THE SPACE OF ANOSOV DIFFEOMORPHISMS ∗ F. THOMAS FARRELL AND ANDREY GOGOLEV 2 1 0 Abstract. WeconsiderthespaceXL ofAnosovdiffeomorphisms 2 homotopic to a fixed automorphism L of an infranilmanifold M. n We show that if M is the 2-torus T2 then XL is homotopy equiva- a lenttoT2. Incontrast,ifdimensionofM islargeenough,weshow J that XL is rich in homotopy and has infinitely many connected 7 components. 1 ] S D 1. Introduction . h t Let M be a smooth compact n-dimensional manifold that supports a m an Anosov diffeomorphism. Recall that a diffeomorphism f: M → M [ is called Anosov if there exist constants λ ∈ (0,1) and C > 0 along 1 with a df-invariant splitting TM = Es ⊕Eu of the tangent bundle of v M, such that for all m ≥ 0 5 9 kdfmvk ≤ Cλmkvk, v ∈ Es, 5 3 kdf−mvk ≤ Cλmkvk, v ∈ Eu. . 1 0 Currently the only known examples of Anosov diffeomorphisms are 2 Anosov automorphisms of infranilmanifolds and diffeomorphisms con- 1 : jugate to them. Furthermore, global structural stability of Franks and v i Manning [Fr70, M74] asserts that any Anosov diffeomorphism f of an X infranilmanifold is conjugate to an Anosov automorphism L with the r a conjugacy being homotopic to identity. See, e.g., [KH95] for the back- ground on Anosov diffeomorphisms. In the light of the above discussion we fix an infranilmanifold M and an Anosov automorphism L: M → M. We shall study the space X L of Anosov diffeomorphisms of M that are homotopic to L. In other words, an Anosov diffeomorphism f belongs to X if and only if there L exists a continuous path of maps f : M → M such that f = L and t 0 f = f. If one has a smooth path of diffeomorphisms (rather than 1 maps) connecting L and f then we say that f is isotopic to L. We equip X with Cr-topology, r = 1,2,...∞. L ∗ Both authors were partially supported by NSF grants. 1 THE SPACE OF ANOSOV DIFFEOMORPHISMS 2 Denote by Diff (M) the group of diffeomorphisms of M that are 0 homotopic to identity. Also denote by Top (M) the group of homeo- 0 morphisms of M that are homotopic to identity. Equip Top (M) with 0 compact-open topology. The group Diff (M) acts on X by conjugation. Assume for a mo- 0 L ment that L has only one fixed point and M = Tn. This guarantees, by [W70], uniqueness of the conjugacy given by global structural sta- bility. That is, for every f ∈ X there exists unique h ∈ Top (Tn) such L 0 that f = h◦L◦h−1. Therefore we have the following inclusions Diff (Tn) ֒→ X ֒→ Top (Tn) (∗) 0 L 0 with the composition Diff (Tn) ֒→ Top (Tn) being the natural inclu- 0 0 sion. Therefore, one gets topological information about the space X L from that on Diff (Tn) and Top (Tn). We will make precise state- 0 0 ments and arguments below which are valid for the general case; i.e., when M is perhaps not Tn or when L has possibly more than one fixed point. 2. Results Our goal is to provide some information on homotopy type of the space of Anosov diffeomorphisms X . We start by recalling the defini- L tion an Anosov automorphism. An infranilmanifoldis a double coset space M d=ef G\N⋊G/Γ, where N is a simply connected nilpotent Lie group, G is a finite group, and Γ is a torsion-free discrete cocompact subgroup of the semidirect product N ⋊G. When G is trivial M is called nilmanifold. An automorphism L: N → N is called hyperbolic if the differential DL: n → n does def not have eigenvalues of absolute value 1. If an affine map L = v · L e e commutes with Γ then it induces an affine diffeomorphism L on M. It e is easy to show that if L is hyperbolic then L is Anosov. And in this case we call