On bundles that admit fiberwise hyperbolic dynamics
aa r X i v : . [ m a t h . D S ] J u l ON BUNDLES THAT ADMIT FIBERWISE HYPERBOLICDYNAMICS
F. THOMAS FARRELL ∗ AND ANDREY GOGOLEV ∗∗ Abstract.
This paper is devoted to rigidity of smooth bundles which areequipped with fiberwise geometric or dynamical structure. We show that thefiberwise associated sphere bundle to a bundle whose leaves are equipped with(continuously varying) metrics of negative curvature is a topologically trivialbundle when either the base space is simply connected or, more generally, whenthe bundle is fiber homotopically trivial. We present two very different proofsof this result: a geometric proof and a dynamical proof. We also establisha number of rigidity results for bundles which are equipped with fiberwiseAnosov dynamical systems. Finally, we present several examples which showthat our results are sharp in certain ways or illustrate necessity of variousassumptions. Negatively curved bundles
The results.
Recall that a smooth bundle is a fiber bundle M → E p → X whose fiber M is a connected closed manifold and whose structure group is thediffeomorphism group Diff( M ). A negatively curved bundle is a smooth bundle M → E p → X whose fibers M x = p − ( x ), x ∈ X , are equipped with Riemannian metrics g x , x ∈ X , of negative curvature. Furthermore, we require that the Riemannianmetrics g x vary continuously with x in the C ∞ topology.We say that a smooth bundle M → E p → X is topologically trivial if thereexists a continuous map r : E → M such that the restriction r | M x : M x → M , is ahomeomorphism for each x ∈ X . Note that if M → E p → X is topologically trivialthen ( p, r ) : E → X × M is a fiber preserving homeomorphism. Conjecture 1.1 (Farrell-Ontaneda) . Let X be a compact simply connected man-ifold or a simply connected finite simplicial complex and let M → E p → X be anegatively curved bundle. Then the bundle p : E → X is, in fact, topologicallytrivial.Remark . Techniques developed in [FO09, FO10a, FO10b] can be used to verifythis conjecture in certain special cases (when dim M ≫ dim X and X is a sphere).However the general conjecture above seems to be out of reach to these topologicaltechniques.In this paper classical dynamical systems techniques are used to verify relatedresults in full generality which we proceed to describe. Given a smooth closed ∗ The first author was partially supported by NSF grant DMS-1206622. ∗∗ The second author was partially supported by NSF grants DMS-1204943, 1266282. He alsowould like to acknowledge the support provided by Dean’s Research Semester Award at SUNYBinghamton. manifold M we define its tangent sphere bundle SM as the quotient of T M \ M byidentifying two non-zero tangent vectors u and v based at a common point x ∈ M if and only if there exists a positive number λ such that u = λv . Similarly givena smooth bundle M → E p → X we define its (fiberwise) associated sphere bundle SM → SE S ( p ) → X as the quotient of the space of non-zero vectors tangent to thefibers of p : E → X . Given a (class of a) vector v ∈ SE with a base point x ∈ E we define S ( p )( v ) = p ( x ). Then it is clear that the fibers of the associated spherebundle are diffeomorphic to SM . Also note that if the fibers of a smooth bundleare equipped with continuously varying Riemannian metrics then the total spaceof the associated sphere bundle can be realized as the space of unit vectors tangentto the fibers of the original bundle. Theorem 1.3.
Let p : E → X be a smooth negatively curved bundle whose basespace X is a simply connected closed manifold or a simply connected finite sim-plicial complex. Then its (fiberwise) associated sphere bundle S ( p ) : SE → X istopologically trivial. We say that a smooth bundle M → E p → X is fiber homotopically trivial if thereexists a continuous map q : E → M such that the restriction q | M x : M x → M is ahomotopy equivalence for each x ∈ X . Proposition 1.4. If X is a simply connected closed manifold or a simply connectedfinite simplicial complex and M → E p → X is negatively curved bundle then it isfiber homotopically trivial. Because of the above proposition Theorem 1.3 is a consequence of the followingmore general result.
Theorem 1.5.
Let X be a closed manifold or a finite simplicial complex and p : E → X be a smooth negatively curved, fiber homotopically trivial bundle. Thenits (fiberwise) associated sphere bundle S ( p ) : SE → X is topologically trivial. It was shown in [FO10a] that there are “lots” of smooth bundles E → S k , ( k > M supports a negatively curved Riemannian metric, but whichare not negatively curved. So Theorem 1.3 does not apply to these bundles. Henceit is a priori conceivable that the conclusion of Theorem 1.3 holds under the weakerassumption that the abstract fiber M supports a negatively curved Riemannianmetric. However the next result shows that this is not the case. Theorem 1.6.
For each prime number p > , there exists a smooth bundle M → E → S p − such that the (abstract) fiber M is real hyperbolic, i.e., M supports a Riemannianmetric of constant negative sectional curvature; the (fiberwise) associated sphere bundle SM → SE → S p − is not topologi-cally trivial. We will prove Theorem 1.6 in Section 5. Now we proceed to give concise proofsof Proposition 1.4 and Theorem 1.5.1.2.
Proofs.
Proof of Proposition 1.4.
We first consider the case where X as a finite simplicialcomplex. Let X ⊂ X be the 1-skeleton of X . First note that the inclusion map UNDLES WITH HYPERBOLIC DYNAMICS 3 σ : X → X is homotopic to a constant map with value, say, x ∈ X . By thecovering homotopy theorem, this homotopy is covered by a bundle map homotopystarting with the inclusion map p − ( X ) ⊂ E and ending with a continuous map q : p − ( X ) → M x which is a homotopy equivalence when restricted to each fiber.Denote by G ( M ) the space of self homotopy equivalences of M . Then π n ( G ( M )) = ( Center ( π ( M )) , if n = 10 , if n ≥ , (1.1)since M is aspherical (see [G65]). Note that center of π ( M ) is trivial because M is negatively curved. Hence π n ( G ( M )) is trivial for n ≥ q to succesive skeletons of X until we obtain a fiber homotopytrivialization q .We now consider the case where X is a closed manifold. If X is smooth, thenwe are done by the first case since every smooth manifold can be triangulated.But there are many examples of closed topological manifolds that cannot be tri-angulated. However, every closed manifold is an ANR and, hence, is homotopyequivalent to finite simplicial complex K (see [W70]). Let r : K → X be a homo-topy equivalence and σ : K → X be its homotopy inverse. Also let( ξ K ) M → E → K denote the pullback bundle along r : K → X of the given bundle( ξ X ) M → E p → X Then bundle ( ξ K ) is negatively curved and, hence, is fiber homotopically trivial bythe argument in the first case.But if ξ K is fiber homotopically trivial, then so is ξ X since ξ X = id ∗ ( ξ X ) = σ ∗ ( r ∗ ( ξ X )) = σ ∗ ( ξ K ) . (cid:3) Proof of Theorem 1.5.
Denote by q : E → M the fiber homotopy trivialization.Choose a base point x ∈ X . We identify the abstract fiber M of the bundle withthe fiber M x over x . We choose this identification so that q | M : M → M inducesthe identity map on π ( M ). Therefore q : π ( E ) → π ( M ) splits the short exactsequence 0 → π ( M ) → π ( E ) → π ( X ) → . Hence π ( E ) ≃ π ( X ) × π ( M ).Let ρ : ˜ E → E be a covering map such that ρ π ( ˜ E ) = π ( X ) ⊂ π ( E ). Let˜ p = p ◦ ρ . Note that ρ intertwines the long exact sequences for p and ˜ p : . . . / / π ( X ) id (cid:15) (cid:15) / / π ( ˜ M ) ρ (cid:15) (cid:15) / / π ( ˜ E ) ρ (cid:15) (cid:15) isom. / / π ( X ) id (cid:15) (cid:15) / / π ( ˜ M ) / / . . . / / π ( X ) / / π ( M ) / / π ( E ) / / π ( X ) / / M is a connected simply connectedmanifold.Recall that p : E → X is a negatively curved bundle. The negatively curvedRiemannian metrics on the fibers of p : E → X lift to negatively curved Riemannian UNDLES WITH HYPERBOLIC DYNAMICS 4 metrics on the fibers of ˜ p : ˜ E → X . For each fiber ˜ M x , x ∈ X , consider the “sphereat infinity” ˜ M x ( ∞ ), which is defined as the set of equivalence classes of asymptoticgeodesic rays in ˜ M x . Given a point y ∈ ˜ M x , consider the map r y : S y ˜ M x → ˜ M x ( ∞ )defined by v [ γ v ], where γ v is the geodesic ray with γ v (0) = y , ˙ γ v (0) = v , and [ γ v ]is the equivalence class of γ v . Map r y is a bijection that induces sphere topologyon ˜ M x ( ∞ ). (This topology does not depend on the choice of y ∈ ˜ M x .)Let ˜ E ( ∞ ) = ⊔ x ∈ X ˜ M x ( ∞ ). Charts for ˜ p : ˜ E → X induce charts for the bundle˜ p ∞ : ˜ E ( ∞ ) → X whose fiber is the sphere at infinity ˜ M ( ∞ ). Using the MorseLemma one can check that any diffeomorphism (and more generally any homotopyequivalence) f : M → M of a negatively curved manifold M induces a homeo-morphism f ∞ : ˜ M ( ∞ ) → ˜ M ( ∞ ) that only depends on f : π ( M ) → π ( M ) (see, e.g., [Kn02]). It follows that ˜ M ( ∞ ) → ˜ E ( ∞ ) ˜ p ∞ → X is a fiber bundle whose struc-ture group is T op ( ˜ M ( ∞ )). The action of π ( M ) on ˜ E extends to ˜ E ( ∞ ) in thenatural way.Denote by ˜ q : ˜ E → ˜ M a lifting of fiber homotopy trivialization q : E → M . Again,by the Morse Lemma, ˜ q induces a continuous map ˜ q ∞ : ˜ E ( ∞ ) → ˜ M ( ∞ ) such thatthe restriction ˜ q ∞ | ˜ M x ( ∞ ) : ˜ M x ( ∞ ) → ˜ M ( ∞ ) is a homeomorphism for each x ∈ X . Remark . Note that S ˜ M is (equivariently) homeomorphic to ˜ M × ˜ M ( ∞ ) via ˜ M × ˜ M ( ∞ ) ∋ ( y, ξ ) r − y ( ξ ) ∈ S ˜ M . Therefore the map S ˜ E ∋ v (˜ q ( y ) , ˜ q ∞ ( r y ( v )) ∈ ˜ M × ˜ M ( ∞ ) induces a fiber homotopy trivialization SE → SM for the associatedsphere bundle.Now we use an idea of Gromov [Gr00] to finish the proof. We code each vectorin S ˜ E by a pair ( γ, y ), where y is the base point of the vector and γ is the orientedgeodesic in ˜ M y in the direction of the vector. Furthermore, we code each orientedgeodesic γ in ˜ M y by an ordered pair of distinct points ( γ ( −∞ ) , γ ( ∞ )) ∈ ˜ M y ( ∞ ) × ˜ M y ( ∞ ). Define ˜ r : S ˜ E → S ˜ M by˜ r ( γ ( −∞ ) , γ ( ∞ ) , y ) = (˜ q ∞ ( γ ( −∞ )) , ˜ q ∞ ( γ ( ∞ )) , pr (˜ q ( y ))) , where pr : ˜ M → ˜ q ∞ ( γ ) is the orthogonal projection on the geodesic ˜ q ∞ ( γ ) with end-points ˜ q ∞ ( γ ( −∞ )) and ˜ q ∞ ( γ ( ∞ )). Because the restrictions ˜ q ∞ | ˜ M x ( ∞ ) are homeo-morphisms, the restrictions ˜ r | S ˜ M x induce homeomorphisms on the correspondingspaces of geodesics. However ˜ r | S ˜ M x may fail to be injective along the geodesics. Toovercome this difficulty we follow Gromov and consider the maps ˜ r K : S ˜ E → S ˜ M given by˜ r K ( γ ( −∞ ) , γ ( ∞ ) , y ) = ˜ q ∞ ( γ ( −∞ )) , ˜ q ∞ ( γ ( ∞ )) , K Z K pr (˜ q ( γ ( t ))) dt ! , for K >
0. For each x ∈ X there exists a sufficiently large K such that the restric-tion ˜ r K | S ˜ M x is an equivariant homeomorphism (see [Kn02] for a detailed proof).In fact, because q is continuous, the same argument yields a stronger result: foreach x ∈ X there exists a sufficiently large K such that the restrictions ˜ r K | S ˜ M y areequivariant homeomorphisms for all y in a neighborhood of x . Hence, because X is compact, for a sufficiently large K the restrictions of ˜ r to the fibers are home-omorphisms. Project ˜ r back to E to obtain the posited topological trivialization r : E → M . (cid:3) UNDLES WITH HYPERBOLIC DYNAMICS 5
Further geometric questions.
