Smooth models for certain fibered partially hyperbolic systems

Abstract

We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.

1 Introduction

This paper establishes the existence of smooth models for certain fibered partially hyperbolic systems. An ingredient in our proof is a classification of Anosov homeomorphisms on nilmanifolds. We explain now through an example what some of these terms mean.

The three-dimensional Heisenberg group is given by

$$ \begin{align*} \operatorname{\mathrm{Heis}}:= \left\{ A_{(x,y,z)}= \begin{pmatrix} 1&x&z\\ 0&1&y\\0&0&1 \end{pmatrix} : \ x,y,z\in {\mathbb R} \right\} \end{align*} $$

with matrix multiplication as the group operation. The group $\operatorname {\mathrm {Heis}}$ is a nilpotent Lie group whose center consists of the matrices $A_{(0,0,z)}$ , with $z\in {\mathbb R}$ . Any automorphism of $\operatorname {\mathrm {Heis}}$ must preserve this center.

Nilmanifolds are quotients of nilpotent Lie groups by discrete subgroups. For the Heisenberg group, a compact quotient can be obtained as follows. Let

$$ \begin{align*} \Gamma= \{ (x,y,z)\in H : \ x,y,2z\in {\mathbb Z} \}, \end{align*} $$

where we use $(x,y,z)$ to denote the matrix $A_{(x,y,z)}$ . The quotient $\operatorname {\mathrm {Heis}} /\Gamma $ is a compact nilmanifold, an example of a Heisenberg nilmanifold. It is a fiber bundle over the $2$ -torus ${\mathbb T}^2$ with fiber the circle ${\mathbb T}$ , where the fibers lie in the ‘z-direction,’ tangent to the center of the Lie algebra of $\operatorname {\mathrm {Heis}}$ .

Any automorphism of $\operatorname {\mathrm {Heis}}$ that preserves $\Gamma $ descends to a diffeomorphism of $\operatorname {\mathrm {Heis}}/\Gamma $ : since the automorphism preserves the center of $\operatorname {\mathrm {Heis}}$ , the quotient diffeomorphism preserves the bundle structure. The interesting quotient diffeomorphisms are examples of fibered partially hyperbolic diffeomorphisms. An example is the map $f_0:\operatorname {\mathrm {Heis}}/\Gamma \to \operatorname {\mathrm {Heis}}/\Gamma $ given by

$$ \begin{align*} f_0(x,y,z)=\bigg(2x+y,x+y,z+x^2+\frac{y^2}{2}+xy\bigg). \end{align*} $$

Since $f_0$ preserves the smooth fibration ${\mathbb T}\hookrightarrow \operatorname {\mathrm {Heis}}/\Gamma \twoheadrightarrow {\mathbb T}^2$ , it induces a diffeomorphism of the base ${\mathbb T}^2$ , in this case the hyperbolic linear automorphism $(x,y)\mapsto (2x+y,x+y)$ .

This example is partially hyperbolic, meaning that the tangent bundle $TN$ to ${N=\operatorname {\mathrm {Heis}}/\Gamma }$ splits as a $df_0$ -invariant direct sum $TN=E^s\oplus E^c\oplus E^u$ such that for all $x\in N$ and all unit vectors $v^s\in E^s(x), \ v^c\in E^c(x), $ and $v^u\in E^u(x)$ , we have that

$$ \begin{align*} \|d_xf(v^s)\|<\|d_xf(v^c)\|<\|d_xf(v^u)\| \quad \text{and} \quad \|d_xf(v^s)\|<1<\|d_xf(v^u)\|. \end{align*} $$

That is, the center direction is dominated by the stable and unstable directions.

Hammerlindl and Potrie proved that if $f\colon N\to N$ is partially hyperbolic and homotopic to $f_0$ , then f is leaf conjugate to $f_0$ , meaning that there exists a homeomorphism $h:N\to N$ , an f-invariant foliation $W^c_{f}$ tangent to the center direction of f, and an $f_0$ -invariant foliation $W^c_{f_0}$ tangent to the center direction of $f_0$ such that h maps center leaves of $f_0$ to center leaves of f (that is, $h(W^c_{f_0}(x))=W^c_f(h(x))$ ) and $h(f_0(W^c_{f_0}(x)))=f(h(W^c_{f_0}(x)))$ [19]. In this example, $f_0$ is a smooth model, and any partially hyperbolic diffeomorphism homotopic to it is leaf conjugate to it.

In this paper, we consider the class of fibered partially hyperbolic diffeomorphisms. These are partially hyperbolic diffeomorphisms that, like $f_0$ above, have an integrable center bundle $E^c$ , tangent to an invariant fibration by compact submanifolds. More precisely, $f\colon M\to M$ , where M is a closed Riemannian manifold, is a fibered partially hyperbolic diffeomorphism (this is also known as a fibered partially hyperbolic system with $C^k$ fibers) with ( $C^k$ , $k\geq 1$ ) fiber X if there exists an f-invariant continuous fiber bundle $\pi :M\to B$ (for some manifold B) with $C^k$ fibers modeled on X, which are tangent to $E^c$ and such that the k-jets along fibers are continuous in M. (In other words, M is a continuous X-bundle with structure group $\operatorname {\mathrm {Diff}}^k(X)$ .) We say that a fibered partially hyperbolic system $(f,\pi ,M)$ is $C^k$ if the fiber bundle $\pi :M\to B$ is $C^k$ . Note the distinction between a $C^k$ fibered partially hyperbolic system and a fibered partially hyperbolic system with $C^k$ fibers: in a $C^k$ fibered partially hyperbolic system, we require that the bundle $\pi :M\to B$ is $C^k$ , whereas in a fibered partially hyperbolic system with $C^k$ fibers, the bundle $\pi :M\to B$ is merely required to be continuous.

Fibered partially hyperbolic systems form a rich class of dynamical systems. Beyond the relatively simple examples of skew products on trivial bundles, fibered systems appear as automorphisms of nilmanifolds and play a role in the construction of exotic partially hyperbolic systems (e.g. [16]). The classification of fibered partially hyperbolic diffeomorphisms up to leaf conjugacy is an open question in general, except in low dimension. Such diffeomorphisms appear in several rigidity contexts (e.g. [3, 10, 36]).

Here is our main result.

Theorem A. Let $f:M\to M$ be a fibered partially hyperbolic system with quotient a nilmanifold B and $C^1$ fibers F (where F is a closed manifold). Suppose that the structure group of the F-bundle M is $G\subset \operatorname {\mathrm {Diff}}^1(F)$ and that there exists a Riemannian metric on F and a subgroup I of $\operatorname {\mathrm {Isom}}(F)\cap G$ such that the inclusion $I\hookrightarrow G$ is a homotopy equivalence.

Then f is leaf conjugate to a $C^\infty $ fibered partially hyperbolic system $g:\widehat {M}\to \widehat {M}$ such that:

1. (1) the projection of the leaf conjugacy to B is a map homotopic to the identity;

2. (2) the F-bundles M and $\widehat M$ are isomorphic (this is implicit from the definition of leaf conjugacy, but we state it explicitly for clarity);

3. (3) the structure group of $\widehat M$ is $\operatorname {\mathrm {Isom}}(F)$ ; and

4. (4) the projection of g to B is a hyperbolic nilmanifold automorphism.

Remark 1.1

• • The base manifold B in Theorem A a priori might not have a smooth structure. In our results, we assume B is a topological nilmanifold, meaning that B is homeomorphic to a nilmanifold. This case can easily be reduced to the case where B is a nilmanifold by replacing the projection map for the bundle with the projection map composed with the homeomorphism. Thus, our results easily extend to a topological nilmanifold B.

