New partially hyperbolic dynamical systems I

c

2016 by Institut Mittag-Leffler. All rights reserved

New partially hyperbolic dynamical systems I

by

Andrey Gogolev

Binghamton University Binghamton, NY, U.S.A.

Pedro Ontaneda

Binghamton University Binghamton, NY, U.S.A.

Federico Rodriguez Hertz

Pennsylvania State University State College, PA, U.S.A.

1. Introduction

Let M be a smooth compact d-dimensional manifold. A diffeomorphism F is called Anosovif there exist a constantλ>1 and a Riemannian metric along with aDF-invariant splitting T M=E^s⊕E^u of the tangent bundle of M, such that for any unit vectors v^s andv^u in E^sandE^u, respectively, we have

kDF(v^s)k6λ⁻¹ λ6kDF(v^u)k.

All known examples of Anosov diffeomorphisms are supported on manifolds which are homeomorphic to infranilmanifolds. The classification problem for Anosov diffeomor- phisms is an outstanding open problem that goes back to Anosov and Smale. The great success of the theory of Anosov diffeomorphisms (and flows) [A] motivated Hirsch–Pugh–

Shub [HPS1], [HPS2] and Brin–Pesin [BP] to relax the definition as follows.

A diffeomorphismF is calledpartially hyperbolicif there exist a constantλ>1 and a Riemannian metric along with aDF-invariant splittingT M=E^s⊕E^c⊕E^uof the tangent bundle ofM, such that for any unit vectorsv^s,v^candv^uinE^s,E^c andE^u, respectively,

The first author was partially supported by NSF grant DMS-1266282. The second author was partially supported by NSF grant DMS-1206622. The last author was partially supported by NSF grant DMS-1201326. A. Gogolev would also like to acknowledge the excellent working environment provided by the Institute for Mathematical Sciences at Stony Brook University.

we have

kDF(v^s)k6λ⁻¹, kDF(v^s)k<kDF(v^c)k<kDF(v^u)k,

λ6kDF(v^u)k.

In recent years the dynamics of partially hyperbolic diffeomorphisms has been a popular subject, see e.g. [PS], [RRU]. The pool of examples of partially hyperbolic diffeomor- phisms is larger than that of Anosov diffeomorphisms, in particular, due to the fact that extensions (e.g.F×idN) of partially hyperbolic diffeomorphisms are partially hyperbolic.

However, the collection of basic building blocks for partially hyperbolic diffeomorphisms is still rather limited. Up to homotopy, all previously known examples of irreducible(¹) partially hyperbolic diffeomorphisms are either affine diffeomorphisms on homogeneous spaces or time-1 maps of Anosov flows. The affine examples go back to Brin–Pesin [BP]

and Sacksteder [Sa].

Theorem 1.1. (Main theorem) For any d>6 there exists a closed d-dimensional simply connected manifold M that supports a volume-preserving partially hyperbolic dif- feomorphism F:M!M. Moreover, F is ergodic with respect to the volume.

Remark 1.2. There are no previously known examples of partially hyperbolic dif- feomorphisms on simply connected manifolds. It is easy to show that simply connected compact Lie groups do not admit partially hyperbolic automorphisms (use e.g. [HM, Theorems 6.61 and 6.63]). However, to the best of our knowledge, the possibility that some simply connected manifolds support Anosov flows is open.

Burago and Ivanov proved that simply connected 3-manifolds (i.e. the sphere S³) do not support partially hyperbolic diffeomorphisms [BI]. Simply connected 4-manifolds have non-zero Euler characteristic and hence do not admit line fields. Consequently simply connected 4-manifolds do not support partially hyperbolic diffeomorphisms.

Question 1.3. Do simply connected 5-manifolds support partially hyperbolic diffeo- morphisms?

Remark 1.4. It is easy to see, for topological reasons, that the 5-sphere S⁵ does not admit partially hyperbolic diffeomorphisms. Indeed, the splitting TS⁵=E^s⊕E^c⊕ E^u is either a 3-1-1 or a 2-2-1 splitting. By an old result of Eckmann [E] (see also Whitehead [W]), the sphereS⁵does not admit two linearly independent vector fields, and therefore the splitting must be a 2-2-1 splitting. Because the structure group GL⁺(2,R)

(¹) See§12.5for our definition of irreducible.

retracts toO(2,R), the 2-plane bundles over S⁵ are in one-to-one correspondence with circle bundles, which are classified by homotopy classes of maps S⁴!Diff(S¹) by the clutching construction. But Diff(S¹) is homotopy equivalent toS¹, and hence all 2-plane bundles overS⁵ are trivial. Therefore the existence of a 2-2-1 splitting of TS⁵ implies that S⁵ is parallelizable, which is a contradiction. It is an interesting open problem to decide whetherS⁷ supports partially hyperbolic diffeomorphisms.

In the next section we briefly (and very informally) outline our approach. Then we proceed with a detailed discussion leading to the proof of the main theorem in§11. The authors would like to thank the referee for his/her careful reading.

2. Informal description of the construction

Our approach is to consider a smooth fiber bundleM!E−!^p X, whose baseX is a closed manifold and whose fiberM is a closed manifold which admits a partially hyperbolic diffeomorphism. The idea now is to equip the total spaceE with a fiberwise partially hyperbolic diffeomorphismF:E!E, which fibers over a diffeomorphism f:X!X, i.e.

the following diagram commutes:

E

p

F //E

p

X ^f //X.

Then the diffeomorphismF is partially hyperbolic provided thatf is dominated by the action (on extremal subbundles) of F along the fibers. However, for non-trivial fiber bundles the bundle mapp:E!X intertwines the dynamics in the fiber with dynamics in the base, which makes it difficult to satisfy

(1) F is fiberwise partially hyperbolic and (2) f is dominated byF

at the same time. In particular, ifXis simply connected andf is homotopic to idX, then such constructions seem to be out of reach (cf. [FG, Question 6.5]). Moreover, assuming thatf=idX, it was shown in [FG] that such construction is, in fact, impossible in certain more restrictive setups. However, in this paper, we show that iff^∗:H^∗(X)!H^∗(X) is allowed to be non-trivial then our method works in the setup of principal torus bundles over simply connected 4-manifolds.