Given a closed smooth Riemannian manifold M denote by St k M the bundle of ordered orthonormal k -frames in the tangentbundle T M . As before, given a negatively curved bundle M → E p → X we candefine its (fiberwise) associated k -frame bundle St k M → St k E St k ( p ) → X . Theorem 1.8.
Let X be a closed manifold or a finite simplicial complex and p : E → X be a smooth negatively curved, fiber homotopically trivial bundle. Thenits (fiberwise) associated -frame bundle St ( p ) : St E → X is topologically trivial. The proof is the same except that one has to use the idea of Cheeger [Gr00] whosuggested to code 2-frames in St ˜ M by ordered triples of points in ˜ M ( ∞ ). Question 1.9.
Assume that M and N are closed negatively curved manifolds withisomorphic fundamental groups. Is it true that St k M is homeomorphic to St k N for k ≥ ? Question 1.10.
Let X be a finite simplicial complex and p : E → X be a smoothnegatively curved, fiber homotopically trivial bundle. Let k ≥ . Is it true that its(fiberwise) associated k -frame bundle St k ( p ) : St k E → X is topologically trivial? Potentially, our techniques could be useful for this question, cf. discussion inSection 7.2. 2.
Anosov bundles: an overview
The rest of the paper is mainly devoted to topological rigidity of smooth bundlesequipped with fiberwise Anosov dynamics in various different setups. Some of theseresults directly generalize Theorems 1.3 and 1.5 from the preceding section. Weinformally summarize our topological rigidity results in the following table. ❳❳❳❳❳❳❳❳❳❳❳❳❳
FiberwiseDynamics Topology Simply connectedbase Fiberwise homotopicallytrivial bundleAnosov diffeomorphisms Theorem 3.1examples: 3.7 Theorem 3.2 ( ⋆ ) ,strong form: Addendum 3.5 ( ⋆ ) examples: 3.7, 3.9Anosov flows Theorem 4.1examples: 1.6 Theorem 4.2 ( ⋆⋆ ) ,strong form: ?examples: 1.6, 4.5Partially hyperbolicdynamics ? ?discussion: sections 1.3 and 7.2.We now proceed with some comments on the table and contents of our paper.Then, after presenting the background on structural stability, we outline a generalstrategy which is common to all the proofs of topological rigidity results whichappear further in the paper. UNDLES WITH HYPERBOLIC DYNAMICS 6
Comments.
1. The cells of the table contain references to topological rigidity results. Thefirst row of the table indicates the assumption on the fiber bundle underwhich corresponding result holds and the first column of the table lists thecorresponding dynamical assumption.2. Under the label “examples” we refer to statements which provide examplesto which corresponding topological rigidity result applies. All these examplesare fairly explicit and are not smoothly rigid.3. The topological rigidity results from the last column require extra assump-tions: ( ⋆ ) infranilmanifold fibers; ( ⋆⋆ ) absence of homotopic periodic orbits.We do not know if these extra assumptions can be dropped.4. Theorems 1.6 and 3.15 (which do not appear in the table) show that we can-not dispose of the dynamical assumption and still have topological rigidity.5. In Section 7.1 we point out that examples given by Theorem 3.7 can beviewed as partially hyperbolic diffeomorphisms with certain peculiar prop-erties.2.2. Structural stability.
All our proofs of topological rigidity results which ap-pear in further sections strongly rely on structural stability of Anosov dynamicalsystems [An69].
Structural stability for Anosov diffeomorphisms.
Let f : M → M be anAnosov diffeomorphism. Then, for every δ > there exists ε > such that if adiffeomorphism ¯ f : M → M is such that d C ( f, ¯ f ) < ε then ¯ f is also Anosov and there exist a homeomorphism h : M → M such that h ◦ ¯ f = f ◦ h and d C ( h, id M ) < δ. Moreover, such homeomorphism h is unique. Structural stability for Anosov flows.
Let g t : M → M be an Anosovflow. Then, for every δ > there exists ε > such that if a flow ¯ g t : M → M is C ε -close to g t then ¯ g t is also Anosov and there exist an orbit equivalence h : M → M ,( i.e., h is a homeomorphism which maps orbits of ¯ g t to orbits of g t ) such that d C ( h, id M ) < δ. The homeomorphism h given by structural stability is merely H¨older continuousand, in general, not even C . Note that the uniqueness part of structural stabilityfails for flows. Indeed, if h is an orbit equivalence which is C close to id M then g β ◦ h ◦ ¯ g α is another orbit equivalence which is C close to identity provided thatthe functions α, β : M → R are C close to the zero function.2.3. Scheme of the proof of topological rigidity.
Below we outline our ap-proach to topological rigidity of Anosov bundles in a broad-brush manner. Defini-tions and precise statements of results appear in further sections.Assume that M → E → X is a smooth fiber bundle over a compact base X whosetotal space E is equipped with a fiberwise Anosov diffeomorphism or a fiberwise UNDLES WITH HYPERBOLIC DYNAMICS 7
Anosov flow. To show that such Anosov bundle is topologically trivial we firstenlarge the structure group of the bundleDiff( M ) ⊂ Top( M )and, from now on, view M → E → X as a topological bundle. Employing Anosovdynamics in the fibers, we can identify nearby fibers using homeomorphisms whichcome from structural stability. This allows to reduce the structure group of thebundle Diff( M ) ⊂ Top( M ) ⊃ F, where F is the group of certain self conjugacies of a fixed Anosov diffeomorphismof M (self orbit equivalences in the flow case), which are homotopic to id M .The rest of the proof strongly depends on particular context. We only remarkthat, in the case when E is equipped with a fiberwise Anosov diffeomorphism, group F is discrete because of uniqueness part of structural stability. (Indeed, the onlyself conjugacy of Anosov diffeomorphism which is close to identity is the identityself conjugacy.) In the case of flows, the group F is larger, however we are able toshow that F is contractible, i.e., we further reduce the structure group to a trivialgroup which implies that the bundle is trivial as Top( M )-bundle.3. Bundles admitting fiberwise Anosov diffeomorphisms
Let M → E → X be a smooth bundle. A fiber preserving diffeomorphism f : E → E is called a fiberwise Anosov diffeomorphism if the restrictions f x : M x → M x , x ∈ X , are Anosov diffeomorphisms. A smooth bundle M → E → X is called Anosov if it admits a fiberwise Anosov diffeomorphism.
Theorem 3.1.
Let X be a closed simply connected manifold or a finite simplyconnected simplicial complex. Assume M → E p → X is an Anosov bundle. Thenthe bundle p : E → X is, in fact, topologically trivial. Theorem 3.2.
Let X be a closed manifold or a finite simplicial complex. Assumethat M → E p → X is a fiber homotopically trivial Anosov bundle, whose fiber M isan infranilmanifold. Then the bundle p : E → X is, in fact, topologically trivial.Remark . In the above theorem the assumption that the bundle is fiber homo-topically trivial is a necessary one. To see this let
A, B : T n → T n be two commutinghyperbolic automorphisms. Then the bundle T n → E A p → S , whose total space isdefined as E A = T n × [0 , / ( x, ∼ ( Ax, , is Anosov. Indeed, the map E A ∋ ( x, t ) ( Bx, t ) ∈ E A is a fiberwise Anosov diffeomorphism. Remark . Theorem 3.2 can be generalized to a wider class of bundles whose fiber M is only homeomorphic to an infranilmanifold. The modifications required in theproof are straightforward.We will explain in Section 3.4 how results of [FG13] imply that there are “lots” ofsmooth, fiber homotopically trivial bundles M → E → S d whose infranilmanifoldfiber M supports an Anosov diffeomorphism, but which are not Anosov bundles(and are not topologically trivial). UNDLES WITH HYPERBOLIC DYNAMICS 8
Two fiber bundles M → E p → X and M → E p → X over the same base X are called fiber homotopically equivalent if there exists a fiber preserving continuousmap q : E → E covering id X such that the restriction q | M ,x : M ,x → M ,x is ahomotopy equivalence for each x ∈ X . Similarly, two fiber bundles M → E p → X and M → E p → X over the same base X are called topologically equivalent if thereexists a fiber preserving homeomorphism h : E → E covering id X such that therestriction h | M ,x : M ,x → M ,x is a homeomorphism for each x ∈ X . Theorem 3.2can be generalized to the setting of non-trivial fiber bundles in the following way. Addendum 3.5.
Let X be a closed manifold or a finite simplicial complex. Assumethat M → E p → X and M → E p → X are fiber homotopically equivalent Anosovfiber bundles over the same base X whose fibers M and M are nilmanifolds. Thenthe bundles p : E → X and p : E → X are, in fact, topologically equivalent.Remark . Addendum 3.5 can be generalized to a wider class of bundles whosefiber M and M are only homeomorphic to nilmanifolds. The modifications re-quired in the proof are straightforward.We prove Theorem 3.1 now and postpone the proofs of Theorem 3.2 and Adden-dum 3.5 to Section 6. Proof of Theorem 3.1.
Fix x ∈ X . For each x ∈ X consider a path γ : [0 , → X that connects x to x . Because Anosov diffeomorphisms are structurally stable, foreach t ∈ [0 ,
1] there exists a neighborhood of t , U ⊂ [0 ,
1] and a continuous familyof homeomorphisms h s,t : M γ ( t ) → M γ ( s ) for s ∈ U such that h t,t = id M γ ( t ) and h s,t ◦ f γ ( s ) = f γ ( t ) ◦ h s,t . Since [0 ,
1] is compact we can find a finite sequence 0 = t < t < . . . < t N = 1 andhomeomorphisms h t i − ,t i that conjugate f γ ( t i − ) to f γ ( t i ) . Clearly h x = h t N − ,t N ◦ h t N − ,t N − ◦ . . . ◦ h t ,t is a conjugacy between f x and f x . Because the “local” conjugacies h s,t are uniqueand depend continuously on s , the resulting homeomorphism h x : M x → M x doesnot depend on the arbitrary choices that we made. Moreover, two homotopic pathsthat connect x to x will yield the same homeomorphism h x . Hence, since X issimply connected, h x depends only on x ∈ X . It is also clear from the constructionthat h x depends on x continuously. Therefore E ∋ y h p ( y ) ( y ) ∈ M x gives theposited topological trivialization. (cid:3) We will say that a smooth bundle M → E p → X is smoothly trivial if thereexists a continuous map r : E → M such that the restriction r | M x : M x → M is adiffeomorphism for each x ∈ X . If the base X is a smooth closed manifold thenthe total space is also a smooth manifold. (This can be seen by approximatingfiberwise smooth transition functions between the charts of the bundle by smoothtransition functions.) In this case, the trivialization r can be smoothed out so thatthe restrictions r | M x : M x → M depend smoothly on x ∈ X . Hence, then map E → X × M given by y ( p ( y ) , r p ( y ) ( y )) is fiber preserving diffeomorphism. Theorem 3.7.
