Solvable Lie Flows of Codimension 3

Algebraic & Geometric Topology 16 (2016) 2751–2778 msp

Solvable Lie ﬂows of codimension 3

NAOKI KATO

In Appendix E of Riemannian foliations [Progress in Mathematics 73, Birkhäuser, Boston (1988)], É Ghys proved that any Lie g–ﬂow is homogeneous if g is a nilpotent Lie algebra. In the case where g is solvable, we expect any Lie g–ﬂow to be homogeneous. In this paper, we study this problem in the case where g is a 3–dimensional solvable Lie algebra.

57R30; 53C12, 22E25

1 Introduction

Throughout this paper, we suppose that all manifolds are connected, smooth and orientable and all foliations are smooth and transversely orientable. In this paper, flows mean orientable 1–dimensional foliations. Lie foliations were first defined by E Fedida[4]. A classical example of a Lie foliation is a homogeneous one. Through the results of several authors, it is recognized that the class of homogeneous Lie foliations is a large class in the class of Lie foliations, though of course these classes do not coincide. Therefore deciding which Lie foliations belong to the class of homogeneous Lie foliations is an important problem in Lie foliation theory. P Caron and Y Carrière[2] proved that any Lie Rq–flow without closed orbits is diffeomorphic to a linear flow on the .q 1/–dimensional torus, which is homoge- C neous. Carrière[3] proved that any Lie a.2/–flow is homogeneous. S Matsumoto and N Tsuchiya[13] proved that any Lie a.2/–foliation of a 4–dimensional manifold or its double covering is homogeneous. In the case where g is semisimple, M Llabrés and A Reventós constructed an example of Lie sl2.R/–flow which is not homogeneous[12, Example 5.3]. In the case where g is nilpotent, É Ghys[7] proved that any Lie g–flow is homogeneous. In the case where g is solvable, we conjecture that any Lie g–flow is homogeneous. In this paper, we study this problem in the case where g is a 3–dimensional solvable Lie algebra.

Published: 7 November 2016 DOI: 10.2140/agt.2016.16.2751 2752 Naoki Kato

If a Lie g–ﬂow F on M has a closed orbit, then any orbit is closed and M is an oriented S 1–bundle. In this case, the base space is diffeomorphic to a homogeneous space G n and hence g is unimodular. The total space M is, in general, not diffeomorphic to a homogeneous space. However, in the case where g is of type (R) or g is 3–dimensional, we can prove that the total space is a homogeneous space. More precisely, we obtain the following theorem.

Theorem A Let g be a solvable Lie algebra and F be a Lie g–ﬂow on a closed manifold M . Suppose that F has a closed orbit.

(i) If g is of type (R) and unimodular, then F is diffeomorphic to the ﬂow in Example 3.1. 0 (ii) If the dimension of g is three and g is isomorphic to g3 , then F is diffeomorphic to the ﬂow in Example 3.1.

In particular, if g is a 3–dimensional solvable Lie algebra and F has a closed orbit, then F is diffeomorphic to the ﬂow in Example 3.1.

In the case where F has no closed orbits, we obtain the following theorem.

Theorem B Let g be a 3–dimensional solvable Lie algebra and F be a Lie g–ﬂow on a closed manifold. Suppose that F has no closed orbits.

(i) If g is isomorphic to either R3 or n.3/, then F is diffeomorphic to the flow in Example 3.1. (ii) If g is isomorphic to a.3/, then F is isomorphic to the flow in Example 3.3. k (iii) If g is isomorphic to g2 , then F is isomorphic to the flow in Example 3.4. (iv) If g is isomorphic to gh and h 0, then F is isomorphic to the flow in 3 6D Example 3.5. 0 (v) If g is isomorphic to g3 , then F is isomorphic to the flow in Example 3.6.

Since (see Llabrés and Reventós[12]) there does not exist a Lie g1–ﬂow on a closed manifold, we have the following corollary.

Corollary 1.1 For any 3–dimensional solvable Lie algebra g, any Lie g–ﬂow on a closed manifold is homogeneous.