3. Preliminaries on principal bundles

In this section we review some of the concepts and facts about principal fiber bundles that will be needed later. For more details consult [Hu].

Standing assumption. In this and further sections we will always assume that all topological spaces are connected countable CW complexes. Given a space X, we will writeH_∗(X;A) andH^∗(X;A) for its homology and cohomology groups with coefficients in an abelian groupA. If we abbreviate toH_∗(X) andH^∗(X) then we assume that the coefficient group isZ.

Let X be a space and G be a topological group. Recall that a (locally trivial) principal G-bundleπ:E!X is a (locally trivial) fiber bundle with fiberGand structure group G, where (the group) G acts on (the fiber)G by left multiplication. Moreover, letφα:Uα×G!π⁻¹Uα, for α∈A, be a complete collection of trivializing charts of the principal bundle E. Denote by Uαβ the intersection Uα∩Uβ, for α, β∈A. Also define φαβ:Uαβ!Gin the following way:

(φ⁻¹_β φ_α)(x, g) = (x, φ_αβ(x)·g), x∈U_αβ6=∅ and g∈G.

This collection oftransition functions {φαβ}U_αβ6=∅ satisfies the followingcocycle condi- tion:

φαβ(x)·φβγ(x)·φγα(x) =e, x∈Uα∩Uβ∩Uγ, whereeis the identity inG.

Conversely, let {φαβ}U_αβ6=∅ be a cocycle of transition functions over a covering {Uα}α; that is, assume that we have

(i) an open covering{Uα}α of the spaceX;

(ii) a collection of mapsφαβ:Uαβ!G,Uαβ6=∅, that satisfy the cocycle condition.(²) Then we can construct a principalG-bundleE!X by gluing the spacesUα×Gusing the transition functions{φαβ}. The cocycle condition ensures that the gluings are consistent.

We will need the following facts:

(1) Every principalG-bundleE!X has a (right) actionE×G!E. This action is free and the orbits are exactly the fibers. This can be seen from the construction ofE using a cocycle of transition functions: defineφ_α(x, g). h=φ_α(x, g . h). This is well defined because right and left translations onGcommute. (There are other equivalent ways of defining principal bundles. In some of them the action is included in the definition.)

(²) In what follows we will for notational simplicity write{Uα}and{φ_αβ}, and let the range ofα andβbe implied as above.

(2) Two G-bundles π:E!X and π⁰:E⁰!X are called equivalent, and we write E∼=E⁰, if there exists a homeomorphismf:E!E⁰ which fits into the commutative dia- gram

E

π@@@@@@

@@ ^f //E⁰

X

π⁰

>>

}} }} }} }}

and commutes with theG-action, i.e.f(y . g)=f(y). g.

(3) From (1) we get a canonical (up to right translation) way of identifying a fiber of a principalG-bundle withG.

(4) For everyGthere is a principal G-bundleEG!BGsuch that for any spaceX and any principalG-bundleE!X there is a unique, up to homotopy, map %:X!BG, such thatE∼=%^∗EG. We say that the map%classifies the bundleE!X. The spaceBG is called the classifying space of G, and the G-bundleEG!BGis called the universal principal G-bundle.

(5) The classifying space of the topological group S¹ is CP^∞=S

n>0CPⁿ. The universal principalS¹-bundle is ES¹=S^∞!CP^∞. Here S^∞=S

n>0Sⁿ. This bundle is the limit ofS¹!S²ⁿ⁻¹!CPⁿ, whereS¹⊂Cacts onS²ⁿ⁻¹⊂Cⁿ by scalar multiplication.

(6) We have B(G×H)=BG×BH, provided that bothBGandBH are countable CW complexes. Moreover,E(G×H)=EG×EH and the action and projections respect the product structure. It follows that BT^k=CP^∞×...CP^∞

| {z }

k

, where T^k=S¹×...×S¹

| {z }

k

is thek-torus andET^k=S^∞×...×S^∞

| {z }

k

.

4. A(E) construction

Letπ1:E1!X1andπ2:E2!X2be principalG-bundles. A fiber-preserving mapF:E1! E2 covering f:X1!X2 (i.e. fπ1=π2F) is a principal G-bundle map if F commutes with the right action ofG, that is, F(y . g)=F(y). g for all y∈E1 and g∈G. Hence F restricted to a fiber is a left translation.

More generally, let A:G!Gbe an automorphism of the topological group G and letE1 andE2 be as above. We say that a mapF:E1!E2coveringf:X1!X2is anA- bundle map (or simply anA-map) ifF(y . g)=F(y). A(g) for ally∈E1 andg∈G. Hence F restricted to a fiber is the automorphism Acomposed with a left translation.

Remark 4.1. Of course, an idG-map is just a principalG-bundle map. In particular, an idG-map that covers the identity idX:X!X is a principalG-bundle equivalence.

Remark 4.2. Note that the composition of anA-map and aB-map is aBA-map.

Now letπ:E!X be a principalG-bundle and let {φαβ} be a cocycle of transition functions for E. Note that {Aφαβ} is also a cocycle of transition functions. This is because

A(φαβ(x))·A(φβγ(x))·A(φγα(x)) =A(φαβ(x)·φβγ(x)·φγα(x)) =A(e) =e.

Therefore the new cocycle of transition functions{Aφαβ}defines a principalG-bundle overX. We denote this bundle byA(E). Next we show thatA(E) is well defined.

Proposition 4.3. The principal G-bundle A(E)does not depend on the choice of the cocycle of transition functions {φαβ}.

Proof. Let{φαβ}and{ψab}be two cocycles of transition functions over the coverings {Uα} and {Va}, respectively, both defining equivalent principal G-bundles. Denote the corresponding bundles byEandE⁰, respectively.

Special case. The cocycle {ψab} is a refinement of {φαβ}. That is, the covering {Va} is a refinement of{Uα}(i.e everyVa is contained in someUα) and everyψabis the restriction of someφαβ.

Recall that in this case the principal bundle equivalence betweenEandE⁰ is simply given by inclusions: the element (x, g)∈Va×G maps to (x, g)∈Uα×G, where Uα is a fixed (for eacha) element of{Uα} such thatVa⊂Uα.