L an Anosov automorphism. e Theorem 1. Let L: T2 → T2 be an Anosov automorphism of the 2- torus. Then the space X of Cr, r > 1, Anosov diffeomorphisms ho- L motopic to L is homotopy equivalent to T2. The proof relies on some standard results and techniques from hy- perbolic dynamics. The outline of the proof is given in Appendix A. Next we collect information about homotopy of Diff (M) for higher 0 dimensional M. BelowZ∞ standsforthedirect sum ofcountablymany p copies of Z d=ef Z/pZ. p THE SPACE OF ANOSOV DIFFEOMORPHISMS 3 Proposition 2. If n ≥ 10, then n n n π (Diff (Tn)) ≃ Z∞ ⊕ Z ⊕ Γ , 0 0 2 (cid:18)2(cid:19) 2 (cid:18)i(cid:19) i+1 Xi=0 where Γ , i = 0,...,n, are the finite abelian groups of Kervaire-Milnor i “exotic”spheres. Moreover, Z∞ mapsmonomorphicallyinto π (Top (Tn)) 2 0 0 via the map induced by the inclusion Diff (Tn) ֒→ Top (Tn). 0 0 Proof. This result is contained in Theorem 4.1 of [H78] and Theo- rem2.5of[HS76]withonecaveat. Theproofsofbothofthesetheorems depended strongly on a formula given in [HW73] and [H73]; cf. Theo- rem 3.1 of [H78]. Igusa found that this formula and its proof were se- riously flawed, and he corrected this formula in Theorem 8.a.2 of [I84]. Using Igusa’s formula, thetwo proofsof Proposition2 mentioned above (cid:3) are valid with minor modifications. Proposition 3. Let p be a prime number different from 2 and k be an integer satisfying 2p − 4 ≤ k < n−7. Then π (Diff (Tn)) contains a 3 k 0 subgroup S such that 1. S ≃ Z∞ and p 2. S maps monomorphically into π (Top (Tn)) via the map induced k 0 by the inclusion Diff (Tn) ֒→ Top (Tn). 0 0 We postpone the proof of the above proposition to Appendix B. Proposition 4. If M is an infranilmanifold of dimension n ≥ 10 then Z∞ < π (Diff (M)). 2 0 0 Moreover, Z∞ maps monomorphically into π (Top (M)) via the map 2 0 0 induced by the inclusion Diff (M) ֒→ Top (M). 0 0 Proof. This result follows from a slightly augmented form of Proposi- tion 2.2(A) in [HS76] with the same caveat made in the proof of our Proposition 2. Since M is an infranilmanifold, π (M) contains a nor- 1 mal nilpotent subgroup N with a finite quotient group π (M)/N. Now 1 note that the center Z(N) of N is a finitely generated, infinite abelian group. Hence Z(π (M)) = π (Aut(M)) is also finitely generated (but 1 1 perhaps not infinite); thus verifying one of the hypotheses of Propo- sition 2.2(A). And since the π (M) conjugacy class of any element in 1 Z(N) is finite, π (M) contains an infinite number of distinct conjugacy 1 classes; therefore Wh (π (M);Z ) = Z∞. 1 1 2 2 Then ∞ ∞ Wh (π (M);Z )/{c+εc¯} = H (Z ;Z ) ≃ Z 1 1 2 0 2 2 2 THE SPACE OF ANOSOV DIFFEOMORPHISMS 4 by the simple algebraic argument given on page 287 of [F02]. That is, we don’t need to know whether “π (M) contains infinitely many 1 conjugacy classes distinct from their inverse classes” as hypothesized in Proposition 2.2 of [HS76] to complete the argument given in that paper which produces a subgroup Z∞ of π (Diff (M)). (The diffeo- 2 0 0 morphisms representing the elements of Z∞ are all homotopic to id 2 M since they are constructed to be pseudo-isotopic to id .) Since Propo- M sition 2.2 is also true in the topological category (cf. footnote (i) on page 401 of [HS76]), this subgroup Z∞ maps monomorphically into 2 π (Top (M)). (cid:3) 0 0 Proposition 5. Let M be an n-dimensional infranilmanifold and p be a prime number different from 2. Assume that the first Betti number of M is non-zero, i.e., H (M,Q) 6= 0 and that n > max{9,6p − 5}. 