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2753

The contents of this paper are the following: Section 2 is devoted to recalling some basic definitions and properties of Lie foliations and Lie algebras. In Section 2A, we recall some basic definitions of Lie algebras. In Section 2B, we recall the classification of 3–dimensional solvable Lie algebras. In Section 2C, we recall the definition and some properties of Lie foliations. In Section 3, we construct some important examples of Lie g–flows, which are models of codimension-3 solvable Lie g–flows. In Section 4, we prove Theorem A. In Section 5, we construct a diffeomorphism between Lie flows without closed orbits according to the construction of Ghys[7]. In Section 6, by using the diffeomorphism constructed in Section 5, we prove Theorem B.

2 Preliminaries

2A Solvable Lie groups and solvable Lie algebras

Let g be a q–dimensional real Lie algebra. The descending central series of g is deﬁned inductively by C 0g g and C k g Œg; C k 1g: D D Similarly the derived series of g is deﬁned inductively by D0g g and Dk g ŒDk 1g; Dk 1g: D D A Lie algebra g is nilpotent if there exists an integer k such that C k g 0 , and a D f g connected Lie group G is nilpotent if the Lie algebra of G is nilpotent. Similarly a Lie algebra g is solvable if there exists an integer k such that Dk g 0 , and a connected D f g Lie group G is solvable if the Lie algebra of G is solvable. Let H and G be Lie groups, and let ˆ H Aut.G/ be a homomorphism. Then W ! we can construct a new Lie group H Ëˆ G , which is called the semidirect product of H and G with respect to ˆ, as follows. The semidirect product H Ëˆ G is the direct product of the sets H and G endowed with the group structure via

.h1; g1/ .h2; g2/ .h1 h2; g1 ˆ.h1/.g2//: D The Lie group H is naturally a subgroup of H Ëˆ G , and G is naturally a normal subgroup of H Ëˆ G . Let g be a Lie algebra and ad g gl.g/ be the adjoint representation of g. W ! Deﬁnition 2.1 A solvable Lie algebra g is said to be of type (R) if all the eigenvalues of ad.X / gl.g/ are real for any X g. A simply connected solvable Lie group is 2 2 said to be of type (R) if the Lie algebra of G is of type (R).

Algebraic & Geometric Topology, Volume 16 (2016) 2754 Naoki Kato

It is well known that simply connected solvable Lie groups of type (R) have similar properties of simply connected nilpotent Lie groups; see[9].

2B Classiﬁcation of 3–dimensional solvable Lie algebras

It is well known that 1–dimensional Lie algebras are isomorphic to R and that 2–dimensional Lie algebras are isomorphic to either R2 or a.2/, where t x ˇ a.2/ ˇ t; x R D 0 0 ˇ 2 is the Lie algebra of A.2/, which is the afﬁne transformation group of the real line.

Let V T; X; Y R be a 3–dimensional vector space and consider the following Lie D h i brackets on V : R3 (abelian): ŒT; X ŒT; Y ŒX; Y 0; D D D n.3/ (Heisenberg): ŒT; Y X and ŒT; X ŒX; Y 0; D D D a.3/ (afﬁne): ŒT; X X and ŒT; Y ŒX; Y 0; D D D g1 : ŒT; X X Y;ŒT; Y Y and ŒX; Y 0; D C D D gk : ŒT; X X;ŒT; Y kY and ŒX; Y 0 where k 0; 2 D D D 6D gh : ŒT; X Y;ŒT; Y X hY and ŒX; Y 0 where h2 < 4. 3 D D C D Then any 3–dimensional solvable Lie algebra is isomorphic to one of the above Lie algebras. It is well known that n.3/ is the Lie algebra of the 3–dimensional Heisenberg group 801 t x19 < = N.3/ @0 1 yA : D : 0 0 1 ; We will need an explicit description of simply connected Lie groups corresponding to k h the Lie algebras a.3/, g2 and g3 . These Lie groups are given by 80et 0 x1 ˇ 9 < ˇ = A.3/ @ 0 1 yA ˇ t; x; y R ; D ˇ 2 : 0 0 1 ˇ ; 80et 0 x1 ˇ 9 k < ˇ = G @ 0 ekt yA ˇ t; x; y R ; 2 D ˇ 2 : 0 0 1 ˇ ; Gh 0 SO.2/ Ë R2; 3D D

