξ ) if s < i ≤ m, Dx i = 0 if i > mD ¯ e m + k − = De k = 0 . Proof.
For the proof we consider the case m = 2 l + 1 and k = 2 s + 1 . The proofs of theothers cases are similar.By 3.4, the model of ˜ ρ : P m ξ × SO ( m ) V m (cid:0) R m + k (cid:1) −→ B is of the form ( A, d A ) ֒ → ( A ⊗ Λ ( x , · · · , x l + s , ¯ e m + k − , b , · · · , b s ) , D ) where D | A = d A Db i = 0 Dx i = φ (( Bµ ) ∗ ( c i )) − ( Bν ) ∗ ( c i ) . By [10, Theorem 7.1] ( Bµ ) ∗ ( b i ) = p i and ( Bν ) ∗ ( c i ) = b i . On the other hand the rationalmodel of f is completely determined by the Pontryagin classes of ξ denoted p i ( ξ ) . That is φ : (Λ( p , · · · , p l ) , −→ ( A, d A ) defined by φ ( p i ) = p i ( ξ ) . Since ξ if of rank m , then p i ( ξ ) = 0 for i > m . Consequently, we have D | A = d A Db i = 0 Dx i = p i ( ξ ) − b i . Now let ( A, d A ) −→ ( A ⊗ Λ( x s +1 , · · · , x l + s , ¯ e m + k − ) , D ) Da = d A a if a ∈ ADx i = p i ( ξ ) if i ≤ m , Dx i = 0 if i > m D ¯ e m + k − = 0 .
10e want to prove that it’s the model of ( A, d A ) ֒ → ( A ⊗ Λ ( x , · · · , x l + s , ¯ e m + k − , b , · · · , b s ) , D ) where D | A = d A Db i = 0 Dx i = p i ( ξ ) − b i . for this we define Φ : ( A ⊗ Λ ( x , · · · , x l + s , ¯ e m + k − , b , · · · , b s ) , D ) −→ ( A ⊗ Λ( x s +1 , · · · , x l + s , ¯ e m + k − ) , D ) by Φ | A = id Φ ( x i ) = 0 for ≤ i ≤ s Φ ( x i ) = x i for i > s Φ ( b i ) = 0Φ ( p i ) = p i ( ξ ) it’s clear that Φ commutes with the differentials and define a quasi-isomorphism. This endthe proof. Remark 3.6.
From this construction we deduce the minimal model of the Stiefel manifoldof m frame in R m + k . Thus we have • if k = 2 s + 1 we have V m (cid:0) R m + k (cid:1) ≃ Q ( (Λ ( x s +1 , · · · , x l + s , ¯ e m + k − ) , if m = 2 l + 1 , (Λ ( x s +1 , · · · , x l + s ) , if m = 2 l • if m = 2 l + 1 and k = 2 s we have V m (cid:0) R m + k (cid:1) ≃ Q (Λ ( x s , · · · , x l + s , e k ) , d ) where dx s = e k , de k = 0 dx i = 0 if i > s • if m = 2 l and k = 2 s we have V m (cid:0) R m + k (cid:1) ≃ Q (Λ ( x s , · · · , x l + s − , ¯ e m + k − , e k ) , d ) where dx i = 0 if i > sdx s = e k , d ¯ e m + k − = de k = 0 From this proposition we deduce the following result
Corollary 3.7. If p i ( ξ ) = 0 , ∀ i ≥ s the Framed bundle associated to ξ with fiber V m (cid:0) R m + k (cid:1) , Framed m ( τ M ) , is rationally trivial. Model of space of immersions
In this section we give the proofs of Theorem 4.2, Theorem 4.3 and Corollary 4.5. We startwith a review of Smale-Hirsch theorem which identifies the space of immersions and thespace of section.