1 Then there exists a subgroup S of π2p−4(Diff0(M)) such that 1. S ≃ Z∞ and p 2. S maps monomorphically into π2p−4(Top0(M)) via the map in- duced by the inclusion Diff (M) ֒→ Top (M). 0 0 Remark. The first Betti number of any nilmanifold is different from zero. We postpone the proof of the above proposition to Appendix B. Theorem 6. Let M be an n-dimensional infranilmanifold, L: M → M be an Anosov automorphism and X be the space of Anosov diffeomor- L phisms homotopic to L then the following is true 1. If M = Tn and n ≥ 10, then X has infinitely many connected L components. 2. If M = Tn, p is a prime number different from 2 and k is an integer satisfying 2p−4 ≤ k < n−7, then 3 Z∞ < π (X ). p k L 3. Let M be an infranilmanifold of dimension n ≥ 10, then X has L infinitely many connected components. 4. If p is a prime number different from 2 and M is an infranil- manifold of dimension n > max{9,6p−5} with a non-zero first Betti number then Z∞p < π2p−4(XL). Proof. We prove the third statement only. The proofs of the other statements are easier and follow the same lines. The proof will rely on the following statement whose proof we postpone to Appendix C. THE SPACE OF ANOSOV DIFFEOMORPHISMS 5 Proposition 7. Let M be an infranilmanifold. Assume that a homeo- morphism h: M → M is homotopic to identity and commutes with an Anosov automorhism L: M → M. Then h is isotopic to id . M Take h ,h ∈ Diff (M) that represent different elements [h ], [h ] 1 2 0 1 2 of Z∞ < π (Diff (M)). Let L = h ◦L◦h−1 and L = h ◦L◦h−1. 2 0 0 1 1 1 2 2 2 Assume that there is a path of Anosov diffeomorphisms L , t ∈ [1,2], t connecting L and L . By structural stability and local uniqueness we 1 2 ˜ ˜ get a continuous path {h ,t ∈ [1,2]} in Top (M) such that h = h t 0 1 1 and L = ˜h ◦L◦h˜−1. Hence we get t t t h ◦L◦h−1 = h˜ ◦L◦h˜−1. 2 2 2 2 The homeomorphism h˜−1 ◦h commutes with L. Hence Proposition 7 2 2 implies that [h˜−1 ◦h ] = [id ]. Therefore 2 2 M ˜ [h ] = [h ] = [h ] in π (Top (M)), 2 2 1 0 0 which gives us a contradiction. We conclude that L and L represent 1 2 different connected components of X . (cid:3) L Remarks. 1. It is not clear whether or not there are other connected compo- nents of X that we are not detecting. Question: for which h ∈ L Diff (M), does the connected component of h◦L in Diff(M) 0 contain an Anosov diffeomorphism? 2. By Moser’s homotopy trick the space of Diff (M)vol consisting 0 of all volume preserving diffeomorphisms homotopic to id is M a deformation retraction of Diff (M). Hence, we have analo- 0 gous results for the space of volume preserving Anosov diffeo- morphisms Xvol. L 3. See page 10 of [H78] for a conjectural geometric description for representatives of non-zero elements in Z∞ < π Diff (M) of 2 0 0 Propositions 2 and 4. Appendix A. Sketch of the proof of Theorem 1 Convention. When we say that an object is C1+ we mean that it is C1 and the first derivative is H¨older continuous with some positive exponent. Westartbyintroducingsomenotationandrecallingsomedefinitions. Given an Anosov diffeomorphism g: T2 → T2 we denote by Ws(g) and Wu(g) the stable and unstable foliations of g. We assume that a background Riemannian metric a is fixed. The logarithms of stable and unstable jacobians of g will be denoted by ϕs(g) and ϕu(g). THE SPACE OF ANOSOV DIFFEOMORPHISMS 6 Two H¨oldercontinuous functionsϕ ,ϕ : T2 → Rarecalledcohomol- 1 2 ogous up to an additive constant over g ifthereexist aconstant C anda H¨oldercontinuous functionu: T2 → R such thatϕ = ϕ +C+u◦g−u. 1 2 In this case we write hϕ i = hϕ i. We remark that even though ϕs(g) 1 2 