• • Item (3) implies that there exists a smooth Riemannian metric on $\widehat M$ adapted to g such that g is isometric on fibers. This will be clear from the construction of g in the proof of Theorem A.

• • The fibered partially hyperbolic system $g:\widehat M \to \widehat M$ may act differently on the fibers than the original fibered partially hyperbolic system $f:M\to M$ ; that is, if ${h:M\to \widehat M}$ is the leaf conjugacy from Theorem A, then $h\circ f$ and $g\circ h$ may not be homotopic.

The main assumption in Theorem A is that the structure group of the F-bundle contains a homotopy equivalent subgroup of $\operatorname {\mathrm {Isom}}(F)$ . We discuss the necessity of this assumption to our proof in Remark 5.3. Finding circumstances where this assumption applies (and thus we can apply Theorem A) comes down to studying the relationship between $\operatorname {\mathrm {Diff}}^1(F)$ , $\operatorname {\mathrm {Isom}}(F)$ , and their subgroups. (In the following discussion, we often replace $\operatorname {\mathrm {Diff}}^1(F)$ with $\operatorname {\mathrm {Diff}}^\infty (F)$ , which we can do since $\operatorname {\mathrm {Diff}}^\infty (F)\hookrightarrow \operatorname {\mathrm {Diff}}^1(F)$ is a homotopy equivalence (Proposition 2.5).) Namely, for which manifolds F and which subgroups G of $\operatorname {\mathrm {Diff}}^\infty (F)$ is there a subgroup $H\subset \operatorname {\mathrm {Isom}}(F)\cap G$ such that the inclusion $H\hookrightarrow G$ is a homotopy equivalence?

Note that even without the assumption that the structure group of the F-bundle contains a homotopy equivalent subgroup of $\operatorname {\mathrm {Isom}}(F)$ , our argument gives that the initial fibered partially hyperbolic system $f:M\to M$ is leaf conjugate to an extension over a hyperbolic nilmanifold automorphism. For further discussion and details, see Remark 5.3. Notably, in the case where the F-bundle from the fibered partially hyperbolic system $f:M\to M$ is trivial (that is, $M=B\times F$ ), we can construct this extension over a hyperbolic nilmanifold automorphism to be partially hyperbolic.

Proposition B. Let $f:M\to M$ be a fibered partially hyperbolic system with quotient a nilmanifold B and $C^1$ fibers F (where F is a closed manifold). Suppose that the F-bundle M is trivial (that is, the F-bundle M is isomorphic to $B\times F$ ).

Then f is leaf conjugate to a $C^\infty $ fibered partially hyperbolic system $g:\widehat {M}\to \widehat {M}$ such that items (1), (2), and (4) from Theorem A hold.

The proof of Proposition B is given at the end of §5.

The following two corollaries come from answering the above question about subgroups $H\subset \operatorname {\mathrm {Isom}}(F)\cap G$ for specific F and G. They are by no means the only such corollaries. (For example, the conclusion of Corollary D holds for any F such that $\operatorname {\mathrm {Diff}}^1_0(F)$ is contractible.)

Corollary C. Let $f:M\to M$ be a fibered partially hyperbolic system with quotient a nilmanifold B and $C^1$ fibers F, where F is:

1. (1) a n-sphere $S^n$ for $n=1,2,3$ ; or

2. (2) a hyperbolic $3$ -manifold.

Then f is leaf conjugate to a $C^\infty $ fibered partially hyperbolic system $g:\widehat {M}\to \widehat {M}$ , which induces a hyperbolic nilmanifold automorphism on the base.

Corollary D. Let $f:M\to M$ be a fibered partially hyperbolic system with quotient a nilmanifold B and $C^1$ fibers F and with structure group $\operatorname {\mathrm {Diff}}^1_{0}(F)$ . Suppose that F is:

1. (1) the two or three torus, $\mathbb {T}^2, \ \mathbb {T}^3$ ;

2. (2) a hyperbolic surface; or

3. (3) a Haken 3-manifold.

Then f is leaf conjugate to a $C^\infty $ fibered partially hyperbolic system $g:\widehat {M}\to \widehat {M}$ , which induces a hyperbolic nilmanifold automorphism on the base.

Remark 1.2. As noted before, these are by no means the only cases where Theorem A can be applied to get results analogous to the results of Corollaries C and D. Notably, the conclusion of Corollary C holds for any fiber F such that the inclusion $\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}(F)$ is a homotopy equivalence. For the sake of conciseness, we have not listed in Corollary C more of the known examples of closed manifolds F for which $\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}(F)$ is a homotopy equivalence. Other examples include lens spaces, prism and quaternionic manifolds, tetrahedral manifolds, octahedral manifolds, and icosahedral manifolds [4].

Theorem A builds on previous work by Hirsch, Pugh, and Shub, and by Hammerlindl and Potrie. Hirsch, Pugh, and Shub [25] proved that perturbations of fibered partially hyperbolic systems are fibered partially hyperbolic systems, and that the perturbed system is leaf conjugate to the original system. Hammerlindl [18] proved that a partially hyperbolic diffeomorphism of ${\mathbb T}^3$ is leaf conjugate to a linear automorphism of ${\mathbb T}^3$ . Hammerlindl and Potrie [19] proved an analogous result for partially hyperbolic diffeomorphisms of three-dimensional nilmanifolds.

Corollaries C and D give specific examples of cases where we can apply Theorem A in dimensions 1, 2, and 3, and Proposition B gives an analog of Theorem A in the case of a trivial bundle. Note that Corollary C shows that the hypothesis of Theorem A holds for all fibered partially hyperbolic systems with a one-dimensional fiber. Corollaries C and D also show that the conclusion of Theorem A holds for surface bundles, although in the cases of surfaces other than $S^2$ , an added assumption on the structure group of the bundle is needed. The question remains of whether, and in what conditions, for other types of fibers in nontrivial bundles.

Remark 1.3. If for the fibered partially hyperbolic system $f:M\to M$ , $\dim E^s = 1$ (or $\dim E^u =1$ ), then the base space B will always be a nilmanifold [5]. This allows us to replace the assumption that the quotient of the fibered partially hyperbolic system $f:M\to M$ is a nilmanifold in Theorem A or in Proposition B with the assumption that $\dim E^s=1$ (or $\dim E^u =1$ ).

One ingredient in the proof of Theorem A, which is of independent interest, is the following generalization of the works of Franks and Manning, and of Hiraide to Anosov homeomorphisms of nilmanifolds.

Theorem E. [41, Theorem 2(1)]

An Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism via a conjugacy that is homotopic to the identity.

This theorem was originally proved by Sumi in [41]. A proof of Theorem E, which follows the same structure as Sumi and Hiraide’s proofs in [24, 41], is provided in §A for the sake of completeness.

Anosov homeomorphisms are generalizations of Anosov diffeomorphisms. Many of the important properties of Anosov diffeomorphisms come directly from the fact that Anosov diffeomorphisms are expansive and have the shadowing property. A homeomorphism $f:X\to X$ of a metric space is expansive if there exists a constant $c>0$ such that for all $x,y\in X$ , if $d(f^n(x),f^n(y))<c$ for all $n\in {\mathbb Z}$ , then $x=y$ . Such a constant c is called an expansive constant for f.