In this part we describe the Smale-Hirsch theorem and give its connection with the Stiefelbundle.Let M and N be smooth manifolds of dimension m and m + k respectively. We suppose that k > and that M and N are simply connected. We denote by T M and
T N the tangentof M and N respectively. We set T m = P m T M and T m N = P m T N , P m T M and P m T N are the space of m - frame in T M and
T N defined in section 2.1. Then
Framed m ( τ M ) = (cid:0) T m M × SO ( m ) T m N, p, M (cid:1) denotes the framed bundle associated to
T M with fiber T m N .Let Γ (Framed m ( τ M )) be the space of sections of (cid:0) T m M × SO ( m ) T m N, p, M (cid:1) . That is
Γ (Framed m ( τ M )) = { s : M −→ T m M × SO ( m ) T m N | ps = id } . We will construct a map from
Imm ( M, N ) to Γ ( p ) . For this, let f ∈ Imm ( M, N ) it inducesan SO ( m ) equivariant map from f ∗ : T m M −→ T m N the map which send the m frames of T m M into m - frames of T m N . Then the map T m M −→ T m M × T m NX ( X, f ∗ ( X )) is also SO ( m ) -equivariant. Passing to SO ( m ) − orbits yields s f : M −→ T m M × SO ( m ) T m N which is in fact a section of Framed m ( τ M ) = (cid:0) T m M × SO ( m ) T m N, p, M (cid:1) . Morris Hirschproved in [6] the following result.
Theorem 4.1.
The map
Imm ( M, N ) −→ Γ (Framed m ( τ M )) f s f is a homotopy equivalence. .2 Rational model of the space of immersions In this part we give the proofs of Theorem 4.2, and Theorem 4.3.
Theorem 4.2.
Let M be simply connected manifold of dimension m and k = 2 s + 1 with s = 0 . Suppose that p i ( τ M ) = 0 , ∀ i ≥ s + 1 . Then, each component Imm (cid:0) M, R m + k (cid:1) has therational homotopy type of the product of Eilenberg Maclane spaces, which depends only onthe rational Betti numbers of M and k . More precisely we have: Imm (cid:0) M, R m + k , f (cid:1) ≃ Q (Q s ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) si m = 2 l Q s ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × K si m = 2 l + 1 where K = Y ≤ j ≤ m + k − K (cid:0) H m + k − − j ( M, Q ) , j (cid:1) and Imm (cid:0) M, R m + k , f (cid:1) is the component of Imm (cid:0) M, R m + k (cid:1) contains f .Proof. Let f : M R m + k be an immersion and s f : M −→ T m M × SO ( m ) V m (cid:0) R m + k (cid:1) the section of Framed m ( τ M ) induced by f . By Theorem 4.1 we have: Imm (cid:0) M, R m + k , f (cid:1) ≃ Γ (Framed m ( τ M ) , s f ) where Γ ( p, s f ) is the space of sections of p : T m M × SO ( m ) V m (cid:0) R m + k (cid:1) −→ M homotope to s f .Denote Framed m ( τ m ) ( Q ) = (cid:16)(cid:0) T m M × SO ( m ) V m (cid:0) R m + k (cid:1)(cid:1) ( Q ) , p ( Q ) , M (cid:17) the fiberwise rationalization of p : T m M × SO ( m ) V m (cid:0) R m + k (cid:1) −→ M and ( s f ) ( Q ) : M −→ (cid:0) T m M × SO ( m ) V m (cid:0) R m + k (cid:1)(cid:1) ( Q ) the induced section.Since V m (cid:0) R m + k (cid:1) is simply connected, because k > , and M is simply connected and offinite type, the bundle (cid:0) T m M × SO ( m ) T m (cid:0) R m + k (cid:1) , p, M (cid:1) is nilpotent. Then