It is straightforward to verify that the same rule defines an equivalence between {Aφαβ}and{Aψαβ}. This proves the special case.

Because of the special case, we may now assume that both cocycles {φαβ} and {ψab}are defined over the same covering{Uα}. Then the existence of a principal bundle equivalence betweenE and E⁰ is equivalent to the existence of a collection of functions {rα},rα:Uα!G, such that

φαβ(x)·rα(x) =rβ(x)·ψαβ(x) (4.1) forx∈Uαβ(see [Hu, Chapter 5, Theorem 2.7]). ApplyingAto equation (4.1), we obtain

(Aφαβ)(x)·(Arα)(x) = (Arβ)(x)·(Aψαβ)(x).

Therefore, the collection{Arα}defines a principal bundle equivalence between{Aφαβ} and{Aψαβ}.

Proposition 4.4. Let E!X be a principal G-bundle. Also let A and B be auto- morphisms of G. Then

(AB)(E) =A(B(E)) and idG(E) =E.

Proof. Direct from the definition ofA(E).

Proposition 4.5. Let E!X be a principal G-bundle, let A be an automorphism of Gand let f:Z!X be a map. Then

f^∗(A(E)) =A(f^∗(E)).

Proof. Let{φαβ} be a cocycle of transition functions forE defined over a covering {Uα}. Then {Aφαβf} is a cocycle of transition functions over {f⁻¹Uα} for both f^∗(A(E)) andA(f^∗(E)).

Proposition4.6. Let E!Xbe a principalG-bundle and letAbe an automorphism of G. Then there is an A-map FA:E!A(E)covering the identity idX:X!X.

Proof. Let {φαβ} be a cocycle of transition functions for E over a covering {Uα}.

Then{Aφαβ}is a cocycle of transition functions for A(E) over {Uα}. Define the map F_A in each chart as follows:

U_α×G3(x, g)7−!(x, A(g))∈U_α×G,

where the latter copy ofU_α×Gis a chart ofA(E). The mapF_A is well defined because the following diagram commutes

G

A

L_φαβ(x)

//G

A

G

L_A(φαβ(x))

//G.

HereL_h denotes left multiplication byh.

Corollary 4.7. Let E!X be a principal G-bundle and let Abe an automorphism of G. Then there is an A-map F_A−1:A⁻¹(E)!E covering the identity id_X:X!X.

Proof. This follows from Propositions4.6and Remark 4.2.

Let EG!BG be the universal principalG-bundle and let A be an automorphism of G. Then A(EG) is a principal G-bundle, and hence (see (4) in §3) there is a map

%A:BG!BGsuch that

A(EG)∼=%^∗_A(EG). (4.2) Moreover, this map is unique up to homotopy.

5. Principal T^k-bundles

We now take G=T^k=S¹×...×S¹ and recall that by §3(6) BT^k=(CP^∞)^k. Therefore π2BT^k is canonically identified with Z^k (by identifying ith generator of Z^k with the canonical generator of the second homotopy group of theith copy ofCP^∞).

LetA∈SL(Z, k). The matrix Ainduces automorphisms A:Z^k!Z^k and A:T^k!T^k for which we use the same notation.

The next proposition is a key result and its proof occupies the rest of this section (except for a lemma at the end). Recall that%A is characterized by equation (4.2).

Proposition5.1. Let g:BT^k!BT^k be a map such thatπ₂(g)=A∈SL(Z, k). Then g is homotopic to %_A, that is,

A(ET^k)∼=g^∗(ET^k).

The proof will require some lemmas and claims.

We considerCP^∞=S

nCPⁿwith the usualCW-structure, i.e. one cell in each even dimension. This structure induces a product CW-structure on BT^k=(CP^∞)^k. Then the 2-skeleton ofBT^k is the wedge Wk

i=1S²i of k copies of the 2-sphere S². Denote by Y this 2-skeleton and by E!Y the restriction of ET^k!BT^k to Y. We first prove the proposition for the principalT^k-bundleE!Y.

Lemma5.2. Let g_Y:Y!Y be a map such that π₂(g_Y)=A∈SL(Z, k). Then A(E)∼=g_Y^∗(E).

Let pbe the wedge point of Y. Then we have S²i∩S²j={p} for i6=j. We identify p with the south pole of each S²i. Denote by D_i⁺ and D⁻_i the closed upper and lower hemispheres ofS²i, respectively.

LetEi!Si be the restriction ofE!Y toSi,i=1, ... k.

Claim5.3. The principal T^k-bundle E_i!Siis obtained by identifyingD_i⁻×T^k with D_i⁺×T^kalong their boundaries using the gluing map ω_i:S¹!T^k,ω_i(u)=(1, ...,1, u,1, ...1), that is, all coordinates of ω_i(u)are equal to 1∈S¹, except for the i-th coordinate, which is equal to u.

Proof. The claim follows from putting together the following two facts (see also (6) in§3).

(1) The 2-skeleton ofBS¹=CP^∞ isCP¹=S²and the restriction of ES¹=S^∞ toS² is the Hopf bundle S¹!S³!S². Moreover, S³ is obtained by identifying two copies of D²×S¹ along the boundaries using the identity map id_S1:S¹!S¹ as gluing map.

(2) LetF1!E1!X1andF2!E2!X2be two fiber bundles. Consider the inclusion X1,!X1×X2given byx7!(x,∗), for some fixed∗∈X2. Then the restriction (E1×E2)|X₁

of the product bundle F1×F2!E1×E2!X1×X2 to X1⊂X1×X2 is the bundle F1× F2!E1×F2!X1.

Write A=(a_ij)∈SL(Z, k). Since A=π₂(g_Y), after performing a homotopy, we can assume thatg_Y satisfies the following property.