For each d ≥ there exists a smooth Anosov bundle T n → E p → S d which is not smoothly trivial. UNDLES WITH HYPERBOLIC DYNAMICS 9
Remark . The construction of examples of Anosov bundles for the above theoremrelies on the idea of construction of “exotic” Anosov diffeomorphism from [FG12].
Addendum 3.9.
There exists smooth fiber homotopically trivial Anosov bundle T n → E p → S which is not smoothly trivial.Remark . Note that Anosov bundles posited above are topologically trivial byTheorems 3.1 and 3.2.Because the proof of Addendum 3.9 is virtually the same as the proof of Theo-rem 3.7 but requires some alternations in notation, we prove Theorem 3.7 only. Toexplain the construction of our examples of Anosov bundles we first need to presentsome background on the Kervaire-Milnor group of homotopy spheres.3.1.
Gromoll filtration of Kervaire-Milnor group. A homotopy m -sphere Σ isa smooth manifold which is homeomorphic to the standard m -sphere S m . Kervaireand Milnor showed that if m ≥ m -spheres forms a finite abelian group Θ m under the connected sumoperation [KM63].One way to realize the elements of Θ m is through so called twist sphere con-struction. A twist sphere Σ g is obtained from two copies of a closed disk D m bypasting the boundaries together using an orientation preserving diffeomorphism g : S m − → S m − . We will denote by Diff( · ) the space of orientation preservingdiffeomorphisms. It is easy to see that the map Diff( S m − ) ∋ g Σ g ∈ Θ m factorsthrough to a homomorphism π Diff( S m − ) → Θ m , which was known to be a group isomorphisms for m ≥ l ∈ [0 , m −
1] and view the ( m − D m − as the product D l × D m − − l .Let Diff l ( D m − , ∂ ) be the group of diffeomorphisms of the ( m − l -coordinates and are identity in a neighborhood of the boundary ∂ D m − . Thenwe have Diff l ( D m − , ∂ ) ֒ → Diff( D m − , ∂ ) ֒ → Diff( S m − ) , where the last inclusion is induced by a fixed embedding D m − ֒ → S m − . These in-clusions induce homomorphisms of corresponding groups of connected components π Diff l ( D m − , ∂ ) ֒ → π Diff( D m − , ∂ ) ֒ → π Diff( S m − ) ≃ Θ m . The image of π Diff l ( D m − , ∂ ) in Θ m does not depend on the choice of the embed-ding D m − ֒ → S m − and is called the Gromoll subgroup Γ ml +1 . Gromoll subgroupsform a filtration Θ m = Γ m ⊇ Γ m ⊇ . . . ⊇ Γ mm = 0 . Cerf [C61] proved that Γ m = Γ m for m ≥
6. Higher Gromoll subgroups are alsoknown to be non-trivial in many cases, see [ABK70]; see also [FO09]. In particular,Γ u − u − = 0 for u ≥ Proof of Theorem 3.7: construction of the bundle.
Fix an Anosovautomorphism L : T n → T n whose unstable distribution E u has dimension k . Let q be a fixed point of L . Choose a small product structure neighborhood U ≃ D k × D n − k in the proximity of fixed point q so that L ( U ) and U are disjoint. Alsochoose an embedded disk D d − ֒ → S d − . Then we have the product embedding i : D d − × D k × D n − k ֒ → S d − × T n . UNDLES WITH HYPERBOLIC DYNAMICS 10
Now pick a diffeomorphism α ∈ Diff k + d − ( D n + d − , ∂ ), consider i ◦ α ◦ i − andextend it by identity to a diffeomorphism α : S d − × T n → S d − × T n . Now we em-ploy the standard clutching construction, that is, we paste together the boundaries D d − × T n and D d + × T n using α : ∂ D d − × T n → ∂ D d + × T n . This way we obtain asmooth bundle T n → E α p → S d . Clearly this bundle does not depend on the choiceof the product neighborhood U and the choice of embedding D d − ֒ → S d − , i.e., onthe choice of embedding i . Proposition 3.11. If [ α ] = 0 in the group Γ n + dk + d , d ≥ , then T n → E α p → S d isnot smoothly trivial.Proof. Detailed proofs of similar results were given in [FJ89] and [FO09]. So wewill only sketch the proof of this result; but enough to show how it follows fromarguments in [FJ89] and [FO09].Our proof proceeds by assuming that there exists a fiber preserving diffeomor-phism f : E α → S d × T n and showing that this assumption leads to a contradiction.One may assume that f | D d − × T n is the identity map (we view S d × T n as D d − × T n and D d + × T n glued together using the identity map). Also note that, by the Alexan-der trick, there is a “canonical” homeomorphism g : E α → S d × T n which is fiberpreserving and satisfies g | D d − × T n = id . One also easily sees that E α is diffeomor-phic to S d × T n α and that under this identification g becomes the “obvioushomeomorphism” S d × T n α → S d × T n . Here, Σ α is the exotic twist sphere of α .Now f and g are two smoothings of the topological manifold S d × T n . Recallthat a homeomorphism ϕ : N → S d × T n , where N is a closed smooth ( d + n )-dimensional manifold, is called a smoothing of S d × T n . Two smoothings ϕ i : N i → S d × T n , i = 0 ,
1, are equivalent if thereexists a diffeomorphism ψ : N → N such that the composite ϕ ◦ ψ is topologicallypseudo-isotopic (or concordant ) to ϕ ; i.e., there exists a homeomorphismΦ : N × [0 , → S d × T n × [0 , | N ×{ i } : N × { i } → S d × T n × { i } is ϕ if i = 0 and ϕ ◦ ψ if i = 1.Since S d × T n is stably parallelizable, it is well-known that the smoothing g : S d × T n α → S d × T n is inequivalent to the “trivial smoothing” id : S d × T n → S d × T n ;cf. [FJ89] or [FO09]. But the smoothing f : E α → S d × T n is obviously equivalentto id : S d × T n → S d × T n , since f is a diffeomorphism. Therefore, f and g areinequivalent smoothings of S d × T n .We now proceed to contradict this assertion. Since f | D d − × T n = g | D d − × T n , it suf-fices to show that f | D d + × T n is topologically pseudo-isotopic to g | D d + × T n relative toboundary. But this follows from the basic Hsiang-Wall topological rigidity re-sult [HW69] that the homotopy-topological structure set S ( D k × T n , ∂ ) containsexactly one element when k + n ≥ π i G ( T n ) = 0 for all i > (cid:3) In the proof above we only used the assumption that diffeomorphism f : E α → S d × T n is fiber preserving in order to be able to assume that f D d − × T n is identity. UNDLES WITH HYPERBOLIC DYNAMICS 11
Notice that this step also works for diffeomorphisms which are fiber preserving onlyover the southern hemisphere D d − of S d . Hence, from the above proof, we concludethat there does not exist a diffeomorphism E α → S d × T n which is fiber preservingover the southern hemisphere D d − of S d . This observation justifies the followingdefinition. A smooth bundle T n → E → S d is weakly smoothly trivial if thereexists a diffeomorphism E → S d × T n which is fiber preserving over the southernhemisphere D d − of S d . By the above observation, we have the following somewhatstronger result. Addendum 3.12. If [ α ] = 0 in the group Γ n + dk + d , d ≥ , then T n → E α p → S d isnot even weakly smoothly trivial. This result will be used in Section 4 while proving Theorem 4.5.3.3.
Proof of Theorem 3.7: construction of the fiberwise Anosov diffeo-morphism.
For each ( x, y ) ∈ D d − × T n define f ( x, y ) = ( x, L ( y )) . Because of clutching we automatically have f ( x, y ) = α ◦ ¯ L ◦ α − ( x, y ) , (3.1)where ( x, y ) ∈ ∂ D d + × T n and ¯ L : S d − × T n → S d − × T n is given by ( x, y ) ( x, L ( y )). Hence it remains to extend f to the interior of the disk D d + .Take a smooth path of embeddings i t : D d − × D k × D n − k ֒ → S d − × T n , t ∈ [0 , i t = ¯ L ◦ i for all t ∈ [0 , δ ], where δ > i = i ;3. p ◦ i t = p ◦ i for all t ∈ [0 , i t ), t ∈ [0 , S d − × { q } ,but do not intersect S d − × { q } ;5. i t ( x, · , · ) : D k × D n − k → { x } × T n is a local product structure chart for L : T n → T n for all t ∈ [0 ,
1] and x ∈ D d − .Denote by α t : S d − × T n → S d − × T n the composition i t ◦ α ◦ i − t extended byidentity to the rest of S d − × T n .Now we use polar coordinates ( r, ϕ ) on D d + , r ∈ [0 , ϕ ∈ S d − , to extend f tothe interior of D d + . Let f (( r, ϕ ) , y ) = α r ◦ ¯ L ◦ α − (( r, ϕ ) , y ) . When r = 1 this definition is clearly consistent with (3.1). When r ∈ [0 , δ ] we have f (( r, · ) , · ) = α r ◦ ¯ L ◦ α − = i r ◦ α ◦ i − r ◦ ¯ L ◦ i ◦ α ◦ i − ◦ ( ¯ L ◦ i ) ◦ α − ◦ i − = i ◦ α ◦ i − ◦ i ◦ α − ◦ i − = i ◦ i − = ¯ L, for points in the image of i and it is obvious that f (( r, · ) , · ) = ¯ L outside of the imageof i . Note that because the path α r is locally constant at r = 0 diffeomorphism f is smooth at r = 0.Therefore we have constructed a fiber preserving diffeomorphism f : E α → E α .The restrictions to the fibers f x : T nx → T nx are clearly Anosov when x ∈ D d − orclose to the origin of D d + . Note that our construction depends on the choice ofthe diffeomorphism α ∈ Diff k + d − ( D n + d − , ∂ ), the choice of embedding i and thechoice of isotopy i t , t ∈ [0 , UNDLES WITH HYPERBOLIC DYNAMICS 12
Proposition 3.13.
For each number d ≥ , each Anosov automorphism L : T n → T n whose unstable distribution is k -dimensional and each diffeomorphism α ∈ Diff k + d − ( D n + d − , ∂ ) there exists an embedding i : D d − × D k × D n − k ֒ → S d − × T n and an isotopy i t , t ∈ [0 , such that the fiber preserving diffeomorphism f = f ( α, i, i t ) : E α → E α of the total space of the fiber bundle T n → E α p → S d is afiberwise Anosov diffeomorphism. Clearly Propositions 3.11 and 3.13 together with non-triviality results for Gro-moll subgroups imply Theorem 3.7.
Sketch of the proof of Proposition 3.13.