The shadowing property says that we can approximate pseudo-orbits by actual orbits. More formally, a sequence of points $\{x_i\}_{i\in {\mathbb Z}}\subset X$ is called a $\delta $ -pseudo-orbit if $d(f(x_i),x_{i+1})<\delta $ for all $i\in {\mathbb Z}$ . A point $z\in X$ is said to $\varepsilon $ -shadow a sequence of points, $\{x_i\}_{i\in {\mathbb Z}}$ , if $d(f^i(z),x_i)<\varepsilon $ for all $i\in {\mathbb Z}$ . We say that f has the shadowing property if for any $\varepsilon>0$ , there exists $\delta>0$ such that any $\delta $ -pseudo-orbit is $\varepsilon $ -shadowed by a point in X. An expansive homeomorphism with the shadowing property is known as an Anosov homeomorphism.

In contrast to some other weakenings of Anosov diffeomorphisms (e.g. hyperbolic homeomorphisms [29, §IV.9]), Anosov homeomorphisms are not assumed to have invariant foliations.

In §A, we examine some of the similarities between Anosov homeomorphisms and Anosov diffeomorphisms and give a proof Theorem E.

Theorem E is useful in the proof of our main theorem because given a fibered partially hyperbolic diffeomorphism $f:M\to M$ with associated bundle $\pi :M\to B$ , the map induced by f on B is an Anosov homeomorphism. (This follows from a result of Bohnet and Bonatti [6], as we will explain in §5.)

After proving Theorem E in §A, we spend the rest of the paper proving Theorem A. The proof is split into four parts. Due to the fact that much of the proof will take place in the quotient leaf space B, we denote the partially hyperbolic diffeomorphism on M by $\hat f:M\to M$ . We denote the homeomorphism that $\hat f$ descends to on B by $f:B\to B$ .

The strategy of the proof is to first construct:

• • a conjugacy $h:B\to B$ between $f:B\to B$ and a hyperbolic nilmanifold automorphism $A:B\to B$ ; and

• • a smooth F-bundle $\widehat {M}$ over B that is isomorphic to the original F-bundle M.

Then lift:

• • the conjugacy $h:B\to B$ to a homeomorphism $\hat h: \widehat M\to \widehat M$ ; and

• • the hyperbolic nilmanifold automorphism $A:B\to B$ to a partially hyperbolic diffeomorphism $g:\widehat M \to \widehat M$ .

The construction of the conjugacy $h:B\to B$ and the hyperbolic nilmanifold automorphism $A:B\to B$ relies almost entirely on Theorem E. The construction of the smooth bundle $\widehat M$ relies on tools developed in §2. Lifting the conjugacy $h:B\to B$ to a homomorphism $\hat h:\widehat M\to \widehat M$ takes place in §3, and finally lifting A to a partially hyperbolic diffeomorphism g relies on tools developed in §4. The entire proof of Theorem A is given in §5.

2 Preliminaries about fiber bundles

The goal of this section is to provide several results about fiber bundles that are necessary for the proof of Theorem A.

The structure of a fiber bundle is given by its transition functions. Given a continuous fiber bundle $\pi :E\to B$ with fiber F and structure group G, let $\{ (U_i,\phi _i:\pi ^{-1}(U_i)\to U_i\times F) \}$ be locally trivializing charts for E. The functions $\tau _{ij}:=\phi _i\circ \phi _j^{-1}:U_i\cap U_j\to G$ are transition functions for the bundle E.

All the data in a fiber bundle are contained in the transition functions. More precisely, we have the following lemma.

Lemma 2.1. (Fiber bundle construction theorem)

Let $F, B$ be $C^k$ manifolds ( $0\leq k\leq \infty $ ), and let G be a topological group with the structure of a $C^k$ manifold that has a $C^k$ left action on F. Given an open cover $\{U_i\}_{i\in A}$ of B and a set of $C^k$ functions $\tau _{ij}:U_i\cap U_j\to G$ such that the cocycle condition, $\tau _{ij}(x)\tau _{jk}(x)=\tau _{ik}(x)$ holds for all $x\in U_i\cap U_j\cap U_k$ . Then there exists a $C^k\ F$ -bundle $\pi :E\to B$ with transition functions $\tau _{ij}$ .

This means that any set of functions $\{\tau _{ij}:U_i\cap U_j\to G\}$ that satisfy the cocycle condition, $\tau _{ij}(x)\tau _{jk}(x)=\tau _{ik}(x)$ holds for all $x\in U_i\cap U_j\cap U_k$ , are transition functions for a fiber bundle. Additionally, the smoothness of a bundle is defined by the smoothness of the transition functions. This is a standard result about fiber bundles (e.g. see [27, §5.3]).

Transition functions also classify fiber bundles up to isomorphism. Two fiber bundles are isomorphic if they have cohomologous transition functions. Two sets of transition functions (that is, two sets of functions that satisfy the cocycle condition) ${\{\tau _{ij}:U_i\cap U_j\to G\}}$ and $\{ \tau ^{\prime }_{ij}:U_i\cap U_j\to G \}$ are said to be cohomologous if there exist continuous functions $t_i:U_i\to G$ such that $\tau ^{\prime }_{ij}(x)=t_i(x)^{-1}\tau _{ij}(x)t_j(x)$ for all $x\in U_i\cap U_j$ .

Corollary 2.2. Let $F,B$ be topological spaces and G a topological group that has a continuous left action on F. Suppose $\pi :E\to B$ and $\tilde \pi :\tilde E\to B$ are continuous F-bundles with structure group G. Let $\{(U_i,{\phi _i}:\pi ^{-1}(U_i)\to U_i\times F)\}$ and $\{(U_i,\widetilde {\phi _i}:\widetilde \pi ^{-1}(U_i)\to U_i\times F)\}$ be locally trivializing charts for E and $\tilde E$ , respectively, and let $\{\tau _{ij}:U_i\cap U_j\to G\}$ and $\{\widetilde \tau _{ij}:U_i\cap U_j\to G\}$ be the associated transition functions for $\{(U_i,\phi _i)\}$ and $\{(U_i,\tilde \phi _i)\}$ , respectively. Suppose there exist continuous functions ${t_i:U_i\to G}$ such that $\tilde \tau _{ij}(x)=t_i(x)^{-1}\tau _{ij}(x)t_j(x)$ for all $x\in U_i\cap U_j$ . Then, E and $\tilde E$ are isomorphic as F-bundles with structure group G. If, in addition, the left G action on F is faithful, then the converse holds.

For a proof of Corollary 2.2, see [27, Ch. 5.2].

In the proof of Theorem A, we change the structure group of the fiber bundles with which we are working. We now make the notion of changing structure group precise.

Given a continuous homomorphism $\alpha :H\to G$ between two topological groups and a principal H-bundle $q:Q\to B$ , we can construct a principal G-bundle from Q and $\alpha $ in the following way. Consider the space

$$ \begin{align*} Q\times_{\alpha,H}G := Q\times G/(x,g)\sim \{(x\cdot h,\alpha(h)g), h\in H\}. \end{align*} $$

Note that $Q\times _{\alpha ,H}G$ has a free right G-action given by $[x,g]\cdot g'=[x,gg']$ . This makes $Q\times _{\alpha ,H}G$ a principal G-bundle. Note that the projection map for this bundle $Q\times _{\alpha ,H}G\mapsto B$ is given by $[x,g]\mapsto q(x)$ .

We say that the principal H-bundle $q:Q\to B$ induces the principal G-bundle ${p:P\to B}$ if $P\cong Q\times _{\alpha ,H} G$ . Additionally, we say that a principal G-bundle $p:P\to B$ admits a reduction of structure group from G to H if there exists a principal H-bundle $q:Q\to B$ that induces P.