Poperty 5.4. For each j there are k disjoint closed 2-disks Dij⊂D⁺_j, i=1, ..., k, such that

(1) gY: (Dij, ∂Dij)7!(D⁺_i, ∂D⁺_i);

(2) the degree of g_Y: (D_ij, ∂D_ij)!(D⁺_i, ∂D⁺_i) is a_ij. Claim 5.5. The bundle g_Y^∗E|_S2

j is obtained by gluing D_j⁻×T^k with D_j⁺×T^k along their boundaries using the gluing map

fj=

k

Y

i=1

(ωi)^aij:S¹−!T^k, that is,f_j(u)=(u^a1j, ..., u^akj).

Proof. It follows from Claim5.3and Property5.4thatg^∗_YE|_S²

j is obtained by identi- fying`k

i=1Dij×T^k with (S²j\S

iintDij)×T^k along their boundaries (which is the union ofk copies ofS¹×T^k) via the gluing mapsω_i^aij:∂Dij=S¹!T^k, i=1, ..., k. Here we are identifying∂Dij withS¹ using the orientation on∂Dij induced byDij.

The claim now follows from the fact that the inclusion S¹=∂D_j⁺,!D⁺_j is a path in D⁺_j\S

iintD_ij which winds positively around eachD_ij exactly once.

Claim5.6. The principal S¹-bundleA(E)|_S2

j!S²j is obtained by identifyingD⁻_j×T^k with D⁺_j×T^k along their boundaries using the gluing map fj:S¹!T^k.

Proof. By applying Proposition4.5to the inclusion mapS²j,!Y we obtain A(E)|_S²

j=A(E|_S²

j) =A(Ej).

This fact together with Claim 5.3 and the definition of A(Ej) implies that A(Ej) is obtained by identifying D⁻_j×T^k with D⁺_j×T^k along their boundaries using the gluing mapAωj:S¹!T^k. But

A(ωj(u)) =A(1, ...,1, u,1, ...,1) = (u^a1j, ..., u^akj) =fj(u).

Lemma 5.2now directly follows from Claims5.5and5.6.

To finish the proof of Proposition 5.1we need the following lemma.

Lemma5.7. Let E1!BT^k andE2!BT^kbe principal T^k-bundles. Let Zbe a space and leth:Z!BT^k be a map. Assume thath^∗:H²(BT^k;Z)!H²(Z;Z)is injective. Then h^∗E1∼=h^∗E2 implies that E1∼=E2.

Proof. Recall that by§3(6) BT^k=(CP^∞)^k. Hence BT^k is an Eilenberg–MacLane space of type (Z^k,2), i.e.π₂BT^k=Z^k andπ_iBT^k=0 fori6=2. Therefore, we have that for any spaceX the group [X, BT^k] of homotopy classes of maps fromX to (the Eilenberg–

MacLane space)BT^k is isomorphic toH²(X;Z^k) [Ha, Theorem 4.57]. This group splits naturally as follows:

H²(X;Z^k)∼=H²(X;Z)⊕...⊕H²(X;Z). (5.1) Indeed, the splitting Z^k=Z⊕Z⊕...⊕Z induces a natural splitting of the cochain com- plex C^∗(X;Z^k)∼=C^∗(X,Z)⊕...⊕C^∗(X,Z) and (5.1) follows directly from the definition of cohomology.

Let hi:BT^k!BT^k classify Ei. Then hih:Z!BT^k classifies h^∗Ei. But the map h^∗: [BT^k, BT^k]![Z, BT^k] given by f7!fh is the map h^∗:H²(BT^k;Z^k)!H²(Z;Z^k).

This map is injective sinceh^∗:H²(BT^k;Z)!H²(Z;Z) is injective and the splitting (5.1) is natural. Thereforeh1h'h2himplies thath1'h2.

By the cellular approximation theorem [Ha, Theorem 4.8], we may assume that g:BT^k!BT^k is a cellular map. Hencegrestricts to the 2-skeleton Y.

Letι:Y!BT^k be the inclusion map. Note that A(E) =A(ι^∗ET^k)∼=ι^∗A(ET^k), where the last equivalence is by Proposition4.5. Also note that

(g|Y)^∗E=ι^∗g^∗(ET^k).

By Lemma 5.2, A(E)∼=(g|_Y)^∗E, and therefore ι^∗A(ET^k)∼=ι^∗g^∗(ET^k). Now, because ι^∗ is an isomorphism, Lemma 5.7 applies and we conclude that A(ET^k)∼=g^∗(ET^k). This completes the proof of Proposition5.1.

The following is a natural question: given a homomorphism A:Z^k!Z^k, is there a mapf:BT^k!BT^ksuch thatπ2(f)=A? It is well known that the answer to this question is affirmative. Moreover, the mapf is unique up to homotopy. The next lemma is a bit more general, and will be needed later.

Lemma 5.8. Let X be a simply connected space and let A:π₂X!Z^k=π₂BT^k be a homomorphism. Then there is a unique, up to homotopy, map f:X!BT^k such that π₂(f)=A.

Proof. We can equipX with a CW complex structure so thatX has no 1-cells [Ha, Corollary 4.16]. By a simple argument we can definef on the 3-skeleton ofX so that π2(f)=A (see [Ha, Lemma 4.31]). And, since πiBT^k=0, i>2, obstruction arguments show thatf can be extended cell by cell to the whole ofX. The proof of the uniqueness up to homotopy is similar.

6. Principal T^k-bundles that admit A-maps

LetE!X be a principalT^k-bundle andf:X!X. Also letA∈SL(k,Z). In this section we answer the following question.

Question 6.1. When does there exist anA-map E!E coveringf? E ^A-map //

E

X ^f //X.

Recall that by §3(4) every principal T^k-bundle over X is equivalent (as principal bundle) to the pull-backh^∗ET^k for someh:X!BT^k. We will use the notation

Eh

def=h^∗ET^k.

The next result answers Question6.1. It gives a relationship betweenA, f and E=E_h which is equivalent to the existence of anA-mapE!E coveringf. The map%_A, char- acterized by equation (4.2), appears in the next theorem.

Theorem 6.2. Let A∈SL(k,Z), X be a space and f:X!X be a map. Also let h:X!BT^k. Then there exists an A-map Eh!Eh covering f,

E_h

A-map //E_h

X ^f //X

if and only if hf'%Ah. That is,the following diagram homotopy commutes:

X ^f //

h

X

h

BT^k

%_A //BT^k.