Let h be the round metric on S d − and g be the flat metric on T n . Choose and fix an embedding i : D d − × D k × D n − k ֒ → S d − × T n and isotopy i t , t ∈ [0 ,
1] that satisfy properties 1-5 listed in Subsection 3.3.Then for some small ε > t ∈ [0 , i t ) ∈ V h + gε ( S d − × { q } ) , where V h + gε ( S d − × { q } ) is the ε -neighborhood of S d − × { q } in S d − × T n equippedwith Riemannian metric h + g . (Recall that q is a fixed point of L .)For each m ≥ h + mg and corresponding ε -neighborhood V h + mgε ( S d − ×{ q } ). These neighborhoods are isometric in an obviousway I : V h + gε ( S d − × { q } ) → V h + mgε ( S d − × { q } )Then there is a unique choice of embeddings i mt , t ∈ [0 , D d − × D k × D n − k i t / / i mt ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ V h + gε ( S d − × { q } ) I (cid:15) (cid:15) V h + mgε ( S d − × { q } )commute. The following lemma clearly yields Proposition 3.13. Lemma . For a sufficiently large number m ≥ f = f ( α, i m , i mt ) : E α → E α constructed following the procedure of Subsection 3.3 isa fiberwise Anosov diffeomorphism.We do not provide a detailed proof of this lemma because a very similar argumentwas given in Section 3.3 of [FG12].The first step is to notice that the unstable foliation W uL of L : T n → T n isinvariant under the restrictions to the fiber f x : T nx → T nx , x ∈ D d + . This is because i mt ◦ α ◦ ( i mt ) − preserves foliation W uL . Then one argues that for a sufficiently large m the return time (under f x ) to ∪ t ∈ [0 , Im( i t ) is large, which implies that W uL isindeed an expanding foliation for f x .The second step is to employ a standard cone argument to show that there existsan Df x -invariant stable distribution transverse to T W uL . (cid:3) Non-Anosov fiber homotopically trivial bundles.
The following resultdemonstrates that the assumption “ M → E → X is an Anosov bundle” in The-orem 3.2 cannot be replaced by the assumption “the fiber M supports an Anosovdiffeomorphism.” UNDLES WITH HYPERBOLIC DYNAMICS 13
Theorem 3.15 (cf. Theorem 1.6) . Let M be any infranilmanifold that has dimen-sion ≥ , supports an affine Anosov diffeomorphism and has non-zero first Bettinumber. Then for each prime p there exists a smooth bundle M → E → S p − suchthat the bundle M → E → S p − is fiber homotopically trivial; the bundle M → E → S p − is not topologically trivial (and, hence, is notAnosov by Theorem 3.2). For the proof of Theorem 3.15 we need to recall the clutching construction whichwe use as a way to create fiber bundles over spheres. Let S d , d ≥
2, be a sphere,let N be a closed manifold and let α : S d − → Diff( N ) be a map. Let D d − and D d + be the southern and the northern hemispheres of S d , respectively. Considermanifolds D d − × N and D d + × N as trivial fiber bundles with fiber N . Note thatthere boundaries are both diffeomorphic to S d − × N .Now define diffeomorphism ¯ α : S d − × N → S d − × N by the formula( x, y ) ( x, α ( x )( y )) . Clutching with α amounts to pasting together the boundaries of D d − × N and D d + × N using ¯ α . The resulting manifold E is naturally a total space of smooth fiber bundle N → E → S d because diffeomorphism ¯ α maps fibers to fibers. Proof.
We first consider the case when p >
2. The assumptions (on the in-franilmanifold M ) of Proposition 5 of [FG13] are satisfied and we obtain a class[ α ] ∈ π p − (Diff ( M )) whose image under the natural inclusion π p − (Diff ( M )) → π p − (Top ( M )) is non-zero. Then clutching with α : S p − → Diff ( M ) yields asmooth bundle M → E → S p − which is not topologically trivial.On the other hand, by (1.1), the image of α in π p − ( G ( M )) is zero. Hence M → E → S p − is fiber homotopically trivial.The case when p = 2 is analogous and uses Proposition 4 of [FG13]. (cid:3) Bundles admitting fiberwise Anosov flows
Here we generalize Theorems 1.3 and 1.5 to the setting of abstract Anosov flows.
Theorem 4.1.
Let X be a closed simply connected manifold or a finite simplyconnected simplicial complex and let p : E → X be a fiber bundle whose fiber M isa closed manifold. Assume that E admits a C ∞ flow which leaves the fiber M x , x ∈ X , invariant and the restrictions of this flow to the fibers g tx : M x → M x , x ∈ X , are transitive Anosov flows. Then the bundle p : E → X is topologicallytrivial. Theorem 4.2.
Let X be a closed manifold or finite simplicial complex and p : E → X be a fiber homotopically trivial bundle whose fiber M is a closed manifold. As-sume that E admits a C ∞ flow which leaves the fibers M x , x ∈ X , invariant andthe restrictions of this flow to the fibers g tx : M x → M x , x ∈ X , satisfy the followingconditions flows g tx : M x → M x are transitive Anosov flows; flows g tx : M x → M x do not have freely homotopic periodic orbits.Then the bundle p : E → X is topologically trivial. UNDLES WITH HYPERBOLIC DYNAMICS 14
Remark . We only consider free homotopies of periodic orbits that preserve flowdirection of the orbits. This way geodesic flows on negatively curved manifolds donot have freely homotopic periodic orbits.
Remark . By applying structural stability we see that the flows g tx are all orbitequivalent. Hence if assumption 2 of Theorem 4.2 holds for one x ∈ X then itautomatically holds for all x ∈ X .Theorem 4.1 obviously implies Theorem 1.3. Let us also explain how Theo-rem 4.2 implies Theorem 1.5. The associated sphere bundle S ( p ) : SE → X is fiberhomotopically trivial by Remark 1.7. The total space SE is equipped with fiberwisegeodesic flow g t . The flows on the fibers g tx , x ∈ X , are Anosov. Moreover theyare transitive because they are ergodic with respect to fully supported Liouvillemeasures by the work of Anosov [An69]. Finally, assumption 2 of Theorem 4.2 issatisfied because each free homotopy class of loops on a negatively curved manifoldscontains only one closed geodesic (which yields two periodic orbits which are nothomotopic as oriented periodic orbits).Another example of Anosov flow which satisfies assumption 2 of Theorem 4.2 isthe suspension flow of an Anosov diffeomorphism of an infranilmanifold. Theorem 4.5.
For any d ≥ there exists a smoothly non-trivial fiber bundle M → E ′ p ′ → S d which admits fiberwise Anosov flow which satisfies the assumptionsof Theorem 4.2. This result is analogous to (and is based on) Theorem 3.7.For expository reasons we present the proof of Theorem 4.2 ahead of the proofof Theorem 4.1. Then we complete this section with the proof Theorem 4.5.
Proof of Theorem 4.2.
Let q : E → M be a fiber homotopy trivialization. We iden-tify M with a particular fiber M x = p − ( x ). Then we can assume without lossof generality that q | M : M → M is homotopic to identity. We will abbreviate thenotation for the Anosov flow g tx on M x = M to simply g t . We use Y to denotethe Anosov vector field on M defined by Y ( a ) = dg t ( a ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 , a ∈ M. Similarly Y x denotes the generating vector field for g tx , x ∈ X .For each x ∈ X consider the space F x of all homeomorphisms ϕ : M → M x whichsatisfy the following properties.P1. ϕ : M → M x is an orbit conjugacy between g t and g tx , which preserves theflow direction;P2. the map q ◦ ϕ is homotopic to id M ;P3. ϕ : M → M x is bi-H¨older continuous, i.e., both ϕ and ϕ − are H¨older con-tinuous with some positive H¨older exponent (which can depend on ϕ );P4. ϕ : M → M x is differentiable along Y ; moreover, the derivative ϕ ′ : M → R along Y defined by Dϕ ( a ) Y ( a ) = ϕ ′ ( a ) Y x ( ϕ ( a ))is a positive H¨older continuous function (whose H¨older exponent depends on ϕ ). UNDLES WITH HYPERBOLIC DYNAMICS 15
Remark . Note that if ϕ ∈ F x then ϕ − : M x → M is differentiable along Y x .Moreover the derivative of ϕ − along Y x is given by 1 /ϕ ′ ◦ ϕ − and hence is alsoH¨older continuous.In particular, we have defined the space F def = F x of self orbit conjugacies of g t thatare homotopic to identity and satisfy regularity properties P3 and P4. It is routineto check that F is a non-empty group when equipped with composition operation.By the same routine check F acts on F x by pre-composition. Lemma . The action of F on the space F x is free and transitive for each x ∈ X . Proof.
Fix x ∈ X . It is obvious that F acts freely on F x .Now we check that F x is non-empty. Consider a path γ : [0 , → X that connects x to x . Then we have a one parameter family of Anosov flows g tγ ( s ) , s ∈ [0 , s ∈ [0 ,
1] has a neighborhood
U ⊂ [0 ,
1] such thatfor each r ∈ U the flow g tγ ( s ) is orbit conjugate to g tγ ( r ) via orbit conjugacy ξ s,r which is C close to identity (in a chart). This orbit conjugacy is bi-H¨older and itis easy to choose it so that it is differentiable along Y γ ( s ) . The fact that ξ s,r can bechosen so that the derivative of ξ s,r along Y γ ( s ) is H¨older continuous is more subtle.One way to see this is from the proof of structural stability via the implicit functiontheorem, see e.g., [LMM86, Theorem A.1], where continuity of the derivative of ξ s,r along Y γ ( s ) is established. The same proof goes through to yield H¨older continuity,see [KKPW89, Proposition 2.2]. (Note that at this point we do not care aboutdependence of ξ s,r on the parameter r .) Next, since [0 ,
1] is compact, we can find afinite sequence 0 = s < s < . . . < s N = 1 and orbit conjugacies ξ s i − ,s i between g tγ ( s i − ) and g tγ ( s i ) , i = 1 . . . N . Then it is clear that ϕ x def = ξ s N − ,s N ◦ ξ s N − ,s N − ◦ . . . ◦ ξ s ,s belongs to the space F x .We proceed to checking that F acts transitively on F x . Take any ψ ∈ F x . Then,since ψ = ϕ x ◦ ( ϕ − x ◦ ψ ), we only need to check that ϕ − x ◦ ψ ∈ F . Clearly ϕ − x ◦ ψ satisfies properties P1, P3 and P4. Property P2 also holds. Indeed, ϕ − x ◦ ψ ≃ ϕ − x ◦ id M x ◦ ψ ≃ ϕ − x ◦ ( q | M x ) − ◦ q | M x ◦ ψ ≃ id M ◦ id M = id M . (Here ( q | M x ) − stands for a homotopy inverse of q | M x ). (cid:3) Let F ( X ) = G x ∈ X F x . We equip F ( X ) with C topology using charts of p : E → X . Group F acts on F ( X ) by pre-composition. Clearly this action is continuous. Lemma . For each x ∈ X there exists aneighborhood U of x and a continuous family of homeomorphisms ξ z : M x → M z , z ∈ U , such that1. ξ x = id M x ;2. ξ z is an orbit conjugacy between g tx and g tz for each z ∈ U ;3. ξ z is bi-H¨older continuous for each z ∈ U ;4. ξ z is differentiable along Y x for all z ∈ U ; moreover, the derivative of ξ z along Y x is H¨older continuous and depends continuously on z , z ∈ U . UNDLES WITH HYPERBOLIC DYNAMICS 16
Remark . In fact, dependence on z ∈ U is smooth, but we won’t need it.By using a chart about x for the bundle p : E → X one readily reduces Lemma 4.8to a lemma about a smooth multi-parameter family of Anosov flows on M x . Inthis form Lemma 4.8 was proved in [KKPW89, Proposition 2.2] using Moser’simplicit function theorem approach to structural stability. A more detailed proof isin [LMM86, Theorem A.1], however [LMM86] does not address H¨older continuity ofthe derivative along the flow. Both [KKPW89] and [LMM86] treat one parameterfamilies of Anosov flows but the proof can be adapted to multi-parameter settingin a straightforward way. Lemma . The map F ( p ) : F ( X ) → X that assigns to each ϕ ∈ F x ⊂ F ( X ) itsbase-point x is a locally trivial principal fiber bundle with structure group F . Proof.