Algebraic & Geometric Topology, Volume 16 (2016) E Solvable Lie ﬂows of codimension 3 2755 and 80c.t/ cos. t/ c.t/ sin t x1 ˇ 9 h 0 < C ˇ = G ¤ @ c.t/ sin t c.t/ cos. t/ yA ˇ t; x; y R ; 3 D ˇ 2 : 0 0 1 ˇ ; where SO.2/ Ë R2 is the universal covering of the group of rigid motions SO.2/ËR2 , c.t/ .2=˛/eˇt , ˇ tan h=˛, and ˛ p4 h2 ; see[5]. D D D D 0 2 2 Note that G is isomorphic to the semidirect product RË R , where R Aut.R / 3 W ! is given by cos t sin t .t/ : E D sin t cos t

h 0 Note also that the Lie group G36D has another description 80eˇt cos t eˇt sin t x1 ˇ 9 h 0 < ˇt ˇt ˇ = G 6D @e sin t e cos t yA ˇ t; x; y R ; 3 D ˇ 2 : 0 0 1 ˇ ; where ˇ tan h=˛ and ˛ p4 h2 . In this paper, we will use this description. D D D Lie algebras gk and gk0 are isomorphic if and only if k k or k 1=k and 2 2 D 0 D 0 the Lie algebras gh and gh0 are isomorphic if and only if h h or h h . The Lie 3 3 D 0 D 0 algebra gk is unimodular if and only if k 1. The Lie algebra gh is unimodular 2 D 3 if and only if h 1. The Lie algebra gh is not of type (R) for any h and the other D 3 3–dimensional solvable Lie algebras are of type (R).

2C Lie foliations

Let F be a codimension-q foliation of a closed manifold M and g be a q–dimensional real Lie algebra. A g–valued 1–form ! on M is said to be a Maurer–Cartan form 1 if ! satisﬁes the equation d! Œ!; ! 0 and nonsingular if !x TxM g is C 2 D W ! surjective for each x M . 2 Deﬁnition 2.2 A codimension-q foliation F is a Lie g–foliation if there exists a nonsingular g–valued Maurer–Cartan form ! such that Ker.!/ T F . D

Let F1 and F2 be foliations of M1 and M2 , respectively. A smooth map f M1 M2 W ! preserves foliations if f .L/ F2 for every leaf L F1 . We denote such a map 2 2 by f .M1; F1/ .M2; F2/. We call two foliations F1 of M1 and F2 of M2 W ! diffeomorphic if there exists a foliation preserving map f .M1; F1/ .M2; F2/ such W ! that f M1 M2 is a diffeomorphism. W !

Algebraic & Geometric Topology, Volume 16 (2016) 2756 Naoki Kato

In this paper, we call two Lie g–foliations F1 and F2 diffeomorphic if F1 is diffeomorphic to F2 as a foliation. Fedida[4] proved that Lie g–foliations have a special property. Theorem 2.3 [4] Let F be a codimension-q Lie g–foliation of a closed manifold M and G be the simply connected Lie group of g. Let p M M be the universal W ! covering of M . Fix a Maurer–Cartan form ! A1.M g/ of F . Then there exists a 2 I locally trivial fibration D M G and a homomorphism h 1.M / G such that W ! W ! (1) D.˛ x˜/ h.˛/ D.x˜/ for any ˛ 1.M / and any x˜ M, and D 2 2 (2) the lifted foliation F F coincides with the fibers of the fibration D. D The fibration D is called the developing map, the homomorphism h is called the holonomy homomorphism and the image of h is called the holonomy group of the Lie g–foliation F with respect to the Maurer–Cartan form ! . Conversely, if there exist D and h which satisfy condition (1) above, then the set of fibers of D defines a Lie g–foliation F of M such that the developing map is D and the holonomy homomorphism is h.

Example 2.4 Let G be a simply connected Lie group and G a simply connected Lie group with a uniform lattice . Suppose that there exists a short exact sequence D 0 K G 0 G 0: ! ! ! ! Then the map D0 deﬁnes a Lie g–foliation F0 of the homogeneous space G. n We call Lie g–foliations constructed as in Example 2.4 homogeneous Lie g–foliations. Deﬁnition 2.5 A Lie g–foliation F of a closed manifold M is homogeneous if F is diffeomorphic to a homogeneous Lie g–foliation.