Proof. First suppose that there exists anA-mapEh!Eh coveringf. We have the following diagram

A(E_h) ^A

−1-map //

Eh

A-map //Eh

X ^idX //X ^f //X,

where the first square comes from Corollary 4.7 (by taking A⁻¹ instead of A). By composing the consecutive horizontal arrows and using Remark4.1we obtain a principal T^k-bundle mapA(E_h)!E_hcoveringf. Therefore

A(Eh)∼=f^∗Eh (6.1)

and, using Proposition4.5, we obtain the following equivalences:

(%_Ah)^∗ET^k=h^∗(%^∗_A(ET^k))∼=h^∗A(ET^k) =A(E_h)∼=f^∗(E_h) = (hf)^∗ET^k, where the second and fourth equivalences follow from (4.2) and (6.1), respectively, and the third from Proposition4.5. It follows that%_Ah'hf.

Conversely, suppose that

%_Ah'hf (6.2)

Then,

A(Eh) =A(h^∗ET^k) =h^∗A(ET^k)∼=h^∗(%^∗_A(ET^k)) = (%Ah)^∗ET^k∼= (hf)^∗ET^k=f^∗Eh, where the second equivalence follows from Proposition4.5, the third from (4.2) and the fifth from (6.2). Therefore there is a principal bundle equivalence between A(Eh) and f^∗Eh, that is, there is an id_Tk-mapA(Eh)!f^∗Eh covering idX. This gives the second square in the diagram

Eh

A-map //A(Eh)

idT^k-map

//f^∗(Eh)

idT^k-map

//Eh

X ^idX //X ^idX //X ^f //X.

The first square comes from Proposition4.6and the third one from the definition of the pull-back bundle. By composing the consecutive horizontal arrows and using Remark4.2 we obtain anA-mapEh!Eh coveringf. This completes the proof of the theorem.

Our next result says that to verify the condition%_Ah'hf in the theorem above it is enough to verify it algebraically at theH² level.

Proposition 6.3. The following conditions are equivalent:

(1) %Ah'hf;

(2) H²(h)H²(%A)=H²(f)H²(h).

Also,if X is simply connected and H2(X)is free,then (1)and (2)are equivalent to (3) H2(%A)H2(h)=H2(h)H2(f);

(4) π2(%A)π2(h)=π2(h)π2(f).

This proposition follows from the following lemma.

Lemma 6.4. Let X be a space and let φ, ψ:X!BT^k be maps. Then the following are equivalent

(1) φ'ψ;

(2) H²(φ)=H²(ψ).

Also,if X is simply connected and H₂(X)is free,then (1)and (2)are equivalent to (3) H₂(φ)=H₂(ψ),

(4) π₂(φ)=π₂(ψ).

Proof. Clearly (1) implies (2). Recall the splitting (5.1) from Lemma5.7. Assume thatH²(φ)=H²(ψ), then, by naturality of the splitting (5.1), the induced maps on the cohomology withZ^k coefficients also coincide. Now recall that the map

H²(φ;Z^k):H²(BT^k;Z^k)−!H²(X;Z^k)

coincides with the map φ^∗: [BT^k, BT^k]![X, BT^k] given by [λ]7![λφ]. Similarly for H²(ψ;Z^k). Thusλφ'λψfor everyλ. Takingλ=id_BTk, we obtainφ'ψ. This proves that (2) implies (1).

IfX is simply connected andH2(X) is free thenH²(X)∼=H2(X)∼=π2(X). The first isomorphism is by the universal coefficients theorem [Ha, Theorem 3.2] and the second one is by the Hurewicz theorem [Ha, Theorem 4.37]. Therefore,

H²(φ)∼=H2(φ)^T∼=π2(φ)^T, where the superscript^T denotes the transpose.

To prove the proposition apply the above lemma to φ=%_Ahandψ=hf.

7. Simply connected principal T^k-bundles

LetE!X be a principalT^k-bundle. Recall thatE∼=E_h=h^∗ET^k, where the maph:X! BT^k is unique up to homotopy. In this section we deal with the following question.

Question 7.1. When is the total spaceEh simply connected?

Note that the fundamental group of the total space E_h surjects onto the funda- mental group ofX. ThereforeX has to be simply connected. The next result answers Question7.1whenX is simply connected.

Proposition7.2. Let X be a simply connected space and let h:X!BT^k be a map.

Then the following are equivalent:

(1) the total space Eh is simply connected;

(2) the homomorphism π2(h):π2X!π2BT^k is onto;

(3) the homomorphism H2(h):H2X!H2BT^k is onto.

Proof. From the homotopy exact sequence of theT^k-bundle T^k!Eh!X and the fact thatπ1X=0 we obtain the exact sequence

−!π₂X−^∂!π₁T^k−!π₁E_h−!0.

Thereforeπ1Eh=0 if and only if∂ is onto. On the other hand, from the homotopy exact sequence of theT^k-bundleT^k!ET^k!BT^k and the fact thatET^k is contractible, we obtain that

π2BT^{k ∂}

0

−!π1T^k

is an isomorphism. Then the equivalence (1)⇔(2) follows from the following claim.

Claim 7.3. The following diagram commutes:

π₂X

π2(h)@@@@@@@@ ^∂ //π1T^k π₂BT^k.

∂⁰

>>

}} }} }} }

The claim follows from the naturality of the homotopy exact sequence of a pair and the definition of the boundary map.

The equivalence (2)⇔(3) follows from the naturality of the Hurewicz map and Hurewicz theorem. This proves the proposition.

8. The construction

We restrict ourselves to the case whereXis a simply connected 4-manifold andf:X!X is a diffeomorphism. We make the following collection of assumptions (∗).

(1) The second homotopy groupπ₂(X) is a free abelian group withmgenerators.

(2) The group π2(X) splits as a direct sum Z^k⊕Z^m−k in such a way that the first summand isπ2(f)-invariant, i.e.π2(f)|_Zk is an automorphism ofZ^k⊂π2(X).