Now, we note the relationship between the transition functions between a bundle and a bundle it induces. Suppose that $t_{ij}:U_i\cap U_j\to H$ are transition functions for the principal H-bundle $q:Q\to B$ . The transition functions for the induced bundle $Q\times _{\alpha ,H}G$ are given by $\alpha \circ t_{ij}:U_i\cap U_j\to G$ .

Next, we give conditions under which a principal bundle admits a reduction of structure group.

Lemma 2.3. Suppose that $\alpha :H\to G$ is a homomorphism that is also a homotopy equivalence. Then any principal G-bundle admits a reduction of structure group from G to H.

The proof of this lemma relies on the theory of classifying spaces. Let G be a topological group. A principal G-bundle $\pi :EG\to BG$ is a universal principal G-bundle if for all CW-complexes X, the map from the set of homotopy classes of maps $X\to BG$ to the set of isomorphism classes of principal G-bundles over X, given by the map $f\mapsto f^*EG$ is a bijection. The base space of a universal principal G-bundle is known as a classifying space for G.

Proof. See [27, Ch. 6].

Remark 2.4. Note that if $H\subset G$ and the homomorphism $\alpha :H\to G$ is inclusion, then the reduction of structure group from G to H from Lemma 2.3 has transition functions that are cohomologous to the transition functions of the original principal G bundle.

Proposition 2.5. If M is a closed, smooth manifold, then the inclusion $\operatorname {\mathrm {Diff}}^\infty (M)\hookrightarrow \operatorname {\mathrm {Diff}}^1(M)$ is a homotopy equivalence.

Proof. For $0\leq k\leq \infty $ , $\operatorname {\mathrm {Diff}}^k(M)$ is an infinite-dimensional separable Fréchet space [17, §I.4.3]. Since all infinite-dimensional separable Fréchet spaces are homeomorphic to the Hilbert space $\ell ^2$ [1], we see that $\operatorname {\mathrm {Diff}}^k(M)$ is homeomorphic to the Hilbert space $\ell ^2$ . Since $\ell ^2$ has the homotopy type of a CW-complex [40], we get that $\operatorname {\mathrm {Diff}}^k(M)$ is homotopy equivalent to a CW-complex [22]. Thus, by Whitehead’s theorem, to show that the inclusion $\iota :\operatorname {\mathrm {Diff}}^\infty (M)\hookrightarrow \operatorname {\mathrm {Diff}}^1(M)$ is a homotopy equivalence, it is sufficient to show that the induced map on homotopy groups $\iota _*:\pi _*(\operatorname {\mathrm {Diff}}^\infty (M))\to \pi _*(\operatorname {\mathrm {Diff}}^1(M))$ is an isomorphism.

To do this, we first recall that for any map in $\varphi \in C^1(M,M)$ , we can find a smooth map that is arbitrarily close to and homotopic to $\varphi $ and that the choice of this map depends continuously on our original map [28, Theorems 6.21 and 6.28]. This gives us a continuous map $\Phi :\operatorname {\mathrm {Diff}}^1(M)\to C^\infty (M,M)$ such that for any $\varphi \in \operatorname {\mathrm {Diff}}^1(M)$ , $\Phi (\varphi )\simeq \varphi $ and $d(\varphi ,\Phi (\varphi ))<\varepsilon $ . Since $\operatorname {\mathrm {Diff}}^\infty (M)\subset C^\infty (M,M)$ is open (by the inverse function theorem), by choosing $\varepsilon $ small enough, we get that $\Phi (\operatorname {\mathrm {Diff}}^1(M))\subset \operatorname {\mathrm {Diff}}^\infty (M)$ , so we can write $\Phi :\operatorname {\mathrm {Diff}}^1(M)\to \operatorname {\mathrm {Diff}}^\infty (M)$ .

Now, we show that the map induced by $\iota :\operatorname {\mathrm {Diff}}^\infty (M)\to \operatorname {\mathrm {Diff}}^1(M)$ on homotopy groups is an isomorphism. The map $\iota _*$ is surjective because any map $\phi :S^n\to \operatorname {\mathrm {Diff}}^1(M)$ is homotopic to the map $\Phi \circ \phi :S^n\to \operatorname {\mathrm {Diff}}^\infty (M)$ . To see that $\iota _*$ is injective, we take a map $\phi :S^n\to \operatorname {\mathrm {Diff}}^1(M)$ that is null-homotopic. Let $h_t:S^n\to \operatorname {\mathrm {Diff}}^1(M)$ be a null-homotopy for $\phi $ . Then the map $\Phi \circ \phi :S^n\to \operatorname {\mathrm {Diff}}^\infty (M)$ is homotopic to $\phi $ in $\operatorname {\mathrm {Diff}}^1(M)$ , and is null-homotopic in $\pi _n(M)$ via the homotopy $\Phi \circ h_t:S^n\to \operatorname {\mathrm {Diff}}^\infty (M)$ .

3 Lifting the conjugacy on the leaf space

Lemma 3.1. Let $F,M,\tilde M,B$ be closed Riemannian manifolds, and let $\pi :M\to B$ and $\widetilde {\pi }:\tilde M\to B$ be continuous isomorphic F-bundles. Let $h:B\to B$ be a homeomorphism that is homotopic to the identity. Then, $h:B\to B$ lifts to a homeomorphism $\tilde h:M\to \tilde M$ .

Proof. First, we note that M and $h^*M$ are isomorphic bundles. This is because, by assumption, M and $\tilde M$ are isomorphic, and since $h\sim {\text {id}}$ , the bundle $h^* \tilde M$ is isomorphic to the bundle ${\text {id}}^*\tilde M=\tilde M$ . Let $\phi :\tilde M\to h^*\tilde M$ be a bundle isomorphism, that is, the diagram in Figure 1a commutes. From the definition of $h^*\tilde M$ , we have that the commutative diagram in Figure 1b commutes. (Note that $\operatorname {\mathrm {proj}}_2:h^* \tilde M \to \tilde M$ is the projection onto the second coordinate from the definition of the pullback bundle $h^* \tilde M = \{ (b,x)\in B\times M: h(b)=\tilde \pi (x) \}$ .) Combining these diagrams gives us the commutative diagram from Figure 1c. So the continuous map

(3.1) $$ \begin{align} \tilde h:=\operatorname{\mathrm{proj}}_2\circ \phi:M\to \tilde M \end{align} $$

is a lift of h. Since $\tilde h$ is a continuous injection, by invariance of domain, $\tilde h:M\to \tilde M$ is a homeomorphism.

Figure 1 Diagrams showing how $h:B\to B$ is lifted to a homeomorphism $\tilde{h}:M\to \tilde{M}$ in the proof of Lemma 3.1.

4 Lifting the Anosov automorphism on the leaf space to a partially hyperbolic system

Lemma 4.1. Let $F,E_0,E_1, B$ be closed Riemannian manifolds. Assume that ${p_0:E_0\to B}$ and $p_1:E_1\to B$ are $C^k\ F$ -bundles with structure group H, where H is a finite-dimensional Lie group with smooth universal bundle, and that the left action of H on F is $C^k$ , and that H acts on F by isometries. Suppose that $\theta :E_0\to E_1$ is a (continuous) isomorphism of $E_0$ and $E_1$ as F-bundles with structure group H over B. Then, there is a $C^k$ isomorphism $\alpha :E_0\to E_1$ that is an isometry on fibers.

Proof. We begin by constructing principal H-bundles with the same transition data as $E_0$ and $E_1$ using Lemma 2.1. We will call these $q_0:Q_0\to B$ and $q_1:Q_1\to B$ . Let $f_0:B\to BH$ and $f_1:B\to BH$ be classifying maps for $Q_0$ and $Q_1$ . Note that since $E_0$ and $E_1$ (and therefore $Q_0$ and $Q_1$ ) are $C^k$ bundles, the maps $f_0$ and $f_1$ are $C^k$ .