by [11, theorem5.3] Γ (Framed m ( τ M ) , s f ) is nilpotent and we have the following rational equivalence Γ (Framed m ( τ M ) , s f ) ≃ Q Γ (cid:16) Framed m ( τ M ) ( Q ) , ( s f ) ( Q ) (cid:17) . Since p i ( τ M ) = 0 ∀ i ≥ s + 1 Framed m ( τ M ) is rationally trivial. And by Proposition 3.3, Framed m ( τ M ) ( Q ) is rationally trivial , and we have Γ (cid:16) Framed m ( τ M ) ( Q ) , ( s f ) ( Q ) (cid:17) ≃ M ap (cid:0) M, ( V m (cid:0) R m + k (cid:1) ) Q , ( ˜ s f ) ( Q ) (cid:1) . Since k is odd, by corollary 3.6, the Stiefel manifold V m (cid:0) R m + k (cid:1) has the rational homotopytype of product of Eilenberg Maclane spaces V m (cid:0) R m + k (cid:1) ≃ Q (Q s ≤ i ≤ l + s K ( Q , i − if m = 2 l Q s ≤ i ≤ l + s K ( Q , i − × K ( Q , m + k − if m = 2 l + 1 thus M ap (cid:0)
M, V m (cid:0) R m + k (cid:1) , ˜ s f (cid:1) ≃ Q (Q s ≤ i ≤ l + s M ap ( M, K ( Q , i −
1) ˜ s i ) if m = 2 l Q s ≤ i ≤ l + s M ap ( M, K ( Q , i − , ˜ s i )) × K if m = 2 l + 1 where ˜ s i : M −→ K ( Q , i − are conponents of ( ˜ s f ) ( Q ) .By [12] each component of M ap ( M, K ( Q , i − has the rational homotopy type of theproduct of Eilenberg Maclane spaces which depend only with rational cohomology of M .That is M ap ( M, K ( Q , i − , s i ) ≃ Y ≤ q ≤ i − K (cid:0) H i − − q ( M, Q ) , q (cid:1) therefore M ap (cid:0)
M, V m (cid:0) R m + k (cid:1) , ˜ s f (cid:1) ≃ Q (Q s ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) if m = 2 l Q s ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × K if m = 2 l + 1 where K = Y ≤ j ≤ m + k − K (cid:0) H m + k − − j ( M, Q ) , j (cid:1) . Theorem 4.3.
Let M be a simply connected m -manifold of finite type and k and an eveninteger. Then: • If k ≥ m + 1 , Imm (cid:0) M, R m + k (cid:1) is connected and its rational homotopy type is of theform Imm (cid:0) M, R m + k (cid:1) ≃ Q M ap (cid:0)
M, S k (cid:1) × K, where M ap (cid:0)
M, S k (cid:1) is the mapping space of maps from M into S k and K is a productof Eilenberg-Maclane spaces which depend only on the Betti numbers of M . If ≤ k ≤ m , Suppose that p i ( τ M ) = 0 , ∀ i ≥ s then, the connected component of f in Imm (cid:0) M, R m + k (cid:1) has the rational homotopy type of a product Imm (cid:0) M, R m + k , f (cid:1) ≃ Q M ap (cid:16)
M, S k , ˜ f (cid:17) × K, where Imm (cid:0) M, R m + k , f (cid:1) is the component of Imm (cid:0) M, R m + k (cid:1) which contains f , K is the product of Eilenberg-Maclane spaces which depends only on the Betti numbersof M , M ap (cid:16)
M, S k , ˜ f (cid:17) is the path component of the mapping space of M in S k containingthe map ˜ f induced by the immersion f .Proof. By Theorem 4.1, we have a homotopy equivalence between the space of immersionsof M in R m + k and the space of sections of Framed m ( τ M ) Imm (cid:0) M, R m + k (cid:1) ≃ Γ (Framed m ( τ M )) . We distinguish two cases: • If k ≥ m +1 the connectivity of V m (cid:0) R m + k (cid:1) is greater than the dimension of M , thereforethe space of sections of Framed m ( τ M ) is connected and subsequently Imm (cid:0) M, R m + k (cid:1) is connected.In the other hand, for all i ≥ s, p i ( τ M ) = 0 since s > l . By Corollary 3.7, Framed m ( τ M ) is rationally trivial and we have Γ (cid:0) Θ R m + k ( τ M ) Q (cid:1) ≃ M ap (cid:0)