Take any x ∈ X . Then Lemma 4.8 gives a neighborhood U and a continuousfamily of homeomorphisms ξ z : M x → M z , z ∈ U . Consider a chart U × F →F ( p ) − ( U ) given by ( z, ϕ ) ξ z ◦ ϕ x ◦ ϕ. By Lemma 4.8 this chart is continuous. By Lemma 4.7 it is injective and surjective.Also it is easy to see that the inverse is continuous. Finally, Lemma 4.7 also impliesthat F ( p ) : F ( X ) → X is a principal bundle with respect to the fiberwise action of F by pre-composition. (cid:3) Lemma . Group F is contractible.This lemma implies that the bundle F ( p ) : F ( X ) → X has a global (continuous)cross-section σ : X → F ( X ) (and hence is a trivial bundle). Define τ : X × M → E by τ ( x, a ) = σ ( x )( a ) . Clearly τ is a fiber preserving homeomorphism. The inverse τ − is the positedtopological trivialization. Hence in order to finish the proof we only need to estab-lish Lemma 4.11. Proof of Lemma 4.11.
Let γ be a periodic orbit of g t : M → M and let ϕ ∈ F .Then ϕ ( γ ) is also a periodic orbit which is homotopic to γ . Hence, by assumption2 of the theorem, ϕ ( γ ) = γ for each periodic orbit γ and each ϕ ∈ F .Pick a ϕ ∈ F and recall that according to Remark 4.6 ϕ − is differentiable along Y with a H¨older continuous derivative ( ϕ − ) ′ . By applying Newton-Leibniz formulawe obtain the following statement. Claim 4.12.
For every periodic orbit γ Z T ( γ )0 ( ϕ − ) ′ ( γ ( t )) dt = T ( γ ) , where T ( γ ) is the period of γ . Therefore the integral of ( ϕ − ) ′ − g t is a transitive Anosov flow. Hence we can apply the Livshitz Theorem (see, e.g., [KH95, Theorem 19.2.1]) to ( ϕ − ) ′ − α : M → R which isdifferentiable along Y and α ′ = ( ϕ − ) ′ − , (4.1)where α ′ is the derivative of α along Y . UNDLES WITH HYPERBOLIC DYNAMICS 17
Define the following homotopy along the orbits of g t ϕ s ( x ) = g sα ( x ) ( ϕ ( x )) , s ∈ [0 ,
1] (4.2)Clearly ϕ = ϕ . Now we calculate the derivative of ϕ along Yϕ ′ ( x ) = ( g α ( x ) ) ′ ( ϕ ( x )) ϕ ′ ( x ) = (1 + α ′ ( ϕ ( x ))) ϕ ′ ( x ) ( . ) = ( ϕ − ) ′ ( ϕ ( x )) ϕ ′ ( x ) = 1 . Thus ϕ is time preserving self orbit conjugacy of g t , i.e., ϕ is a true self conjugacy.Also it clear from (4.2) that ϕ and ϕ coincide on the space of orbits and hence, ϕ ( γ ) = γ for every periodic orbit γ . Hence we are able to apply the main resultof [JH91] to conclude that ϕ = g t for some t ∈ R .When s = 1 equation (4.2) becomes g t ( x ) = g α ( x ) ( ϕ ( x ))or ϕ ( x ) = g t − α ( x ) ( x ) . Hence for each ϕ ∈ F there exists a continuous function β ϕ : M → R such that ϕ = g β ϕ . Claim 4.13.
The map F ∋ ϕ β ϕ ∈ C ( M, R ) is continuous.Proof. Let ρ > g t .Assume that ϕ is a discontinuity point of the map ϕ β ϕ . Then there exists ε > ψ ∈ F such that d C ( ϕ, ψ ) < ε and d C ( β ϕ , β ψ ) > ε. (4.3)We can assume that ε ≪ ρ .Pick a point x ∈ M such that | β ϕ ( x ) − β ψ ( x ) | > ε . Consider a neighborhood U of size ρ about ϕ ( x ). Let O ( x ) = { g t ( x ) : t ∈ R } . Then ψ ( x ) ∈ O ( x ) ∩ U . By (4.3)points ψ ( x ) and ϕ ( x ) belong to different connected components of O ( x ) ∩ U asindicated on Figure 1. It follows that, in fact, | β ϕ ( x ) − β ψ ( x ) | > ρ . x U ϕ ( x ) ψ ( x ) O ( x ) Figure 1.
UNDLES WITH HYPERBOLIC DYNAMICS 18
By continuity the picture on Figure 1 is robust. Therefore we can perturb x (wecontinue to denote this perturbation by x ) to ensure that O + ( x ) = { g t ( x ) : t ≥ } is dense in M .Note that as a point y moves along O + ( x ) the points ϕ ( y ) and ψ ( y ) sweep thepositive semi-orbits O + ( ϕ ( x )) and O + ( ψ ( x )) respectively. But d ( ϕ ( y ) , ψ ( y )) < ε for all y . Hence the picture in the local product structure neighborhood of ϕ ( y )remains the same as y moves along O + ( x ). We conclude that ψ ( x ) belongs tothe weak stable manifold of ϕ ( x ). Hence the distance from ψ ( y ) to the local orbit { g t ( ϕ ( y )) : t ∈ ( − ε, ε ) } goes to zero as y moves along O + ( x ). Because O + ( x )is dense, we have that, in fact, the distance between ψ ( z ) and the local orbit { g t ( ϕ ( z )) : t ∈ ( − ε, ε ) } is zero for all z ∈ M . This means that d C ( β ϕ , β ψ ) < ε ,which yields a contradiction. (cid:3) Define ∆ : [0 , × F → F by ∆ s ( ϕ ) = g (1 − s ) β ϕ . Claim 4.13 implies that ∆ is continuous. Clearly ∆ = id F and ∆ maps F to id M .Hence ∆ deformation retracts F to a point. This finishes the proof of Lemma 4.11and, hence, the proof of Theorem 4.2. (cid:3) Proof of Theorem 4.1.
A substantial part of the proof is exactly the same as in thepreceding proof, yet there some very different ingredients which, in particular, allowto avoid the use of the assumption on non-homotopic periodic orbits of Theorem 4.2and utilize triviality of π ( X ) instead. In order to keep the exposition coherent wehave allowed a few repetitions from the preceding proof.As before, we fix a base-point x ∈ X and identify M with M x = p − ( x ).Further we abbreviate the notation for the Anosov flow g tx on M x = M to g t . Weuse Y to denote the Anosov vector field on M and Y x for the vector field whichgenerates g tx , x ∈ X .For each x ∈ X consider the space F x of all homeomorphisms ϕ : M → M x which satisfy the following properties. (Cf. the definition of F x in the proof ofTheorem 4.2.)P1. ϕ : M → M x is an orbit conjugacy between g t and g tx , which preserves theflow direction;P3. ϕ : M → M x is bi-H¨older continuous, i.e., both ϕ and ϕ − are H¨older con-tinuous with some positive H¨older exponent (which can depend on ϕ );P4. ϕ : M → M x is differentiable along Y ; moreover, the derivative ϕ ′ : M → R along Y defined by Dϕ ( a ) Y ( a ) = ϕ ′ ( a ) Y x ( ϕ ( a ))is a positive H¨older continuous function (whose H¨older exponent depends on ϕ );Note that, in particular, we have defined the space F def = F x of certain special selforbit conjugacies of g t . Following closely the arguments in the proof of Theorem 4.2we obtain the following claims.1. Space F is a non-empty group when equipped with composition operation.2. The action of F on the space F x by pre-composition is free and transitivefor each x ∈ X . UNDLES WITH HYPERBOLIC DYNAMICS 19
Now let F ( X ) = G x ∈ X F x . We equip F ( X ) with C topology using charts of p : E → X .3. The map F ( p ) : F ( X ) → X that assigns to each ϕ ∈ F x ⊂ F ( X ) its base-point x is a locally trivial principal fiber bundle with structure group F .We call the bijection between the space of orbits of g t and the space of orbitsof g tx induced by a ϕ ∈ F x a marking of orbits of g tx . There is an obvious map F x → M x to the space M x of all possible markings of orbits of g tx . Let M ( X ) = G x ∈ X M x . The assembled map mark : F ( X ) → M ( X ) induces a topology on M ( X ). Thistopology can be described as follows. Markings m x ∈ M x and m y ∈ M y are closeif and only if x is close to y and there exists an orbit conjugacy ξ xy : M x → M y between g tx and g ty which is C close to identity (in a chart) and takes the marking m x to the marking m y . Note that, by structural stability, for any y sufficiently closeto x there exist an orbit conjugacy ξ xy . Moreover, among orbit conjugacies C closeto identity, ξ xy is unique up to homotopy that moves points a short distance alongthe orbits (see [PSW12, Section 5] for the details and discussion of this uniquenessproperty). It immediately follows that the map M ( p ) : M ( X ) → X which takes m ∈ M x ⊂ M ( X ) to its base-point x is a covering map. Because X issimply connected the restriction of M ( p ) to each connected component of M ( X ) isa one sheeted covering. Denote by ¯ M ( X ) the connected component containing thetrivial self marking of g t in M ( X ). Let¯ F ( X ) = ( mark ) − ( ¯ M ( X ))and let ¯ F be the preimage of the trivial self marking, i.e., ¯ F = { ϕ ∈ F : ϕ induces trivial self marking of g t } . Clearly ¯ F ( X ) is open and closed subbundle of F ( X ) and, using the third claimabove, it is straightforward to verify that ¯ F ( X ) → X is a principal fiber bundlewith structure group ¯ F .By the definition of ϕ ∈ ¯ F , we have ϕ ( γ ) = γ for all orbits γ . With thisinformation on the structure group ¯ F we can show that ¯ F is contractible and thendeduce that bundle p : E → X is topologically trivial. The technical conditions P3and P4 guarantee that this last step can be carried out in exactly the same way asin the proof of Theorem 4.2. (cid:3) Proof of Theorem 4.5.
Let T n → E p → S d be the smoothly non-trivial fiber bundleconstructed in Section 3.2 and let f : E → E be the fiberwise Anosov diffeomor-phism constructed in Section 3.3 for the proof of Theorem 3.7. Let E f be themapping torus of f , i.e., E f = E × [0 , / ( x, ∼ ( f ( x ) , . Then E f is the total space of the fiber bundle T nf → E f p f → S d whose fibers T nf,x , x ∈ S d , are the mapping tori of f | T nx , x ∈ S d . It is easy to check that the suspension UNDLES WITH HYPERBOLIC DYNAMICS 20 flow on E f is a fiberwise Anosov flow which satisfies the assumptions of Theorem 4.2.To complete the proof we will show that p f : E f → S d is not smoothy trivial.The fundamental group of the mapping torus fiber T nf,x is Z f ∗ ⋉ Z n , where f ∗ : Z n → Z n is the automorphism of π ( T n ) = Z n induced by f | T nx . The inclusion T nf,x ⊂ E f induces a canonical isomorphism π ( T nf,x ) → π ( E f ). Hence, both π ( E f ) and π ( S d × T nf ) are canonically identified with Z f ∗ ⋉ Z n .Now, assume to the contrary that there exists a fiber preserving diffeomorphism ϕ : E f → S d × T nf . For both, E f and S d × T nf , consider covers that correspond tothe normal subgroup Z n ⊂ Z f ∗ ⋉ Z n , Clearly, the covering spaces are E × R and S d × T n × R , respectively. Claim 4.14.