(3) LetA∈SL(k,Z) be the matrix that representsπ2(f)|_Zk. ThenAalso represents an automorphismR^k!R^k. Assume that there exists anA-invariant splitting

R^k=E^s_A⊕E^c_A⊕E_A^u and a Riemannian metrick · konX such that the numbers

λσ= min

v∈E^σ_A kvk=1

kAvk and µσ= max

v∈E_A^σ kvk=1

kAvk, σ=s, c, u,

satisfy the following inequalities

λs6µs< λc6µc< λu6µu, µs< m(f) and λu>kDfk,

wherem(f) is minimum of the conormm(Dfx) andkDfk is the maximum of the norm kDfxk, i.e.

m(f) = min

v∈T X kvk=1

kDf(v)k and kDfk= max

v∈T X kvk=1

kDf(v)k.

Remark 8.1. We allowE_A^c to be trivial.

Theorem 8.2. Let X be a simply connected closed 4-manifold, let f:X!X be a diffeomorphism that satisfies (∗) and let πh:Eh!X be a principal T^k-bundle. As- sume that Eh admits an A-map F:Eh!Eh. Then F:Eh!Eh is a partially hyperbolic diffeomorphism.

Clearly the splitting of §8(3) descends to a T^k-invariant splitting of the tangent bundleTT^k=E_A^s⊕E_A^c⊕E_A^u. Then the action ofT^konEhinduces aT^kinvariant splitting ofTT^k=E^s⊕E^c⊕E^u; here, abusing notation,TT^kis the subbundle ofT Ehthat consists of vectors tangent to the torus fibers. Since F:Eh!Eh is an A-map, this splitting is DF-invariant.

Addendum8.3. (to Theorem8.2) The subbundles E^sand E^udefined above are the stable and the unstable subbundles for F, respectively. The center subbundle for F has the form E^c⊕H⁰,where H⁰ is a certain subbundle complementary to TT^k.

Proof. We equipT Ehwith a Riemannian metric in the following way. The flat metric on the torus induces a metric onTT^k. Also recall that, by §8(3), we have equippedX with a Riemannian metrick · k. Choose a continuous horizontal subbundleH⊂T Ehsuch thatT Eh=TT^k⊕H. Then,

(Dπ_h)_x:H(x)!T_πh(x)X

is an isomorphism for everyx∈Eh. Set

kvk=kDπh(v)k

forv∈H. Then extend the Riemannian metrick · kto the rest ofT Ehby declaringTT^k andH perpendicular.

Consider the commutative diagram 0 //E^s //

DF|Es

T Eh //

DF

E^c⊕E^u⊕H //

DFp

0

0 //E^s //T Eh //E^c⊕E^u⊕H //0.

(8.1)

The horizontal rows are short exact sequences of Riemannian vector bundles and all vertical automorphisms fiber over f:X!X. The last vertical arrow is defined as the composition of DF and the orthogonal projection p on E^c⊕E^u⊕H. Note that the diagram

E^c⊕E^u⊕H

Dπ_h

Dfp//E^c⊕E^u⊕H

Dπ_h

T X ^DfDf ////T X

commutes, and hence, by our choice of the Riemannian metric, kDF(p(v))k>min(λ_c, m(f))kvk.

Combining this with§8(3), we obtain the following bound on the minimum of the conorm:

m(Dfp)> µs. (8.2)

Lemma8.4. ([HPS2], Lemma 2.18) Let 0 //E1

i //

T1

E2 j //

T2

E3 //

T3

0

0 //E₁ ⁱ //E₂ ^j //E₃ //0

be a commutative diagram of short exact sequences of Riemannian vector bundles, all over a compact metric space X, where Ti:Ei!Ei are bundle automorphisms over the base homeomorphism f:X!X, i=1,2,3. If

m(T3|E₃(x))>kT1|E₁(x)k

for all x∈X, then i(E₁)has a unique T₂-invariant complement in E₂.

By (8.2), we can apply Lemma 8.4 to (8.1) and obtain a DF-invariant splitting T Eh=E^s⊕Eb^s. Exchange the roles of E^s and E^u and apply the same argument to obtain a DF-invariant splitting T Eh=Eb^u⊕E^u. It is easy to see thatE^c⊕E^u⊂Eb^s and E^s⊕E^c⊂Eb^u. Let

Eb^c=Eb^s∩Eb^u.

Then, clearly, we have aDF-invariant splittingT Eh=E^s⊕Eb^c⊕E^u.

To see thatF is partially hyperbolic with respect to this splitting pick a continuous decomposition Eb^c=E^c⊕H⁰, and define a new Riemannian metrick · k⁰ onT Eh in the same wayk · kwas defined, but usingH⁰ instead ofH; i.e. we declare

(1) kvk⁰=kvk ifv∈TT^k, (2) kvk⁰=kDπh(v)k ifv∈H⁰, (3) H⁰ is orthogonal toTT^k.

Now the partial hyperbolicity (with respect tok · k⁰) is immediate from the inequalities of§8(3).

9. The base space—the Kummer surface

A K3 surface is a simply connected complex surface whose canonical bundle is trivial.

All K3 surfaces are pairwise diffeomorphic and have the same intersection form 2(−E8)⊕3

0 1 1 0

.

In this section we recall Kummer’s construction of the K3 surface and describe a holo- morphic atlas on it.

Consider the complex torus

T²_C=C²/(Z⊕iZ)².

Also consider the involution ι:T²_C!T²_C given by ι(z1, z2)=(−z1,−z2). It has 16 fixed points which we call theexceptional set and which we denote byE(T²_C). Note thatT²_C/ι is not a topological manifold because the neighborhoods of the points in the exceptional set are cones overRP³-s. Replace the neighborhoods of the points from the exceptional set with copies of CP² to obtain the blown up torus T²_C#16CP² (here CP² stands for the 2(complex)-dimensional projective space with reversed orientation); see e.g. [Sc, p. 286] for details on complex blow up. The involutionιnaturally induces a holomorphic involutionι⁰ ofT²_C#16CP². The involutionι⁰ fixes 16 copies ofCP¹. One can check that the quotient

X^def=T²_C#16CP²/ι⁰

is a 4-dimensional manifold. This manifold is called theKummer surface. Note that it comes with a map

σ:T²_C\E(T²_C)−!X, (9.1) which is a double cover of its imageX\E(X), whereE(X) is theexceptional setinX, i.e.

the union of 16 copies ofCP¹. One can also check thatX is simply connected. (See [Sc,

§3.3] for more details.)