and ϕu(g) depend on a, the cohomology classes hϕs(g)i and hϕu(g)i are independent of the choice of a. Let p be a fixed point of L, L(p ) = p . For every point p ∈ T2 0 0 0 consider the translation t : T2 → T2 that takes p into p. Then L d=ef p 0 p t ◦L◦t−1 is an Anosov automorphism that fixes point p. p p Recall that the space of all Cr diffeomorphisms of T2 — Diff(T2) — is a separable infinite dimensional manifold modeled on the Banach space or the Fr´echet space of Cr vector fields when r < ∞ or r = ∞, respectively. Hence Diff(T2) is a separable absolute neighborhood re- tract. Since X is an open subset of Diff(T2), we also have that X L L is a separable absolute neighborhood retract. By a result of W.H.C. Whitehead [P66], every absolute neighborhood retract has the homo- topy type of a CW complex. Therefore X has homotopy type of a L CW complex. Our goal is to show that X is homotopy equivalent to the 2-torus L T2 which we identify with {L ,p ∈ T2} ⊂ X . (Note that the map p L T2 ∋ p → L ∈ T2 is a finite covering.) Let Dk be a disk of dimension k p and let α: Dk → X be a continuous map which sends the boundary of L the disk to L. We shall show that α can be homotoped to α: Dk → T2. This implies that the inclusion T2 ֒→ X induces an epimorphism on L homotopy groups. Therefore, since T2 is aspherical, π (T2b) → π (X ) k k L is a trivial isomorphism for k ≥ 2. The homomorphism π (T2) → 1 π (X )ismonicand, hence, isanisomorphismaswell. Thiscanbeseen 1 L from structural stability and the fact that π (T2) → π (Top (T2)) is 1 1 0 monic. Now, since X hashomotopytypeofaCWcomplex, Theorem 1 L will follow from J.H.C. Whitehead’s Theorem. Structural stability gives the continuation ξ: Dk → Top (T2) to the 0 interior of the disk of the conjugacy to the linear model: α(·)◦ξ(·) = ξ(·)◦L. If k ≥ 2 we can assume that ξ(x) = idT2 for x ∈ ∂Dk. Then fix(·) d=ef ξ(·)(p ) defines the continuation of the fixed point p to the interior of 0 0 the disk, α(·)(fix(·)) = fix(·). If k = 1 then ξ is idT2 at one endpoint and a possibly non-trivial translation at the other endpoint. In this case fix is defined as before and gives a path that connects p with a 0 fixed point of L. We shall explain how to construct a homotopy that def connects α to α = Lfix(·). b THE SPACE OF ANOSOV DIFFEOMORPHISMS 7 Take x ∈ Dk and let f d=ef α(x). Let p d=ef fix(x) be the fixed point of f. Next we construct a path of diffeomorphisms that connects f and f , where f is a C1+ Anosov diffeomorphism C1+ conjugate to p p L . The path will consist of Anosov diffeomorphisms of regularity C1+ p that fix p. Choose a simple closed curve T which is transverse to Wu(f) and passesthroughp. Transversal T cutstheleaves oftheunstablefoliation Wu(f) into oriented arcs [y,e(y)] parameterized by y ∈ T. Given a H¨older continuous potential ϕ: T2 → R where exists a unique system of measures {µϕ,y ∈ T}, satisfying the following prop- y erties. 1. µϕ,y ∈ T, are finite measures supported on [y,e(y)]; y d(f∗µϕ) 2. y¯ = eϕ(z)−P(ϕ), where y¯ is the base point of the arc that dµϕ y contains f(z) and P(ϕ) is the pressure of ϕ; 3. the system {µϕ} satisfies certain absolute continuity property y with respect to the stable foliation Ws(f). Measures µϕ are equivalent to the conditional measures on [y,e(y)] y of the equilibrium state of ϕ. Notice that if ϕ = ϕu(f) then µϕ are y absolutely continuous measures induced by the Riemannian metric a. For more details about the system {µϕ} and the proof of existence and y uniqueness, see, e.g., [L00]. Consider the path of potentials