M, V m (cid:0) R m + k (cid:1)(cid:1) . By Remark 3.6 V m (cid:0) R m + k (cid:1) ≃ Q (Q s +1 ≤ i ≤ l + s − K ( Q , i − × S k if m = 2 l + 1 Q s +1 ≤ i ≤ l + s − K ( Q , i − × K ( Q , m + k − × S k if m = 2 l therefore, M ap (cid:0)
M, V m (cid:0) R m + k (cid:1)(cid:1) ≃ Q (Q s +1 ≤ i ≤ l + s M ap ( M, K ( Q , i − × H if m = 2 l + 1 Q s +1 ≤ i ≤ l + s M ap ( M, K ( Q , i − × A × H if m = 2 l where H = M ap (cid:0)
M, S k (cid:1) and A = M ap ( M, K ( Q , m + k − finally , by [12], we have Imm (cid:0) M, R m + k (cid:1) ≃ Q (Q s +1 ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × H si m = 2 l + 1 Q s +1 ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × H × K si m = 2 l K = Y ≤ j ≤ m + k − K (cid:0) H m + k − − j ( M, Q ) , j (cid:1) . • If ≤ k ≤ m , in this cas Imm (cid:0) M, R m + k (cid:1) is not necessarily connected. Since p i ( τ M ) = 0 , ∀ i ≥ s by the same arguments as in the proof of Theorem 4.2 for f ∈ Imm (cid:0) M, R m + k (cid:1) we obtain: Imm (cid:0) M, R m + k , f (cid:1) ≃ Q (Q s +1 ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × G if m = 2 l + 1 Q s +1 ≤ i ≤ l + s Q ≤ q ≤ i − K ( H i − − q ( M, Q ) , q ) × G × K if m = 2 l where G = M ap (cid:0)
M, S k , f k (cid:1) and K = Y ≤ j ≤ m + k − K (cid:0) H m + k − − j ( M, Q ) , j (cid:1) . Corollary 4.4. If H k ( M, Q ) = 0 , then all the path connected of Imm (cid:0) M, R m + k (cid:1) have thesame rational homotopy type, and we can take ˜ f to be the constant map.Proof. It suffices to prove that for each f : M −→ S k we have M ap ( M, S k , f ) ≃ Q M ap ( M, S k , . Let (Λ( x, y ) , d ) be the model of S k and ( A, d ) a finite dimensional model of M . Let φ : (Λ( x, y ) , d ) −→ ( A, d A ) be the model of f . Since H k ( A, d ) = 0 , then φ is homotope to some map which send x to .Replace φ by this map. In this case φ ( y ) = a is a cocycle.Now we consider A ⊗ Λ( x, y ) σ = id ⊗ φ −→ A and proceed to a change of variable in order to have σ ( x ) = σ ( y ) = 0 . Pose y ′ = y − φ ( y ) = y − a and dy ′ = dy .We can suppose with the same differential that σ ( x ) = σ ( y ) = 0 . All the following steps inthe model of M ap ( M, S k , f ) depend only on Λ( x, y ) , A and σ , the result is the same for each f . Henceforth we have M ap ( M, S k , f ) ≃ Q M ap ( M, S k , . Thenceforward all the components of
Imm (cid:0) M, R m + k , f (cid:1) have the same rational homotopytype. 16rom this result we deduce Corollary 4.5. If M is a simply connected manifold of dimension m and k ≥ an integer.Then the rational Betti numbers of each component of the space of immersions of M in R m + k have polynomial growth. Remark 4.6.
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Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, BelgiqueInstitut de Recherche en Mathématique et Physique