In fact, X is a complex surface and we proceed to describe the complex structure onX. For any connected open setV which is disjoint from the exceptional setE(X) and whose preimage underσhas two connected components, a holomorphic chart on T²_Cfor one of the connected components ofσ⁻¹(V) induces a chart onV by composing with σ.

Hence we are left to describe the charts on a neighborhood ofE(X).

Let p∈E(T²_C). We identify a neighborhood of p in T²_C with a neighborhood U of (0,0) inC². Then we blow upp, which amounts to replacing U with

U⁰={(z1, z₂, `(z<sub>1</sub>, z<sub>2</sub>)) : (z<sub>1</sub>, z<sub>2</sub>)∈ U,(z<sub>1</sub>, z<sub>2</sub>)∈`(z₁, z₂)}.

Here`(z1, z2) is a complex line through (0,0) and (z1, z2). Hence, if (z1, z2)6=(0,0) then

`(z1, z2)=[z1:z2] in homogeneous coordinates. Finally, note that

U⁰⁰={(z1, z2, `(z1, z2))∈ U<sup>0</sup>}/(z1, z2, `(z1, z2))∼(−z1,−z2, `(z1, z2)) (9.2) is identified with a neighborhood ofCP¹⊂E(X) in X. We will cover U⁰⁰ by two charts.

Note that the inclusionU,!C² induces the inclusion U⁰,!C²#CP², which in turn induces the inclusionU⁰⁰,!C²#CP²/ι⁰⁰, whereι⁰⁰is induced by (z₁, z₂)7!(−z₁,−z₂). We will define charts forC²#CP²/ι⁰⁰. Then to obtain charts forU⁰⁰ one just needs to take the restrictions of the charts forC²#CP²/ι⁰⁰.

First note that

C²#CP²={(z1, z2, `(z1, z2)) : (z1, z2)∈C<sup>2</sup>,(z1, z2)∈`(z1, z2)} ⊂C²×CP¹. The projective line CP¹ can be covered by two charts u7![u:1] andu⁰7![1:u⁰]. These charts extend to charts forC²#CP² as follows:

ϕ₁: (u₁, u₂)7−!(u₁u₂, u₂,[u₁: 1]) and

ϕ2: (u⁰₁, u⁰₂)7−!(u⁰₂, u⁰₁u⁰₂,[1 :u⁰₁]).

Define ξ:C²!C² by ξ(u1, u2)=(u1, u²₂). By a direct check, we see that the following composition

C²

ξ⁻¹

−−−!C²

ϕ_i

−−−!C²#CP²−!C²#CP²/ι⁰⁰

is independent of the branch ofξand gives a well defined chartψi, fori=1,2, which is a homeomorphism on the image. It is also easy to see that the images ofψ1andψ2 cover C²#CP². Calculating

ψ⁻¹₂ ψ₁(v, w) = 1

v, v²w

(9.3) confirms that the atlas is holomorphic.

Remark 9.1. The formulas ψ1(v, w) = (v√

w,√

w,[v: 1]) and ψ2(v, w) = (√ w, v√

w,[1 :v])

also show that the chartsψ₁andψ₂are compatible with those induced fromC²\{(0,0)}

by the double cover C²\{(0,0)}!(C²#CP²/ι)\CP¹ of the complement of the excep- tional set.

Remark 9.2. Consider the 2-form dz1∧dz2onT²_Cand its push-forward η=σ_∗(dz1∧dz2)

toX\E(X) (it is well defined becausedz₁∧dz₂=(−dz₁)∧(−dz₂)). Calculating the latter in the chartψ₁ yields

d(v√ w)∧d√

w=¹₂dv∧dw.

Together with an analogous calculation in the chartψ₂ this implies thatη extends to a non-vanishing 2-form onX.

Remark 9.1 shows that the charts defined above forE(X) are compatible with the charts induced byσfrom the charts forT²_C. Hence we have equippedX with a holomor- phic atlas.

10. The base dynamics—automorphisms of Kummer surfaces

LetB∈SL(2,Z) be a hyperbolic matrix. ThenBinduces an automorphismB_C:T²_C!T²_C. Note that after appropriately identifying T²_C with the real torus T⁴ the matrix that representsB_C is

B⊕B=

B 0

0 B

.

We use this identification, T²_C∼=T⁴, repeatedly in what follows. The automorphism B_C naturally induces an automorphism ofT²_C#16CP² and, since the latter commutes with ι⁰,B_Cdescends to a homeomorphismf_B:X!X. It is easy to verify thatf_B is, in fact,

a complex automorphism of X. The second integral cohomology group ofX isZ²² and the second rational cohomology group admits a splitting

H²(X;Q)∼=Q⁶⊕Q¹⁶, (10.1) whereQ⁶ is inherited fromH²(T²_C;Q) and the remaining 16 copies ofQcome from the 16 copies ofCP¹ inE(X). See [BHPV, Chapter VIII] for a proof of these facts.

Proposition 10.1. The induced automorphism f_B^∗:H²(X;Z)!H²(X;Z) is repre- sented by the matrix diag(B²,id_Z4, S16),where S16 is a permutation matrix given by the restriction of B_C to E(T²_C).

Proof. Note that, by the universal coefficients theorem, it suffices to show that the induced automorphism of the rational cohomology f_B^∗:H²(X;Q)!H²(X;Q) has the posited form. Then we can use the naturality of the isomorphism (10.1). Under this isomorphism the restrictionf_B^∗|_H2(T²C;Q)corresponds toB^∗

C:H²(T²_C;Q)!H²(T²_C;Q) given by (B⊕B)∧(B⊕B). And the restrictionf_B^∗|_Q16 permutes the coordinates according to the permutationS₁₆ given by the restriction ofB_CtoE(T²_C). After an (integral) change of basis we obtain thatf_B^∗ is given by diag(B²,id_Z4, S₁₆).