Since $E_0$ and $E_1$ are isomorphic as continuous bundles, we get that there is a homotopy $f:B\times [0,1]\to BH$ from $f_0$ to $f_1$ . Since $f|_{B\times \{0,1\}}$ is $C^k$ , then by the Whitney approximation theorem (see [28, Theorem 6.26]), $f_t$ is homotopic to a $C^k$ map ${\overline {f}:B\times [0,1]\to BH}$ relative to $B\times \{0,1\}$ . So, we have a $C^k$ homotopy $\overline {f}:B\times [0,1]\to BH$ from $\overline {f_0}=f_0$ to $\overline {f_1}=f_1$ . If the classifying maps of two $C^k$ principal bundles are homotopic via a $C^k$ homotopy, then the bundles are isomorphic as $C^k$  bundles [27, Ch. 4.9]. (The argument in [27, Ch. 4.9] is only given for continuous bundles, but works for $C^k$ bundles.) Thus, we get that the pullback bundles $f_0^*EH$ and $f_1^*EH$ are isomorphic as $C^k$ principal H-bundles over B. Since $f_0$ and $f_1$ are the classifying maps for $Q_0$ and $Q_1$ , respectively, this means that $Q_0$ and $Q_1$ are isomorphic as $C^k$ principal H-bundles over B. Since $Q_0$ and $Q_1$ have the same transition functions as $E_0$ and $E_1$ , we get that $E_0$ and $E_1$ are isomorphic as $C^k$ bundles with structure group H.

From the definition of a $C^k$ isomorphism of F-bundles with structure group H, we see that this means that there is a $C^k$ isomorphism $\alpha :E_0\to E_1$ , such that, if ${(U_{0,i}, \phi _{0,i})}$ and $\{(U_{1,j},\phi _{1,j})\}$ are trivializing atlases for $E_0$ and $E_1$ , respectively, then there exists functions $d_{ij}:U_{0,i}\cap V_{1,j}\to H$ such that for $x\in U_{0,i}\cap U_{1,j}$ and $y\in F$ , we get that $\phi _{1,j}\circ \alpha \circ \phi _{0,i}^{-1}(x,y)=(x,d_{ij}(x)\cdot y)$ . Since H acts on F by isometries, we get that $\alpha $ is an isometry on fibers.

Figure 2 Diagrams showing how $A:B\to B$ is lifted to $g:M\to M$ in the proof of Corollary 4.2.

Corollary 4.2. Let $F,M, B$ be closed Riemannian manifolds. Assume that ${\pi :M\to B}$ is a smooth F-bundle with structure group H, where H is a finite-dimensional Lie group with smooth universal bundle, and that the left action of H on F is smooth, and that H acts on F by isometries. Suppose $A:B\to B$ is a smooth Anosov diffeomorphism, and suppose that A lifts to a homeomorphism $\widehat {A}:M\to M$ . Then A lifts to a $C^\infty $ diffeomorphism ${g:M\to M}$ that is an isometry on fibers of $\pi :M\to M$ .

Proof. Since A lifts to a homeomorphism $\widehat {A}: M\to M$ , we can construct a (continuous) isomorphism $\theta : M\to A^*M$ of M and $A^*M$ as F-bundles with structure group H given by $\theta (z)=(\pi (z), \widehat {A}(z) )$ (see Figure 2a). By Lemma 4.1, there is a $C^\infty $ isomorphism ${\alpha :M\to A^*M}$ that is an isometry on fibers. We can then use $\alpha :M\to A^*M$ to define a $C^\infty $ diffeomorphism $g:M\to M$ by $g(z)=\operatorname {\mathrm {proj}}_2\circ \alpha (z)$ (see Figure 2b). Since $\alpha $ is an isometry on fibers, we get that g is an isometry on fibers.

Proposition 4.3. Let $F,M,$ and B be closed Riemannian manifolds. Assume that ${\pi :M\to B}$ is a smooth F-bundle and that $f:B\to B$ is an Anosov diffeomorphism. If $g:M\to M$ is a diffeomorphism that is a lift of f and such that g is an isometry on fibers of $\pi :M\to B$ , then $g:M\to M$ is partially hyperbolic.

Proof. We begin by constructing a Riemannian metric on M, with respect to which g is partially hyperbolic. This construction has three ingredients.

• • A smooth family $\langle \cdot ,\cdot \rangle ^F_x$ of Riemannian metrics on the fibers $\pi ^{-1}(x)$ such that ${Dg:T\pi ^{-1}(x)\to T\pi ^{-1}(f(x))}$ is an isometry for all $x\in B$ . (Such a family exists because g is an isometry on fibers of $\pi :M\to B$ .)

• • A Riemannian metric $\langle \cdot , \cdot \rangle ^B$ on B that is adapted to the Anosov diffeomorphism ${f:B\to B}$ .

• • An Ehresmann connection H on M, that is, H is a smooth subbundle of the tangent bundle $TM$ such that for all $p\in M$ , $T_pM=H_p\oplus \ker (D_p\pi )$ . Note that from the definition of an Ehresmann connection, we know that $D_p\pi |_{H_p}:H_p\subset T_pM \to T_{\pi (p)}B$ is an isomorphism and the map $p\mapsto H_p$ is smooth.

We define a Riemannian metric $\langle \cdot , \cdot \rangle $ on M by letting, for all $p\in M$ :

• • $\langle v,v'\rangle =\langle D_p\pi (v), D_p\pi (v')\rangle ^B$ for $v,v'\in H_p$ ;

• • $\langle v,v'\rangle = \langle v,v'\rangle _{\pi (p)}^F$ for $v,v'\in \ker (D_p\pi )$ ; and

• • $H_p\perp \ker (D_p \pi )$ .

Now, we need to show that g is partially hyperbolic with respect to the metric $\langle \cdot , \cdot \rangle $ . To do this, we construct a dominated splitting $TM=E^s\oplus E^c\oplus E^u$ such that g is uniformly contracting on $E^s$ and uniformly expanding on $E^u$ . We begin by letting $E^c=\ker (D\pi )$ .

Next, we construct the unstable bundle $E^u$ using a graph transform argument. We begin by lifting the unstable bundle $E^u_f\subset TB$ for the Anosov diffeomorphism $f:B\to B$ to the bundle $\hat E^u\subset TM$ given by $\hat E^u_p:= H_p\cap D_p\pi ^{-1}(E^u_f(\pi (p)))$ for $p\in M$ . Note that $D_p\pi ^{-1}(E^u_p(\pi (p)))=\hat E^u_p \oplus \ker (D_p\pi )$ , and that $Dg$ preserves $\hat E^u\oplus E^c$ because $Df$ preserves $E^u_f$ and g covers f (so $D_{\pi (p)}f\circ D_p\pi =D_{g(p)}\pi \circ D_pg$ ).

Let

$$ \begin{align*} \Sigma&=\{\sigma:\hat E^u\to E^c: \ \sigma \text{ is fiber preserving over id},\\&\hspace{7pt}\quad \text{and } \sigma_p:\hat E^u(p)\to E^c(p) \text{ is linear } \text{for all } p\in M \}. \end{align*} $$

We put the norm $\|\cdot \|_\Sigma $ on $\Sigma $ given by

$$ \begin{align*} \|\sigma\|_\Sigma=\sup_{p\in M} \|\sigma_p\|, \end{align*} $$

where $\|\sigma _p\|$ is the operator norm. Note that this norm makes $\Sigma $ a Banach space.