ϕ d=ef (1 − 2t)ϕu(f), t ∈ [0,1/2]. t Corresponding system of measures µϕt depends continuously on t (see, y e.g., [C92]). Now we can define a C1+ path of Anosov diffeomorphisms whose logarithmic unstable jacobians are cohomologous up to a constant to ϕ . This is done in the following way. t Consider the functions η : T → R given by η (y) = µϕt([y,e(y)]). t t y Choose a continuous family of Riemannian metrics a , t ∈ [0,1/2], such t that a = a and the induced lengths lt([y,e(y)]) of the arcs [y,e(y)] in 0 y the metric a equal to η (y). Consider the family of homeomorphisms t t h : T2 → T2 that preserve the partition by the arcs [y,e(y)], y ∈ T, t and satisfy the following relation µϕt([y,z]) = lt([y,h (z)]),z ∈ [y,e(y)]. y y t Clearly, the family of homeomorphisms h is uniquely determined by t these properties. Define f d=ef h ◦f ◦h−1,t ∈ [0,1/2]. t t t THE SPACE OF ANOSOV DIFFEOMORPHISMS 8 Then f = f and it is easy to check that f , t ∈ [0,1/2], are C1+ 0 t Anosovdiffeomorphisms withhϕu(f )i = hϕ ◦h−1iover f . Because the t t t t stable foliation Ws(f) is C1+, it follows that hϕs(f )i = hϕs(f)◦h−1i, t t t ∈ [0,1/2]. Now we switch the roles of the stable and the unstable foliations and apply the same construction to f to get a path f , t ∈ [1/2,1], 1/2 t connecting f to f . Then hϕs(f )i = h0i, hϕs(f )i = hϕs(f )i = h0i 1/2 1 1 1 1/2 and it follows that f d=ef f is C1+ conjugate to L [dlL92]. p 1 p It is routine to check that the construction outlined above can be carried out simultaneously for all α(x), x ∈ Dk, and that the resulting homotopy canbe made to be constant on∂Dk. The choices oftransver- sals T = T(x) and families of Riemannian metrics a = a (x) must be t t made continuously in x to make sure that a (continuous) homotopy of α is produced. This homotopy connects α and α¯: D → Diff1+(T2) whose image lies in C1+ conjugacy class of L. Using standard smooth- ing methods we canC1 approximate our homotopy by another one that takes values in X and connects α to α: D → X . L L Themapα¯ canbeC1 approximatedbyamapwhoseimageliesinthe Cr conjugacy class of L simply by apperoximating C1+ conjugacy with a Cr conjugacy. This map is C1 close to α and hence, by performing a short homotopy if needed, we can assume that the map α is Cr conjugate to L. e Finally, map α can be homotoped to the map α: D → T2 byehomo- toping corresponding map h: Dk → Diff (M), h(·)◦L◦h−1(·) = α(·), 0 in the space of Cer conjugacies to a map consistinbg of the translations t: Dk → Diff0(M) given by t(x) = tfix(αe(x)) = tfix(α(x)). The laetter is possible due to a result of Earle and Eells [EE69] who showed that T2 = {t ,p ∈ T2} is a deformation retraction of the space of smooth p conjugacies Diff (T2). 0 Appendix B. Proofs of Propositions 2 and 4 Remark. Wewritek ≪ nformax{3k+7,9} < n; inparticular, 2p−4 ≪ n if and only if max{9,6p−5} < n. Proof of Proposition 5. Consider the following commutative diagram Ps(T) −−−→ Ps(M) −−−t→ Diff (M) 0 (⋆)      t  P(yT) −−−→ P(yM) −−−→ Topy0(M) THE SPACE OF ANOSOV DIFFEOMORPHISMS 9 where T is a closed tubular neighborhood of a smooth simple closed curve α in M such that the homology class represented by α generates an infinite cyclic direct summand of H (M). 