Remark10.2. Note that the basis in which the automorphism has the above diagonal form is not completely canonical because we use the eigenvectors that correspond to unit eigenvalues to write (B⊕B)∧(B⊕B) as diag(B²,id_Z⁴).

The goal now is to perturb fB so that the perturbation satisfies the collection of assumptions (∗) from§8.

SetB= ^{13 8}_{8 5}

and note that because of this choice ofB the automorphismB_Cfixes points inE(T²_C).

Embed the automorphismB:T²!T² into a 2-parameter family of diffeomorphisms ofT² given by

B_ε,d(x, y) = (13x−h_ε,d(x)+8y,8x−h_ε,d(x)+5y), ε>0, d∈Z⁺. Herehε,1:S¹!S¹is aC^∞smooth function that has the following properties:

(1) hε,1(−x)=−hε,1(x);

(2) |h⁰_ε,1(x)|6εfor allx∈S¹; (3) hε,1(x)=hε,1 x+¹₂

=εxforx∈U, where U is a small symmetric neighborhood of 0∈S¹.

The existence of such function for sufficiently small U can be seen by standard C^∞- gluing techniques. To defineh_ε,d:S¹!S¹consider thedsheeted self coverS¹!S¹given

byx7!dxand let hε,d be the lifting ofhε,1 that fixes 0. It is clear thathε,d also satisfies properties (1) and (2) and the following variant of (3):

(3⁰) hε,d(x)=hε,d x+¹₂

=εx for x∈Ud, where Ud is the connected component of 0∈S¹of the set{x:dx∈U}.

Note that B_C:T²_C!T²_C embeds into the 2-parameter family Bε,d⊕Bε,d:T²_C!T²_C. (Recall that we have an identificationT⁴∼=T²_C.)

Proposition 10.3. For sufficiently small ε>0 and all d>1, the diffeomorphisms Bε,d⊕Bε,d:T²_C!T²_C induce volume-preserving Bernoulli diffeomorphisms fε,d:X!X.

Proof. It is easy to see that Bε,d⊕Bε,d fixes points from the finite set E(T²_C) and that the differential is a complex linear map at the points fromE(T²_C). Also,Bε,d(x, y)=

Bε,d(−x,−y), and henceBε,d⊕Bε,dinduces a diffeomorphismfε,d:X!X. The fact that fε,d is smooth boils down to a calculation in charts in the neighborhood of E(X). This is a routine calculation which we omit.

By calculating the Jacobian ofBε,d we see that the diffeomorphismBε,d⊕Bε,d pre- serves the volume vol_T2

C induced by the form dz1∧dz2∧dz1∧dz2. Remark 9.2 implies that volX=σ_∗vol_T2

C is induced byη∧η¯and hence it is indeed a smooth volume. However it is clear from the definition thatfε,d preserves volX.

For sufficiently small ε>0 the diffeomorphism Bε,d⊕Bε,d is Anosov, and hence Bernoulli. Since volX(E(X))=0 the dynamical system (f,volX) is a measure theoretic factor of (Bε,d⊕Bε,d,vol_T2

C), and hence it is also Bernoulli by work of Ornstein [O].

Proposition 10.4. For any sufficiently small ε>0 there exist a sufficiently large d>1such that the diffeomorphism fε,d:X!X satisfies the collection of assumptions(∗) from §8.

The proof of this proposition requires some lemmas.

LetC:C²!C²be the automorphism given by (z1, z2)7!(µz1, µ⁻¹z2),µ>1, and let C_∗:C²#CP²/ι⁰⁰−!C²#CP²/ι⁰⁰

be the automorphism induced by C on the quotient of the blow up (recall that ι⁰⁰ is induced by (z1, z2)7!(−z1,−z2)). It is easy to see that C∗ leaves the projective line CP¹⊂C²#CP²/ι⁰⁰ over (0,0) invariant.

Lemma 10.5. There exists a Riemannian metric k on C²#CP²/ι⁰⁰ such that for any x∈CP¹ and any u∈Tx(C²#CP²/ι⁰⁰)we have

µ⁻²kukk6kDxC_∗(u)kk6µ²kukk, where k · kk=p

k(·,·).

Proof. Clearly it is enough to definek onCP¹⊂C²#CP²/ι⁰⁰, since we can extend it in an arbitrary way.

Recall that in §9 we covered C²#CP²/ι⁰⁰ by two charts ψ₁ and ψ₂. Note that in both chartsCP¹ is given byw=0. We use Remark9.1to calculateC_∗ in charts

C_∗,ψ1: (v, w)−^ψ!¹ (v√ w,√

w,[v: 1])−^C!^∗ (µv√

w, µ⁻¹√

w,[µ²v: 1])^ψ

−1

−1!(µ²v, µ⁻²w), C∗,ψ2: (v, w)−^ψ!² (√

w, v√

w,[1 :v])−^C!^∗ (µ√

w, µ⁻¹v√

w,[1 :µ⁻²v])^ψ

−1

−2!(µ⁻²v, µ²w).

Let us define a Hermitian metric in the chartψ1. Given a point (v,0) define

h_(v,0)=Q(v)dvdv+Q(v)⁻¹dwdw, (10.2) where

Q(v) = 1

1+|v|² 2

.

Define a Hermitian metric in the chart ψ2 by the same formula (10.2). The fact that these definitions are consistent can be seen from the following calculation that uses the transition formula (9.3):

(ψ₂⁻¹ψ₁)^∗h_(v,0)=Q 1

v

d 1

v

d 1

v

+Q 1

v −1

dv²wdv²w

=Q 1

v 1

|v|⁴dvdv+Q(1/v)⁻¹|v|⁴dwdw=h_(v,0), where the second equality follows from

d(v²w) =v²dw+2vw dv=v²dw whenw=0, and the last equality follows from the identity

Q 1

v

=|v|⁴Q(v).

Therefore, (10.2) gives a well-defined Hermitian metric honCP¹⊂C²#CP²/ι⁰⁰. Define the Riemannian metrick as the real part ofh

k=¹₂(h+¯h).

Notice that in chartsk is a warped product. Thus, we only need to prove the posited inequalities for the real parts of dual vectorse_v and e_w. We check the inequality in the