Diffeomorphism ϕ : E f → S d × T nf admits a lifting ˜ ϕ : E × R → S d × T n × R .Proof. Using the fact that f ∗ is hyperbolic one can check that Z n ⊂ Z f ∗ ⋉ Z n isthe commutator subgroup. It follows that the subgroup Z n is invariant under allautomorphisms of Z f ∗ ⋉Z n . Hence ϕ ∗ ( Z n ) = Z n . Therefore, by the lifting criterion, ϕ admits a lifting to the covers that correspond to Z n . (Recall that both π ( E f )and π ( S d × T nf ) are canonically identified with Z f ∗ ⋉ Z n .) (cid:3) Note that ˜ ϕ is a smooth trivialization of the bundle E × R → S d . We cut S d × T nf along ˜ ϕ ( E × { } ). For sufficiently large t the submanifold S d × T n × { t } is disjointwith ˜ ϕ ( E × { } ) and by cutting again along S d × T n × { t } we obtain a cobordism W d + n +1 such that ∂ + W d + n +1 = E and ∂ − W d + n +1 = S d × T n . It is easy to checkthat W d + n +1 is an h -cobordism. Since the Whitehead group W h ( π ( S d × T n ))vanishes the s -cobordism theorem applies (by construction in Section 3.2, d + n ≥ E is diffeomorphic to S d × T n . Moreover, W d + n +1 smoothlyfibers over S d with structure group Diff( T n × [0 , , T n × { } ), i.e., its structuregroup is the group of all smooth pseudo-isotopies P ( T n ) of the torus T n . Thisbundle must of course be trivial over each of the two hemispheres D d ± of S d . Weproceed to use this fact to construct a weakly smooth trivialization of the bundle p : E → S d . (Recall the definition preceding Addendum 3.12.) This will contradictAddendum 3.12 according to which p : E → S d is not weakly smoothly trivial. Andthis contradiction will finish the proof of Theorem 4.5. So it remains to construct g . We obtain g | D d − × T n from the trivialization of the bundle T n × [0 , → W d + n +1 | D d − → D d − mentioned above. Then we apply the s -cobordism theorem to the h -cobordism( W d + n +1 | D d + , D d + × T n ) to extend the product structure on W d + n +1 | D d + ∩ D d − to all of W d + n +1 | D d + . Note that this extension need not be fiber preserving; i.e., g needn’t bea smooth bundle trivialization. But it is, at least, a weak smooth trivialization. (cid:3) Proof of Theorem 1.6
Let M be a smooth closed n -dimensional manifold and let f : M → M be adiffeomorphism. The differential map Df : T M → T M is linear when restricted tofibers and, hence, induces a bundle self map of SM which we still denote by Df .In this way the group Diff( M ) of all self diffeomorphisms of M also acts on SM and, hence, embeds into Diff( SM ). UNDLES WITH HYPERBOLIC DYNAMICS 21
Strategy of the proof.
We will fix a closed orientable real hyperbolic man-ifold M , whose dimension n is an odd integer and n ≫ p − ϕ : S p − → Diff( M ) such that the inducedmap Dϕ : S p − → Diff( SM ) , when composed with the natural inclusion σ : Diff( SM ) ֒ → Top( SM ), representsa non-zero element in π p − (Top( SM )). Applying the clutching construction to α yields the smooth bundle M → E → S p − posited in Theorem 1.6. (The definitionof clutching was given in Section 3.4.) Then notice that SM → SE → S p − is the result of the clutching construction applied to Dϕ . This bundle is not topo-logically trivial if and only if the class of the composition σ ◦ Dϕ in the group π p − (Top( SM )) is not zero. So it remains to do the construction.5.2. The clutching map.
We will use P s ( · ) and P ( · ) to denote the smooth andtopological pseudo-isotopy functors, respectively.Identify S × D n − with a closed tubular neighborhood of an essential simpleclosed curve in M which represents an indivisible element of π ( M ). Abbreviate S × D n − by T ⊂ M and note that ∂T = S × S n − . Denote the “top” of apseudo-isotopy P ( T ) , P ( M ), etc. , by f , i.e., f : M → M is f | M ×{ } . This gives amap P ( T ) → Top( T ).Consider the map v : P s ( T ) → Top( SM ) defined by the following composition P s ( T ) ι → P s ( M ) top −→ Diff( M ) D → Diff( SM ) ֒ → Top( SM ) , (5.1)where ι is induced by the inclusion T ֒ → M ( i.e., ι ( f ) coincides with f on T andis identity on M \ T ). We will show that π p − ( v )( π p − ( P s ( T ))) is a non-trivialgroup. Then Theorem 1.6 follows as discussed in the Section 5.1 above.Since T is parallelizable, we can (and do) fix an identification of ST with T × S n − as bundles over T . This allows us to define a map b : P ( T ) → P ( SM )as follows b f | ( SM \ S ( Int ( T ))) × [0 , = id and b f | T × S n − × [0 , = f × id. Note that f b f is a group homomorphism.Denote by u the composite map P s ( T ) ֒ → P ( T ) b → P ( SM ) top −→ Top( SM ) . (5.2)Let Z ∞ p denote the countably infinite direct sum of copies of the cyclic group Z p of order p . Lemma . There exist a subgroup Z ∞ p ⊂ π p − ( P s ( T )) which maps monomor-phically into π p − Top( SM ) under u ∗ = π p − ( u ), i.e., under the homomorphisminduced by u . Corollary . The image v ∗ ( Z ∞ p ) in π p − (Top( SM )) is not finitely generated and,hence, is non-trivial. UNDLES WITH HYPERBOLIC DYNAMICS 22
Proof.
First notice that each of the two compositions, (5.1) and (5.2), send f ∈ P s ( T ) to two maps v ( f ) : SM → SM and u ( f ) : SM → SM that cover the samemap top( ι ( f ))of M . Hence the map w ( f ) def = u ( f ) ◦ v ( f ) − covers id M and, therefore, is equivalent to a smooth map ϕ : M → GL ( n, R ) suchthat ϕ ( x ) = id R n when x / ∈ T .Notice that u ∗ = w ∗ + v ∗ , where maps u ∗ , w ∗ and v ∗ are the functorially induced group homomorphisms π p − P s ( T ) → π p − Top( SM ).So if w ∗ ( Z ∞ p ) and v ∗ ( Z ∞ p ) were both finitely generated, then, u ∗ ( Z ∞ p ) wouldalso be finitely generated. Since it is not, by Lemma 5.1, we conclude that eitherimage w ∗ ( Z ∞ p ) or image v ∗ ( Z ∞ p ) is not finitely generated. We will presently showthat w ∗ ( Z ∞ p ) is finitely generated and thus conclude that v ∗ ( Z ∞ p ) is not finitelygenerated proving the Corollary.For this consider the bundle GL ( n, R ) → Aut ( T M ) → M, associated to the tangent bundle R n → T M → M in the sense of Steenrod [St51].The fiber of this bundle is the the space of all invertible linear transformations ofthe fiber T x M , the tangent space at x ∈ M . (Caveat: this is not the principalframe bundle associated to T M .) Let Γ be the space of all smooth cross-sectionsto this bundle. Note that there is a canonical mapΓ → Diff( SM )and, as we observed above, w factors through the composite mapΓ → Diff( SM ) → Top( SM ) , where Γ is the subspace of Γ consisting of all cross-sections which map pointsoutside of T to id in the fiber over that point. (Indeed, every element in Γ isequivalent to a smooth map ϕ : M → GL ( n, R ) such that ϕ ( x ) = id R n when x / ∈ T .)Hence, to prove that the image w ∗ ( Z ∞ p ) is finitely generated, it suffices to show thefollowing claim. Claim 5.3.
Let k = 2 p − . Then the group π k Γ is finitely generated. Because T is parallelizable, Γ is clearly homeomorphic to the space of all smoothmaps ( T, ∂T ) → ( GL ( n, R ) , id )and therefore π k Γ ≃ [ D k × T, ∂ ( D k × T ); O ( n, R ) , id ]= [ S × D n + k − , ∂ ( S × D n + k − ); O ( n, R ) , id ]= [ S n + k ∨ S n + k − , wedge pt.; O ( n, R ) , id ]= π n + k O ( n, R ) ⊕ π n + k − O ( n, R ) . Hence, by Serre’s thesis, this group is finitely generated. (cid:3)
UNDLES WITH HYPERBOLIC DYNAMICS 23
Proof of Lemma 5.1.
Let P s ( · ) and P ( · ) denote the stable smooth andtopological pseudo-isotopy (homotopy) functors, respectively; cf. [H78, § b : P ( T ) → P ( SM ), defined in Section 5.2, induces a map P ( T ) → P ( SM )which we still denote by b . (Recall that the are working in the stable dimensionrange.)The argument given in [FO10b, pp. 1419-20] for proving Theorem D of thatpaper carries over, when the manifold N in [FO10b] is replaced by SM , to proveLemma 5.1 provided we can find a subgroup Z ∞ p of π k ( P s ( T )), k = 2 p −
4, suchthat the composite map Z ∞ p ⊂ π k ( P s ( T )) → π k ( P ( T )) b ∗ −→ π k ( P ( SM )) → H ( Z ; π k ( P ( SM ))) (5.3)is one-to-one. For this reduction we use the fact that the homomorphism inducedby inclusion π k ( P s ( T )) → π k ( P ( T ))is an isomorphism modulo the Serre class of finitely generated abelian groups;cf. [FJ87, Lemma 4.1]. To find such Z ∞ p consider the following commutative dia-gram Z ∞ p ⊂ / / π k ( P s ( T )) / / π k ( P ( T )) / / H ( Z ; π k ( P ( T ))) Z ∞ p × O O ⊂ / / π k ( P s ( T )) × O O / / π k ( P ( T )) × O O / / π k ( P ( ST )) q ∗ f f ▼▼▼▼▼▼▼▼▼▼▼ / / i ∗ (cid:15) (cid:15) H ( Z ; π k ( P ( ST ))) H ( q ∗ ) O O H ( i ∗ ) (cid:15) (cid:15) π k ( P ( SM )) / / H ( Z ; π k ( P ( SM ))) Comments on the diagram:
1. The letter q denotes the bundle projections in both of the bundles ST → T and SM → M . (The latter one does not appear in the diagram.)2. The triangle in this diagram commutes because of [H78, Proposition onp. 18]. (Recall that the dimension n is an odd integer.)3. The subgroup Z ∞ p in π k ( P s ( T )) comes from [FO10a, Proposition 4.5] usingthe remarks on the last three lines of page 1420 of [FO10b] and the firstthree lines on the next page. By its construction Z ∞ p maps in a one-to-onefashion into H ( Z ; π k ( P ( N ))) via the composition of the maps given on thetop line of the diagram.4. The functorially induced maps q ∗ : π k ( P s ( ST )) → π k ( P s ( T )) ,q ∗ : π k ( P s ( SM )) → π k ( P s ( M ))are Z -modules isomorphisms because of [H78, Corollary 5.2 and Lemma onp. 18] and the facts that k < n − n − Z on P ( T ) and P ( M ) are induced by “turning a pseudo-isotopy upside down.”(See [H78, pp. 6-7] for more details.) Consequently, the functorially inducedmap H ( q ∗ ) : H ( Z ; π k ( P s ( ST )) → H ( Z ; π k ( P s ( T ))is an isomorphism. UNDLES WITH HYPERBOLIC DYNAMICS 24
5. The inclusion maps of T into M and ST into SM are both denoted by i .Then i ∗ : π k ( P s ( T )) → π k ( P s ( M )) ,i ∗ : π k ( P s ( ST )) → π k ( P s ( SM ))denote the functorially induced group homomorphisms which are both Z -module maps. Furthermore, H ( i ∗ ) : H ( Z ; π k ( P s ( ST ))) → H ( Z ; π k ( P s ( SM )))is the group homomorphism induced functorially by i ∗ .6. The main result of [FJ87] shows that i ∗ is an isomorphism onto a directsummand of π k ( P ( M )) as Z -modules. Hence, so is i ∗ : π k ( P s ( ST )) → π k ( P s ( SM )) by comment 4 above (note that q ◦ i = i ◦ q ). Consequently, H ( i ∗ ) is monic.Now “chasing the diagram” verifies that the composite map (5.3) is one-to-onecompleting the proof of Lemma 5.1, and thus also Theorem 1.6.6. Proof of Theorem 3.2