Now, we want to define a map $\Gamma :\Sigma \to \Sigma $ , called the linear graph transform covering g, so that $D_pg(\operatorname {\mathrm {graph}}(\sigma _p))=\operatorname {\mathrm {graph}}(\Gamma (\sigma _p))$ . We now give $\Gamma :\Sigma \to \Sigma $ explicitly. Since by assumption, $Dg$ preserves $E^c=\ker (D\pi )$ (and $Dg$ preserves $\hat E^u\oplus E^c$ ), we can write for each $p\in M$ ,

$$ \begin{align*} D_pg=\begin{pmatrix} A_p & 0\\ C_p & K_p \end{pmatrix}:\hat E^u(p)\oplus E^c(p) \to \hat E^u(g(p))\oplus E^c(g(p)), \end{align*} $$

where

$$ \begin{align*} A_p:\hat E^u(p)\to \hat E^u(g(p)), \quad C_p:\hat E^u(p)\to E^c(g(p)), \quad K_p:E^c(p)\to E^c(g(p)) \end{align*} $$

are all linear. Also note that since $D_pg$ is invertible, both $A_p$ and $K_p$ are invertible.

Note that we can write a point in the graph of $\sigma _p$ as $(v,\sigma _pv)\in \operatorname {\mathrm {graph}}(\sigma _p)\subset \hat E^u(p)\oplus E^c(p)$ . Applying $D_pg$ to this point gives us

$$ \begin{align*} D_pg(v,\sigma_p v) = \begin{pmatrix} A_p&0\\C_p&K_p \end{pmatrix} \begin{pmatrix} v \\ \sigma_pv \end{pmatrix} = \begin{pmatrix} A_pv\\ C_p v +K_p\sigma_p v \end{pmatrix}. \end{align*} $$

So, we can write

$$ \begin{align*} D_pg(\operatorname{\mathrm{graph}}(\sigma_p))&= \left\{ \begin{pmatrix} A_pv\\ C_p v +K_p\sigma_p v \end{pmatrix} : \ v\in \hat E^u(p) \right\}\\& = \left\{ \begin{pmatrix} w\\ (C_p +K_p\sigma_p )\circ A_p^{-1}w \end{pmatrix} : \ w\in \hat E^u(g(p)) \right\} \end{align*} $$

by reparameterizing. Thus, the requirement that $D_pg(\operatorname {\mathrm {graph}}(\sigma _p))=\operatorname {\mathrm {graph}}(\Gamma (\sigma _p))$ is equivalent to saying that

$$ \begin{align*} \begin{pmatrix} w\\ (C_p +K_p\sigma_p )\circ A_p^{-1}w \end{pmatrix} = \begin{pmatrix} w \\ \Gamma\sigma_p w \end{pmatrix} \end{align*} $$

for all $w\in \hat E^u(g(p))$ . This gives us an equation for $\Gamma $ in terms of $C, \ K,$ and A:

$$ \begin{align*} \Gamma\sigma_p =(C_p +K_p\sigma_p )\circ A_p^{-1}:\hat E^u(g(p))\to E^c(g(p)) \end{align*} $$

for all $\sigma \in \Sigma $ , $p\in M$ , and $v\in \hat E^u(p)$ . Omitting base points, we get that

$$ \begin{align*} \Gamma \sigma = (C +K\sigma )\circ A^{-1}. \end{align*} $$

Now, our goal is to find an invariant section for $\Gamma $ (the graph of which we will then show is the unstable bundle $E^u$ for g). To do this, it suffices to show that $\Gamma :\Sigma \to \Sigma $ is a contraction. Take $\sigma , \sigma '\in \Sigma $ . For $p\in M$ , we have

(4.1) $$ \begin{align} \|\Gamma \sigma_p - \Gamma \sigma_p'\| &= \| ( C_p+K_p \sigma_p )\circ A_p^{-1} - ( C_p+K_p \sigma_p' )\circ A_p^{-1} \|\nonumber\\ &= \| ( C_p+K_p \sigma_p-C_p-K_p\sigma^{\prime}_p )\circ A_p^{-1}\|\nonumber\\ &= \| (K_p \sigma_p-K_p\sigma^{\prime}_p )\circ A_p^{-1}\|\nonumber\\ &= \| K_p \circ (\sigma_p-\sigma^{\prime}_p )\circ A_p^{-1}\|\nonumber\\ &\leq \|K_p\| \|\sigma_p-\sigma^{\prime}_p\|\|A_p^{-1}\|. \end{align} $$

We now bound $\|K_p\|$ and $\|A_p^{-1}\|$ . Since g is an isometry on fibers of $\pi :M\to B$ and $E^c=\ker (D\pi )$ , we have that $D_pg|_{E^c(p)}=K_p:E^c(p)\to E^c(g(p))$ is an isometry. Thus, $\|K_p\|=1$ . Now, we bound $\|A_p^{-1}\|$ by relating the norm of A to the norm of $Df$ on $E^u_f$ . Since $E^u_f$ is the unstable bundle for the Anosov diffeomorphism $f:B\to B$ and the norm $\|\cdot \|^B$ is adapted to f, we know that there exists a constant $\unicode{x3bb}>1$ such that for all $w\in E^u_f$ , $\|Df(w)\|^B\geq \unicode{x3bb} \|w\|^B$ . Take $v\in \hat E^u(p)$ . Since $D_p\pi (v)\in E^u_f(\pi (p))$ , we therefore have that

$$ \begin{align*} \|D_{\pi(p)}f(D_p\pi(v))\|^B\geq \unicode{x3bb} \|D_p(v)\|^B. \end{align*} $$

Since g covers f, we see that $D_{\pi (p)}f( D_p\pi (v))=D_{g(p)}\pi (D_pg(v))$ . This along with the fact that $E^c=\ker (D\pi )$ and $C_p(v)\in E^c(g(p))$ gives that

$$ \begin{align*} D_{\pi(p)}f(D_p\pi(v)) = D_{g(p)}\pi(D_pg(v)) = D_{g(p)}\pi(A_p(v)+C_p(v)) = D_{g(p)}\pi(A_p(v)). \end{align*} $$

Thus,

$$ \begin{align*} \unicode{x3bb} \|D_p(v)\|^B \leq \|D_{\pi(p)}f(D_p\pi(v))\|^B = \| D_{g(p)}\pi(A_p(v)) \|^B. \end{align*} $$

Finally, note that since $v, A_p(v)\in \hat E^u(P)\subset H_p$ , by our definition of the norm on M, we get that

$$ \begin{align*} \|D_p\pi(v)\|^B=\|v\| \quad \text{and} \quad \| D_{g(p)}\pi(A_p(v)) \|^B=\|A_p(v)\|. \end{align*} $$

We have therefore shown that $\unicode{x3bb} \|v\|\leq \|A_p(v)\|$ , which implies that $\|A_p^{-1}\|\leq \unicode{x3bb} ^{-1}$ .

Combining our estimates for the norms of $\|K_p\|$ and $\|A_p^{-1}\|$ with equation (4.1) gives that

$$ \begin{align*} \|\Gamma \sigma_p - \Gamma \sigma_p'\|\leq \unicode{x3bb}^{-1}\|\sigma_p-\sigma^{\prime}_p\|. \end{align*} $$

We have therefore shown that $\Gamma $ is a contraction map. Then, by the contraction mapping principle, we get a $\Gamma $ -invariant section $\sigma ^u\in \Sigma $ . We now define a bundle $E^u\subset TM$ by letting $E^u(p):=\operatorname {\mathrm {graph}}(\sigma ^u_p)$ . Note that $E^u$ is $Dg$ invariant since $\sigma ^u$ is $\Gamma $ invariant and $D_pg(\operatorname {\mathrm {graph}}(\sigma _p))=\operatorname {\mathrm {graph}}(\Gamma (\sigma _p))=\operatorname {\mathrm {graph}}(\sigma _{g(p)})$ .