1 Remark. If M is orientable then T = S1 × Dn−1. In general it is the mapping torus of a self-diffeomorphism of Dn−1. InthisdiagramPs(·)andP(·)arethesmoothandtopologicalpseudo- isotopyfunctors, respectively. Recallthatatopological(smooth)pseudo- isotopy of a compact manifold M is a homeomorphism (diffeomor- phism) f: M ×[0,1] → M ×[0,1] suchthatf(x) = xforallx ∈ M×0∪∂M×[0,1]. ThenP(M) (Ps(M)) is the topological space consisting of all such homeomorphisms (diffeo- morphisms), respectively. Since T ⊂ M is a codimension 0 submani- fold, a pseudo-isotopy f of T canonically induces a pseudo-isotopy F of M by setting F(x) = f(x), when x ∈ T × [0,1], and F(x) = x otherwise. Note that the pseudo-isotopy f must map M ×1 into itself. Hence, after identifying M with M ×1 in the obvious way, the restriction of f to M ×1 determines an element in Top (M) or Diff (M) depending 0 0 on whether f ∈ P(M) or Ps(M). This restriction gives the maps t (standing for top) in diagram (⋆). Now Proposition 5 is clearly implied by the following Assertion. Assertion. Let M be an n-dimensional infranilmanifold and p be a prime number different from 2. Assume that the first Betti number of M is non-zero and that n > max{9,6p − 5}. Then there exists a subgroup S of π2p−4(Ps(T)) such that 1. S ≃ Z∞ and p 2. S maps into π2p−4(Top0(M)) with a finite kernel via the homo- morphism which is functorially induced by the maps in the com- mutative diagram (⋆). This Assertion will be proven by concatenating several facts which we now list. Fact 1. The kernel of the homomorphism π Ps(T) → π P(T) is a k k finitely generated abelian group provided k ≪ n. This fact follows from Corollary 4.2 in [FO10]. Fact 2. Denote the inclusion map T ⊂ M by σ. Then the induced homomorphism π P(σ): π P(T) → π P(M) is monic provided k ≪ n. k k k THE SPACE OF ANOSOV DIFFEOMORPHISMS 10 Fact 2 is proven as follows. Since the class of α in H (M) generates 1 aninfinite cyclic direct summand, thereclearly exists a continuous map γ: M → S1 such that the composition S1 →α T →σ M →γ S1 is homotopic to idS1. Let P(·) denote the stable topological pseudo- isotopy functor. Applying P(·) to the above composition yields that π P(σ): π P(T) → π P(M) k k k is monic since P(·) is a homotopy functor; cf. [H78]. Therefore Igusa’s stability result [I88] completes the proof of Fact 2. There is an involution “–” defined on P(M) which is essentially de- termined by “turning a pseudo-isotopy upside down.” See pages 6 and 18 of [H78] for a precise definition. (Also see page 298 of [FO10].) Since σ commutes with “–”, it induces a homomorphism H (Z ,π P(T)) → H (Z ,π P(M)). 0 2 k 0 2 k Fact 3. This homomorphism is monic provided k ≪ n. Fact 3 is proven by an argument similar to that given for Fact 2 Fact 4. If k ≪ n, then π P(t): π P(M) → π Top (M) factors through k k k 0 a homomorphism ϕ: H (Z ;π P(M)) → π Top (M), 0 2 k k 0 whose kernel contains only elements of order a power of 2. Fact 4 follows from Hatcher’s spectral sequence (see pages 6 and 7 of [H78]) by using that topological rigidity holds for all infranilmani- folds (proven in [FH83]). The argument is a straightforward extension of the one proving Corollary 5 in Section 5 of [F02] Fact 5. There is a subgroup S of π Ps(T), where k = 2p − 4 ≪ n, k satisfying 1. S ≃ Z∞ and p 2. both x 7→ x + x¯ and x 7→ x − x¯ are monomorphisms of S into π Ps(T). k When T is orientable, i.e., T = S1 ×Dn−1, then Fact 5 follows from Proposition 4.6 of [FO10] — which is the analogous result valid for π Ps(S1) — by again using Igusa’s stability theorem [I88]. We note k that Proposition 4.6 depended on important calculations of π Ps(S1) k which can be found in [GKM08]. In the non-orientable case, T = M × Dn−2 where M denotes the M¨obius band and we argue as follows. Since S1 ×D1 is a collar neigh- borhood of the boundary ∂M, we can also identify S1 × Dn−1 with a