Denote by G ( M ) the space of self homotopy equivalences of M and by G ( M ) ⊂ G ( M ) the connected component of id M . Also denote by Top( M ) the group ofself homeomorphisms of M and by Top ( M ) subgroup of Top( M ) which consistsof homeomorphisms homotopic to id M (through a path of maps). The proof ofTheorem 3.2 is based on the following lemma. Lemma . Let M be a (closed) infranilmanifold. Then Top ( M ) contains a con-nected topological subgroup S satisfying the following properties.1. The group S is a connected Lie group.2. The composition σ : S → G ( M ) of the inclusion maps S ⊂ Top ( M ) andTop ( M ) ⊂ G ( M ) induces an isomorphism σ ∗ : π i ( S ) → π i G ( M )for all i .3. If L : M → M is an affine Anosov diffeomorphism of M , then F def= { ϕ ∈ Top ( M ) : ϕ ◦ L = L ◦ ϕ } is a subgroup of S .We proceed with the proof of Theorem 3.2 assuming Lemma 6.1. Let q : E → M be a fiber homotopy trivialization of the Anosov bundle M → E p → X. As in the proof of Theorem 4.2 we identify M with a particular fiber M x = p − ( x ). Recall that we can assume without loss of generality that q | M : M → M is homotopic to identity; moreover, by performing a homotopy in a chart we can(and do) assume that q | M = id M . Also recall that the bundle M → E → X admits a fiberwise Anosov diffeomorphism. Recall that we denote by f x , x ∈ X ,the Anosov diffeomorphisms of the fibers M x , x ∈ X . We will write f : M → M for the restriction f x : M → M of the Anosov bundle map to the fiber M = M x .For each x ∈ X consider the space F x of all homeomorphisms ϕ : M → M x whichare conjugacies between f and f x and which also satisfy the property “ q ◦ ϕ : M → M UNDLES WITH HYPERBOLIC DYNAMICS 25 is homotopic to id M .” It follows from the uniqueness part of the structural stabilitytheorem that each F x is a discrete topological space.Let F ( X ) = G x ∈ X F x . Also let F ( p ) : F ( X ) → X be the map that sends the fibers F x to their base-points, x ∈ X . By an argument similar to that given in the proof of Theorem 4.2, F ( p ) : F ( X ) → X is a covering of X . Moreover, it is a regular covering space; i.e., a principal bundle with structure group F x = { ϕ ∈ Top ( M ) : ϕ ◦ f = f ◦ ϕ } . By work of Franks and Manning [Fr69, M74], the Anosov diffeomorphism f : M → M is conjugate to an affine automorphism L : M → M via a conjugacy homotopicto id M . This conjugacy gives an isomorphism between topological groups F x and F . Thus we can (and do) identify F x and F .Notice that the associated M -bundle F ( X ) × F M is isomorphic to E via ( ϕ, y ) ϕ ( y ), as Top( M )-bundles. Hence we have reduced the structure group of the bundle p : E → X from Top( M ) to the subgroup F and, therefore, to S , where S is thetopological subgroup of Top ( M ) posited in Lemma 6.1. Consequently, it sufficeto show that the principal S -bundle F ( X ) × F S → X associated to p : E → X istrivial.Recall that the total space F ( X ) × F S is the quotient of F ( X ) × S by the actionof F , given by ( ϕ, s ) ( ϕ ◦ ψ, ψ − ◦ s ), ψ ∈ F . Therefore the map η : F ( X ) × F S → G ( M ) given by ( ϕ, s ) η q ◦ ϕ ◦ s is well defined. Notice that because q | M = id M , map η restricted to the fiber over x is the composite map σ : S → Top ( M ) → G ( M ) . Because G ( M ) has the homotopy type of a CW complex, which is a consequenceof [M59, Theorem 1], map σ is a homotopy equivalence by property 2 of Lemma 6.1and, hence, has a homotopy inverse σ − . Then the map σ − ◦ η : F ( X ) × F S → S is ahomotopy equivalence when restricted to the fibers, i.e., the bundle F ( X ) × F S → X is fiber homotopically trivial. Hence it has a cross-section and consequently is atrivial S -bundle.To finish the proof of Theorem 3.2 it remains to establish Lemma 6.1. Proof of Lemma 6.1.
This is a consequence of the construction and argumentsmade in Appendix C of [FG13]. In particular, S is N G , i.e., the connected compo-nent containing identity element of the Lie group N G from Fact 23 of Appendix C.We recall its definition.The universal cover of M is a simply connected nilpotent Lie group N and π ( M ),considered as the group of deck transformations of N , is a discrete subgroup of N ⋊ G , where G is a finite group of automorphisms of N which maps monomorphicallyinto Out ( N ). Also π ( M ) projects epimorphically onto G . Let Γ = π ( M ) ∩ N , i.e., the subgroup of π ( M ) that acts by pure translations. The homogeneous space c M = N/ Γ is a nilmanifold, which is a regular finite-sheeted cover of M whose group UNDLES WITH HYPERBOLIC DYNAMICS 26 of deck transformations can be identified with G . The natural action of N on c M gives a representation ˆ ρ : N → Top ( c M ) , whose kernel is Z ( N ) ∩ Γ, where Z ( N ) denotes the center of N . The image of thisrepresentation is the nilpotent Lie group N , i.e., N = N/ Z (Γ) . (It is a consequence of Mal ′ cev rigidity that Z (Γ) = Z ( N ) ∩ Γ, see [M49].) Now G acts on N by conjugation. Let N G denote the subgroup of N which is fixed by G .The group N G maps isomorphically onto a topological subgroup of Top ( M ) whichwe also denote by N G . Then the connected Lie group S posited in Lemma is theconnected component of identity element N G . The argument on the last 10 lines ofAppendix C (crucially using [W70]) shows that F is a subgroup of S , i.e., verifiesproperty 3 of Lemma.So it remains to verify property 2. Recall (1.1) π n ( G ( M )) = ( Z ( π ( M )) , if n = 10 , if n ≥ π n ( S ).The group S is a connected nilpotent Lie group, hence π n ( S ) = 0 for n ≥
2. Tocalculate π ( S ), we use that ρ : N G → S is a covering space. To see this apply the functor ( · ) G to the following (coveringspace) exact sequence 1 → Z (Γ) → N → N → , which yields the half exact sequence1 → Z (Γ) G → N G → N G → . Then, using a Lie algebra argument, we see that the image of N G in N G is N G = S ,since N G is connected by Fact 22 of [FG13, Appendix C]. Moreover, N G is con-tractible since it is a closed connected subgroup of the simply connected nilpotentLie group N . Consequently, π ( S ) = ker( ρ ) = Z (Γ) G . Claim 6.2.
Group Z (Γ) G is isomorphic to Z ( π ( M )) .Proof. An element v is in Z (Γ) G if and only if the following two conditions aresatisfied.1. v ∈ Z (Γ);2. ∀ h ∈ G , h ( v ) = v .Then, the inclusion Z (Γ) G ⊂ Z ( π ( M )) follows from the following identity( h, u )( id N , v )( h, u ) − = ( id N , uh ( v ) u − ) (6.1)valid for all h ∈ G and u, v ∈ N .To check the opposite inclusion Z ( π ( M )) ⊂ Z (Γ) G take ( g, v ) ∈ Z ( π ( M )).Then uh ( v ) = vg ( u ) (6.2)for all ( h, u ) ∈ Z ( π ( M )). By choosing h = id N we obtain g ( u ) = v − uv for all u ∈ Γ, and, hence, g is an inner automorphism of N . But G maps monomorphically UNDLES WITH HYPERBOLIC DYNAMICS 27 into
Out ( N ); therefore, g = id N and v ∈ Z (Γ). Finally, to see that ( id N , v ) ∈Z ( π ( M )) satisfies the condition 2 as well, we use the fact that the composition π ( M ) ⊂ G ⋉ N → G is an epimorphism in conjunction with (6.1) and (6.2). (cid:3) Therefore, the above claim implies π ( S ) ≃ π ( G ( M )).It remains to show that σ ∗ : π ( S ) → π ( G ( M )) is an isomorphism. However, π ( S ) ⊂ Z (Γ) and, hence, π ( S ) is a finitely generated abelian group. Thereforeit suffices to show that σ ∗ is an epimorphism, but this is a consequence of Fact 22of [FG13, Appendix C]. (cid:3) To prove Addeddum 3.5 we need the following modification of Lemma 6.1.
Lemma . Let M be a (closed) nilmanifold. Then Top( M ) contains a topologicalsubgroup S satisfying the following properties.1. The group S is a Lie group.2. The composition σ : S → G ( M ) of the inclusion maps S ⊂ Top( M ) andTop( M ) ⊂ G ( M ) induces an isomorphism σ ∗ : π i ( S ) → π i G ( M )for all i ≥ L : M → M is an affine Anosov diffeomorphism of M , then F def= { ϕ ∈ Top( M ) : ϕ ◦ L = L ◦ ϕ } is a subgroup of S .We proceed with the proof of Addeddum 3.5 assuming Lemma 6.3. Proof of Addeddum 3.5.
By the argument used to prove Theorem 3.2 we can iden-tify the nilmanifolds M i = N i / Γ i , i = 1 ,
2, with fibers M i,x for a fixed base point x ∈ X . By the argument used in the proof of Theorem 3.2 the Anosov diffeomor-phisms f i : M i → M i can be identified with affine Anosov diffeomorphisms L i . Weproceed as in the proof of Theorem 3.2 separately for each bundle p i : E i → X . For x ∈ X consider the fibers F i,x that consist of all homeomorphisms ϕ : M i → M i,x which are conjugacies between f i and f i,x . (Note we drop the constraint that in-volves fiber homotopy equivalence.) By taking the union of these fibers we stillobtain principal bundles F ( p i ) : F i ( X ) → X, which reduce (inside of Top( M i )) the structure group of each bundle p i : E i → X to the group F i def= { ϕ ∈ Top( M i ) : ϕ ◦ L i = L i ◦ ϕ } . By Mal ′ cev’s rigidity theorem, the given homotopy equivalence M = M ,x to M = M ,x determines an affine diffeomorphism between M and M via whichwe identify M and M to a common nilmanifold M to which we apply Lemma 6.3.In this way we reduce the structure group of each bundle, inside Top( M ), to theLie group S posited in Lemma 6.3.Let k i : X → BS , i = 1 ,
2, be a continuous map which classifies the bundle E i in terms of its reduced structure group S . Here BS denotes the classifyingspace for bundles with structure group S . Similarly, BG ( M ) is the classifying UNDLES WITH HYPERBOLIC DYNAMICS 28 space constructed by Stasheff [St63] for homotopy M -bundles up to fiber homotopyequivalence, cf. [MM79, pages 3-7]. The inclusion map S ⊂ G ( M ) induces a map η : BS → BG ( M ) , which corresponds to the forget structure map at the bundle level. Because ofproperty 2 in Lemma 6.3, η is a weak homotopy equivalence and hence induces abijection η ∗ : [ X, BS ] → [ X, BG ( M )]between homotopy classes of maps, cf. [S66, p. 405, Cor. 23].Since the bundles E and E are fiber homotopically equivalent, the compositemaps η ◦ k and η ◦ k are homotopic. Consequently, k is homotopic to k . Let σ : BS → B Top( M ) denote the map induced by inclusion S ⊂ Top( M ). Then σ ◦ k is homotopic to σ ◦ k and therefore E and E are equivalent as Top( M )-bundles. (cid:3) To finish the proof of Addeddum 3.5 it remains to establish Lemma 6.3.
Proof of Lemma 6.3.