The construction of the bundle $E^s$ is analogous.

We now have a $Dg$ -invariant splitting, $TM=E^s\oplus E^c\oplus E^u$ . We now need to show that this splitting is partially hyperbolic. To do this, we construct a new metric $\langle \cdot , \cdot \rangle '$ on M by letting for all $p\in M$ :

• • $\langle v,v'\rangle ' = \langle D_p\pi (v),D_p\pi (v')\rangle ^B$ for $v,v'\in E^s$ ;

• • $\langle v,v'\rangle ' = \langle D_p\pi (v),D_p\pi (v')\rangle ^B$ for $v,v'\in E^u$ ;

• • $\langle v,v'\rangle ' = \langle v,v' \rangle ^F_{\pi (p)}$ for $v,v'\in E^c$ ; and

• • $E^s$ , $E^c$ , and $E^u$ be pairwise orthogonal.

From our construction of $E^s$ and $E^u$ , we get that $Dg$ is uniformly expanding on $E^u$ and uniformly contracting on $E^c$ with respect to this new metric. Finally, the splitting is dominated because $Dg$ restricted to $E^c$ is an isometry.

5 Proof of Theorem A

First, we recall our setup. Let $\hat {f}:M\to M$ be a fibered partially hyperbolic system with quotient a nilmanifold B, $C^1$ fibers F (where F is a closed manifold), and structure group $G\subset \operatorname {\mathrm {Diff}}^1(F)$ . Suppose that there exists a Riemannian metric on F and a subgroup ${I\subset \operatorname {\mathrm {Isom}}(F)\cap G}$ such that the inclusion $I \hookrightarrow G$ is a homotopy equivalence.

The diffeomorphism $\hat {f}:M\to M$ descends to a homeomorphism $f:B\to B$ . Our first step is to construct a conjugacy $h:B\to B$ between f and a hyperbolic nilmanifold automorphism $A:B\to B$ . This will follow immediately from Theorem E if we can show that $f:B\to B$ is an Anosov homeomorphism. To see why the homeomorphism $f:B\to B$ is Anosov, we first observe that $\hat f$ admits an invariant center foliation $\mathscr {F}^c$ whose leaves are the level sets of $\pi $ and that leaves of $\mathscr {F}^c$ are compact and have trivial holonomy. (For the definition of holonomy, see [7, Ch. 2]. The fact that the leaves of $\mathscr {F}^c$ have trivial holonomy follows immediately from the definition of a fibered partially hyperbolic system and the definition of holonomy.) Thus, by the following result of Bohnet and Bonatti, $f:B\to B$ is an Anosov homeomorphism.

Lemma 5.1. [6, Theorem 2, Proposition 4.20]

If $f:M\to M$ is a partially hyperbolic diffeomorphism with an invariant center foliation $\mathscr {F}^c$ with compact leaves and without holonomy, then the homeomorphism $F:M/\mathscr {F}^c \to M/\mathscr {F}^c$ induced by f on the quotient is an Anosov homeomorphism.

Now, we can apply Theorem E to get that there exists a hyperbolic nilmanifold automorphism $A:B\to B$ and a homeomorphism $h:B\to B$ that is homotopic to the identity such that $A\circ h=h\circ \hat {f}$ .

The next step in the proof is to construct a smooth F-bundle $\widehat {\pi }:\widehat M\to B$ that is isomorphic to the original bundle $\pi :M\to B$ and such that the structure group of ${\widehat {\pi }:\widehat M \to B}$ is $\operatorname {\mathrm {Isom}}(F)$ . To do this, we first construct a principal G bundle $p:P\to B$ with the same transition functions as $\pi :M\to B$ . Since the inclusion of $I\hookrightarrow G$ is a homotopy inclusion, by Lemma 2.3, there exists a continuous principal I-bundle $q':Q'\to B$ that has transition functions cohomologous to those of $p:P\to B$ . Since $I\subset \operatorname {\mathrm {Isom}}(F)$ , we can construct (Lemma 2.1) a continuous principal $\operatorname {\mathrm {Isom}}(F)$ -bundle $q:Q\to B$ with the same transition data as $q':Q'\to B$ .

Now, we find a smooth principal $\operatorname {\mathrm {Isom}}(F)$ bundle $\hat q:\hat Q\to B$ that is isomorphic to $q:Q\to B$ . This follows immediately from the following lemma along with the fact that $\operatorname {\mathrm {Isom}}(F)$ is a locally Euclidean Lie group [35].

Lemma 5.2. [34]

Let K be a Lie group modeled on a locally convex space. Every principal K bundle over a closed manifold is isomorphic to a smooth principal K bundle.

Now, we use the fiber bundle construction theorem (Lemma 2.1) to construct a smooth F-bundle $\widehat \pi :\widehat M\to B$ with the same transition functions as $\hat q: \hat Q\to B$ . Since $\hat q:\hat Q\to B$ has transition functions that are cohomologous to those of the original bundle $\pi :M\to B$ , we get that the bundle $\widehat \pi : \widehat M\to B$ is isomorphic to the original bundle $\pi :M\to B$ .

Next, we lift the conjugacy $h:B\to B$ to a homeomorphism $\hat h: M\to \widehat M$ . This follows immediately from Lemma 3.1.

Finally, we lift the hyperbolic nilmanifold automorphism $A:B\to B$ to a partially hyperbolic diffeomorphism $g:\widehat M \to \widehat M$ . This follows immediately from Corollary 4.2 and Proposition 4.3. To see why we can apply Corollary 4.2 here, we first note that the structure group of $\widehat \pi : \widehat M \to B$ is $\operatorname {\mathrm {Isom}}(F)$ , which is a finite-dimensional compact Lie group [35]. This implies that $\operatorname {\mathrm {Isom}}(F)$ has a smooth universal bundle [34, Lemma I.12]. This completes the proof of Theorem A.

Remark 5.3. The fact that the F-bundle $\widehat M$ has structure group $\operatorname {\mathrm {Isom}}(F)$ is solely used to guarantee that the lift $g:\widehat M\to \widehat M$ of $A:B\to B$ is partially hyperbolic. Without this fact, the arguments given would allow us to lift $A:B\to B$ to a homeomorphism, but we would not be able to guarantee that the lift would be a partially hyperbolic diffeomorphism.

This is the only reason that the assumption that there exists a Riemannian metric on F and a subgroup $I\subset \operatorname {\mathrm {Isom}}(F)\cap G$ such that the inclusion $I\hookrightarrow G$ is a homotopy equivalence is necessary in the proof. Without this assumption, we would be able to get the conjugacy $h:B\to B$ between f and A, and we would be able to construct a smooth F-bundle $\widehat M$ over B that is isomorphic to the original F-bundle M. (To construct $\widehat M$ , we would first use Lemma 2.3 and Proposition 2.5 to get an F-bundle with structure group $\operatorname {\mathrm {Diff}}^\infty (F)$ that is isomorphic to M. We then would apply Lemma 5.2 with $K=\operatorname {\mathrm {Diff}}^\infty (F)$ , which would give us $\widehat M$ .) However, the structure group of $\widehat M$ would only be $\operatorname {\mathrm {Diff}}^\infty (F)$ , not $\operatorname {\mathrm {Isom}}(F)$ .

Finding a way to lift $A:B\to B$ to a partially hyperbolic diffeomorphism $g:\widehat M\to \widehat M$ without requiring that the structure group of $\widehat M$ be $\operatorname {\mathrm {Isom}}(F)$ or be trivial is a question for further research.

Proposition B is an example of a case where we can overcome the difficulty lifting $A:B\to B$ to a partially hyperbolic diffeomorphism that is discussed in the above remark.

Proof of Proposition B

The setup of Proposition B is that we are given a fibered partially hyperbolic system $\hat f:M\to M$ with quotient a nilmanifold B and $C^1$ fibers F. We assume that the F-bundle M is trivial. Note that this means that the structure group of the bundle $\pi :M\to B$ is the trivial group. We can then proceed with an analogous argument to that given in the proof of Theorem A to get the conjugacy $h:B\to B$ between the Anosov homeomorphism $f:B\to B$ induced by $\hat f:M\to M$ and a hyperbolic nilmanifold automorphism $A:B\to B$ , and to get a smooth bundle $\widehat \pi : \widehat M \to B$ with trivial structure group that is isomorphic to the original bundle $\pi : M\to B$ . This means that identifying the smooth F-bundle $\widehat \pi : \widehat M \to B$ with the smooth bundle ${\operatorname {\mathrm {proj}}_1: B\times F \to B}$ (that is, projection onto the first coordinate), we can smoothly lift $A:B\to B$ to the fibered partially hyperbolic diffeomorphism $g:\widehat M\cong B\times F\to \widehat M\cong B\times F$ given by $g: (x,y)\mapsto (Ax,y)$ for $(x,y)\in B\times F$ .

6 Corollaries of Theorem A

We now explain how Corollaries C and D follow from Theorem A.

To prove Corollary C, we apply Theorem A with $G=\operatorname {\mathrm {Diff}}^1(F)$ and with $I=\operatorname {\mathrm {Isom}}(F)$ . To do this, we just need to show that the inclusion $\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}^1(F)$ is a homotopy equivalence when $F=S^n$ for $n=1,2,3$ and for F a hyperbolic 3-manifold. In fact, showing that the inclusion $\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}^1(F)$ is a homotopy equivalence is equivalent to showing that the inclusion $\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}^\infty (F)$ is a homotopy equivalence by Proposition 2.5.

1. (1) When $F=S^1$ , it is a standard fact that $\operatorname {\mathrm {Diff}}^\infty (S^1)$ deformation retracts to $O(2)=\operatorname {\mathrm {Isom}}(S^1)$ . When $F=S^2$ , Smale [38] proved that the inclusion ${\operatorname {\mathrm {Isom}}(S^2)\hookrightarrow \operatorname {\mathrm {Diff}}^\infty (S^2)}$ is a homotopy equivalence. Hatcher [21] proved this for  $S^3$ .

2. (2) When F is a hyperbolic $3$ -manifold, Gabai [15] proved that the inclusion ${\operatorname {\mathrm {Isom}}(F)\hookrightarrow \operatorname {\mathrm {Diff}}^\infty (F)}$ is a homotopy equivalence.

Remark 6.1. When $n\geq 4$ , the inclusion $\operatorname {\mathrm {Isom}}(S^n)\hookrightarrow \operatorname {\mathrm {Diff}}(S^n)$ is not a homotopy equivalence. This was proved for $n=4$ in [43]. To see that $\operatorname {\mathrm {Isom}}(S^n)\hookrightarrow \operatorname {\mathrm {Diff}}(S^n)$ is not a homotopy equivalence for $n\geq 5$ , first note that this statement is equivalent to the statement that $\operatorname {\mathrm {Diff}}(D^n \ \text {rel } \partial D^n)$ is contractible [21, Appendix].

One way to see that $\operatorname {\mathrm {Diff}}(D^n \ \text {rel } \partial D^n)$ is not contractible for many n is to use the fact that $\pi _0(\operatorname {\mathrm {Diff}}(D^n \ \text {rel } \partial D^n))\cong \Theta _{n+1}$ for $n\geq 5$ , where $\Theta _{n+1}$ is the group of exotic ${(n+1)}$ -spheres [8, 22, 39]. For example, this along with the fact that there exist exotic 7-spheres [32] implies that $\operatorname {\mathrm {Isom}}(S^6)\hookrightarrow \operatorname {\mathrm {Diff}}(S^6)$ .

To prove that $\operatorname {\mathrm {Diff}}(D^n \ \text {rel } \partial D^n)$ is not contractible for $n=5$ , we use the fact that the map $\pi _1(\operatorname {\mathrm {Diff}}(D^n \text { rel } \partial D^n))\to \pi _0(\operatorname {\mathrm {Diff}}(D^{n+1} \text { rel } \partial D^{n+1}))$ is surjective for $n\geq 5$ [8]. Thus, since $\pi _0(\operatorname {\mathrm {Diff}}(D^{6} \text { rel } \partial D^{6}))\neq 0$ , we get that $\pi _1(\operatorname {\mathrm {Diff}}(D^5 \text { rel } \partial D^5))\neq 0$ , so $\operatorname {\mathrm {Diff}}(D^5 \text { rel } \partial D^5)$ is not contractible.

For $n\geq 7$ , the fact that $\operatorname {\mathrm {Diff}}(D^n \ \text {rel } \partial D^n)$ is not contractible is proved in [9].

This means that we cannot apply Theorem A as above to get an analogous version of Corollary C for $S^n$ , $n\geq 4$ .

Now, we prove Corollary D. In Corollary D, the structure group of M is $G=\operatorname {\mathrm {Diff}}_0^1(F)$ . We prove each case of Corollary D separately using the same strategy: we apply Theorem A by finding a subgroup $I\subset \operatorname {\mathrm {Isom}}(F)\cap \operatorname {\mathrm {Diff}}^1_0(F)$ such that the inclusion $I\hookrightarrow \operatorname {\mathrm {Diff}}^1_0(F)$ is a homotopy equivalence.

1. (1) When $F={\mathbb T}^n$ for $n=2,3$ , we choose $I={\mathbb T}^n$ , where ${\mathbb T}^n$ acts on itself by Euclidean isometry via translation. When $F={\mathbb T}^2$ , the inclusion $\mathbb {T}^2\hookrightarrow \operatorname {\mathrm {Diff}}^\infty _{0}(\mathbb {T}^2)$ is a homotopy equivalence [33, §4.1.5]. When $F={\mathbb T}^3$ , $\mathbb {T}^3\hookrightarrow \operatorname {\mathrm {Diff}}^\infty _{0}(\mathbb {T}^3)$ is a homotopy equivalence [23, 42].

2. (2) When F is a hyperbolic surface, then $\operatorname {\mathrm {Diff}}^+_0(F)$ is contractible [12, Theorem 1.14], which means that the hypothesis of Theorem A holds for I the trivial subgroup.

3. (3) When F is a Haken manifold, we consider three cases.

   • • When F is not a Seifert manifold with coherently orientable fibers, then the components of $\operatorname {\mathrm {Diff}}(F)$ are contractible [20, 23], [26, §1.3], which means that the hypothesis of Theorem A holds when I is the trivial subgroup.

   • • When F is a Seifert manifold with coherently oriented fibers, that is, not ${\mathbb T}^3$ , we take $I=S^1$ , where $S^1$ acts on F by rotating circle fibers of the Seifert fiber bundle structure. This satisfies the hypothesis of Theorem A because the inclusion $S^1\hookrightarrow \operatorname {\mathrm {Diff}}_0(F)$ is a homotopy equivalence [20, 23], [26, §1.3].

   • • When $F=\mathbb {T}^3$ , we dealt with this case in item (1).