We start by recollecting some terminology. First recall that M = N/ Γ, where N is a simply connected nilpotent Lie group and Γ ⊂ N is adiscrete cocompact subgroup. An affine diffeomorphism of N is the composite ofan automorphism α ∈ Aut ( N ) with a left translation L a : N → N , where a ∈ N ,which is defined by x ax, for all x ∈ N. Likewise, the right translation R a : N → N and the inner automorphism I a : N → N are defined by x xa and x axa − , respectively. Note that L a induces a self diffeomorphism of the left coset space N/ Γ(nilmanifold) given by the formula x Γ ax Γ . The right translation R a also induce a self diffeomorphism when a ∈ Γ. Denotethese diffeomorphisms again by L a and R a , respectively.Mal ′ cev showed that every α ∈ Aut (Γ) induces an automorphism ¯ α ∈ Aut ( N ),which in turn induces a self diffeomorphism (denoted again by) α ∈ Diff( N/ Γ) givenby the formula x Γ ¯ α ( x )Γ . In particular, if a ∈ Γ then the inner automorphism I a induces a self diffeomorphism I a ∈ Diff( N/ Γ). An affine diffeomorphism of N/ Γ is by definition the composite ofmaps of the form α and L a in Diff( N/ Γ). Note that a L a induces an homomor-phism of N onto a closed subgroup of Top( N/ Γ) whose kernel is Z ( N ) ∩ Γ = Z (Γ);while Aut (Γ) → Diff( N/ Γ) induces an embedding of
Out (Γ) onto a discrete sub-group of Top( N/ Γ). We note the following identities α ◦ L a ◦ α − = L α ( a ) ; R a = L a ◦ I a − = I a − ◦ L a , for all a ∈ N and α ∈ Aut ( N ). From these one easily deduces the followingequations1. Aff ( N ) ≃ Aut ( N ) ⋉ N ;2. Aff ( N/ Γ) ≃ Aut (Γ) ⋉ N/σ (Γ),
UNDLES WITH HYPERBOLIC DYNAMICS 29 where σ : Γ → Aff ( N ) is the monic anti-homomorphism given by σ ( a ) = R a , a ∈ Γ. (Here
Aff ( N ) and Aff ( N/ Γ) denote the closed subgroups of Top( N ) andTop( N/ Γ), respectively, consisting of all affine diffeomorphisms. Furthermore, σ (Γ)is a discrete normal subgroup of Aff ( N ).)Under the canonical projection Aut (Γ) ⋉ N to Aut (Γ), σ (Γ) maps onto the groupof all inner automorphisms of Γ, Inn (Γ) with kernel Z (Γ). Hence the canonicalexact sequence 1 → N → Aut (Γ) ⋉ N → Aut (Γ) → → N/ Z (Γ) → Aff ( N/ Γ) → Out (Γ) → Aff ( N/ Γ) ⊂ G ( M/ Γ) inducesan isomorphism on the homotopy groups π i for i = 0 , , , . . . . We now define S to be this subgroup Aff ( N/ Γ) of Top( N/ Γ) = Top( M ). Hence, we have alreadyverified the properties 1 and 2 asserted in Lemma 6.3. The remaining property 3follows immediately from Theorem 2 in Walters’ paper [W70]. (cid:3) Remark . We conjecture that Addendum 3.5 remains true in the more generalsituation where the fibers of the Anosov bundles p i : E i → X , i = 1 ,
2, are onlyassumed to be infranilmanifolds. The proof given above for Addendum 3.5 wouldwork in this more general setting provided Lemma 6.3 remained true when M isonly assumed to be an infranilmanifold.7. Final remarks
A partially hyperbolic diffeomorphism of ( S d × T n ) α . Here we explainthat the fiberwise Anosov diffeomorphism f : E α → E α constructed in Section 3 canbe viewed as a partially hyperbolic diffeomorphism and point out certain propertiesof this diffeomorphism.First we need to recall some definitions from partially hyperbolic dynamics.Given a closed Riemannian manifold N a diffeomorphism f : N → N is called partially hyperbolic if there exists a continuous Df -invariant non-trivial splitting ofthe tangent bundle T N = E sf ⊕ E cf ⊕ E uf and positive constants ν < µ − < µ + < λ , ν < < λ , and C > n > k D ( f n ) v k ≤ Cν n k v k , v ∈ E sf , C µ n − k v k ≤ k D ( f n ) v k ≤ Cµ n + k v k , v ∈ E cf , C λ n k v k ≤ k D ( f n ) v k , v ∈ E uf . It is well known that the distributions E sf and E uf integrate uniquely to foliations W sf and W uf . If the distribution E cf also integrates to an f -invariant foliation W cf then f is called dynamically coherent . Furthermore, f is called robustly dynami-cally coherent if any diffeomorphism sufficiently C close to f is also dynamicallycoherent.Using Hirsch-Pugh-Shub structural stability theorem [HPS77, Theorem 7.1] weestablish the following result. Proposition 7.1.
Let X be a closed simply connected manifold and let M → E p → X be a smoothly non-trivial bundle. Assume that f : E → E is a fiberwise UNDLES WITH HYPERBOLIC DYNAMICS 30
Anosov diffeomorphism. Then diffeomorphism f is partially hyperbolic and robustlydynamically coherent. Moreover, for any g which is sufficiently C close to f , thecenter foliation W cg is not smooth. Note that this proposition applies to f : E α → E α constructed in Section 3.Recall that, by construction, the total space E α is diffeomorphic to ( S d × T n ) α ,where Σ α is a homotopy ( d + n )-sphere, obtained by clutching using diffeomorphism α ∈ Diff k + d − ( D n + d − , ∂ ).In general, the center foliation W cf of a dynamically coherent partially hyper-bolic diffeomorphism is continuous foliation with smooth leaves. To the best ofour knowledge, all previously known examples of partially hyperbolic dynamicallycoherent diffeomorphisms f have the following property:( ⋆ ) There exists a path f t , t ∈ [0 , , of partially hyperbolic dynamically coherentdiffeomorphism such that f = f and the center foliation W cf is a smooth foliation. Conjecture 7.2.
The partially hyperbolic diffeomorphism f : E α → E α constructedin Section 3 does not have property ( ⋆ ) . Note that Proposition 7.1 supports this conjecture.
Proof of Proposition 7.1.
Let T fib E def = [ x ∈ X T M x be the bundle of vectors tangent to the fibers of p : E → X . We have a Df -invariant splitting T fib E = E sf ⊕ E uf , where E sf is exponentially contracting and E uf is exponentially expanding.Recall that the proof of Theorem 3.1 yields a topological trivialization h : E → M. Hence we have a fiber preserving homeomorphism ( p, h ) : E → X × M .For each y ∈ E consider the the preimage W cf ( y ) def = ( p, h ) − ( X × { h ( y ) } ).Because of our contruction of h , which locally comes from parametric families ofhomeomorphisms given by structural stability theorem, and smooth dependenceof these homeomorphisms on parameter ( i.e., base point) [LMM86, Theorem A.1],we obtain that, in fact, W cf ( y ) are smoothly embedded submanifolds of E whosetangent space E cf ( y ) def = T y W cf ( y ) depends continuously on y ∈ E . Hence W cf isa continuous foliation by smooth compact leaves. Moreover, this foliation is f -invariant. This follows from the fact that W cf is the pullback of the “horizontal”foliation on X × M and ( p, h ) is a conjugacy between f and id X × f | M (recall,that in the proof of Theorem 3.1 we have identified M with a particular fiber M x ).Therefore the distribution E cf is Df -invariant. Then, because the dynamics inducedby f on the base is id X , one can check that f is partially hyperbolic with respectto the Df -invariant splitting T E = E sf ⊕ E cf ⊕ E uf .Since W cf is a compact foliation, Hirsch-Pugh-Shub structural stability theo-rem [HPS77, Theorem 7.1] in conjunction with the result which verifies plaque ex-pansivity hypothesis for applying Hirsch-Pugh-Shub structural stability (see e.g., [PSW12]) yields that f is robustly dynamically coherent. Now, for any g sufficiently C close to f the center foliation W cg is C close to W cf and, hence, is transverseto the fibers of p : E → X . Therefore W cg defines holonomy homeomorphisms M x → M (= M x ). These holonomy homeomorphisms assemble into a topologicaltrivialization of p : E → X . Note that if W cg is smooth then holonomies are smooth UNDLES WITH HYPERBOLIC DYNAMICS 31 and the trivialization will be smooth as well. Hence W cg cannot be smooth, because p : E → X is smoothly non-trivial. (cid:3) Partially hyperbolic setup.
It would be very interesting to generalize ourresults ( e.g.,
Theorems 3.1, 3.2, and 4.2) to the more general setting of bundlesthat admit fiberwise partially hyperbolic diffeomorphisms (see Section 7.1 for thedefinition of a partially hyperbolic diffeomorphism).
Conjecture 7.3.
Let X be a closed manifold or finite simplicial complex and let p : E → X be a fiber homotopically trivial bundle whose fiber M is a closed manifold.Assume that E admits a fiberwise partially hyperbolic diffeomorphism whose centerdistribution is lower dimensional (1 or 2). Then p : E → X is topologically trivial. The above conjecture can be viewed as a possible generalization of Theorem 4.2.Theorem 4.2 depended on the Livshitz Theorem for Anosov flows in a crucial way.For the above conjecture one may try replacing the argument that uses LivshitzTheorem with a similar argument that would use a heat flow along the centerfoliation instead.On the other hand, Theorem 3.1 does not admit straightforward (conjectural)generalization. This can be seen via the following example. Let A : T → T bean Anosov diffeomorphism, then f = ( A, id S ) : T × S → T × S is partiallyhyperbolic. By multiplying the Hopf fibration by T we can view T × S as thetotal space of the fiber bundle T → T × S → S . It is clear that f is a fiberwisepartially hyperbolic diffeomorphism, whose restrictions to the fibers have the form A × id S . Hence, for a generalization of Theorem 3.1 we need to further restrictthe class of fiberwise partially hyperbolic diffeomorphism. One natural class is thefollowing one. A partially hyperbolic diffeomorphism g : N → N is called irreducible if it verifies the following conditions:1. diffeomorphism g does not fiber over a (topologically) partially hyperbolic (orAnosov) diffeomorphism ˆ g : ˆ N → ˆ N of a lower dimensional manifold ˆ N ; thatis, one cannot find a fiber bundle p : N → ˆ N and a (topologically) partiallyhyperbolic (or Anosov) diffeomorphism ˆ g : ˆ N → ˆ N such that p ◦ g = ˆ g ◦ p ;2. if g ′ is homotopic to g then g ′ also verifies 1;3. if ˜ g is a finite cover of g then ˜ g also verifies 1 and 2. Conjecture 7.4.
Let X be a simply connected closed manifold or a simply con-nected finite simplicial complex and let p : E → X be a fiber bundle whose fiber M isa closed manifold. Assume that E admits a fiberwise irreducible partially hyperbolicdiffeomorphism with low dimensional (1 or 2) center distribution. Then p : E → X is topologically trivial. Question 7.5.
Can one construct an example of a simply connected manifold X and a non-trivial fiber bundle p : E → X whose fiber M is a closed manifold suchthat the total space E can be equipped with a fiberwise irreducible partially hyperbolicdiffeomorphism?Remark . If one modifies the definition of “fiberwise partially hyperbolic dif-feomorphism” to allow diffeomorphisms that permute the fibers ( i.e., factor overa non-trivial diffeomorphism of X ) then the answer to the above question is posi-tive [GORH14]. UNDLES WITH HYPERBOLIC DYNAMICS 32
Acknowledgements.
We thank Rafael de la Llave and Federico RodriguezHertz for very useful communications. Also we thank the referee for commentswhich helped to improve our presentation.
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