Let D M G be the developing map and h 1.M / G be the holonomy homo- W ! W ! morphism of a Lie g–foliation F . Let h.1.M // be the holonomy group of F . D Since the developing map D is h–equivariant, the map D induces a fibration D M G; W ! n where is the closure of . This fibration D is called the basic fibration, the homogeneous space G the basic manifold, and the dimension of G the basic n n dimension of F . Let F be the foliation of M defined by the fibers of the fibration D. By the definition of D, we can see that any leaf F of F is saturated by F and the foliation F F is j a minimal foliation of F . Moreover the basic fibration D M G induces a W ! n diffeomorphism from the leaf space M=F to G . n

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2757

3 Models of Lie ﬂows

In this section, we construct some examples of homogeneous Lie ﬂows which are important examples in this paper.

Example 3.1 Let D 1 R G 0 G 1 ! ! ! ! be a central exact sequence of Lie groups and be a uniform lattice of G. Then the surjective homomorphism D0 G G defines a Lie g–flow F0 on G. W ! n The Lie g–flow construction in Example 3.1 is a special case of the construction of homogeneous Lie g–flows. In the case in which the dimension of g is three, by using the classification of 4–dimensional solvable Lie algebras (see[1]), we have more explicit descriptions of G and D0 .

Example 3.2 Let g be a unimodular 3–dimensional solvable Lie algebra and G be the simply connected Lie group with the Lie algebra g. Then any central extension D 1 R G 0 G 1 ! ! ! ! of G by R is given as follows:

(1) If g is abelian, then G is isomorphic to either R4 R R3 or R N.3/. If G 4 4 3 3 D is isomorphic to R , then D0 R R R R is given by the natural projection W D ! D0 .t; x; y; z/ .x; y; z/: W 7! 3 If G is isomorphic to R N.3/, then D0 R N.3/ R is given by W ! 01 t x1! 0s1 D0 s; @0 1 yA A @t A ; W 7! 0 0 1 y where A GL.3 R/. 2 I (2) If g is isomorphic to n.3/, then G is isomorphic to R N.3/ or to 80 1 9 1 t 1 t 2 x ˆ 2 ˇ > <ˆB0 1 t yC ˇ => N.4/ B C ˇt; x; y; z R : D B0 0 1 zC ˇ 2 ˆ@ A ˇ > :ˆ 0 0 0 1 ;>

Algebraic & Geometric Topology, Volume 16 (2016) 2758 Naoki Kato

If G is isomorphic to R N.3/, then D0 R N.3/ N.3/ is given by the natural W ! projection. If G is isomorphic to N.4/, then D0 N.4/ N.3/ is given by W ! 01 t 1 t 2 x1 2 01 at cz 1 .abt 2 cdz2/ .ad 1/tz y1 B0 1 t yC C 2 C C C B C @0 1 bt dz A ; @0 0 1 zA 7! C 0 0 1 0 0 0 1 where either a b 0 1 a b 1 0 or for k R: c d D 1 0 c d D k 1 2 (3) If g is isomorphic to g 1 , then G is isomorphic to R G 1 or the semidirect 2 2 product R Ëˆ N.3/ with respect to the homomorphism ˆ R Aut.N.3// deﬁned W ! by 01 t x1 01 est x 1 s ˆ.s/ @0 1 yA @0 1 e yA : W 7! 0 0 1 0 0 1

1 1 1 If G is isomorphic to R G , then D0 R G G is given by the natural 2 W 2 ! 2 projection. If G is isomorphic to R Ëˆ N.3/, then the homomorphism 1 D0 R Ë N.3/ G W ! 2 is given by 01 t x1! 0es 0 t 1 s D0 s; @0 1 yA @ 0 e yA : W 7! 0 0 1 0 0 1

(4) If g is isomorphic to g0 , then G is isomorphic to either R G0 or the semidirect 3 3 product R Ë‰ N.3/ with respect to the homomorphism ‰ R Aut.N.3// deﬁned W ! by 01 t x1 01 t cos s y sin s x ty sin2 s 1 .t 2 y2/ sin 2s1 C 4 @0 1 yA @0 1 t sin s y cos s A : 7! C 0 0 1 0 0 1

0 0 0 If G is isomorphic to G R, then D0 R G G is given by the natural projection. 3 W 3 ! 3 If G is isomorphic to R Ë‰ N.3/, then the homomorphism 0 2 D0 R Ë‰ N.3/ G R Ë R W ! 3 D is given by 01 t x1! t D s; 0 1 y s; : 0 @ A y W 0 0 1 7!

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2759

k h Define two classes of 4–dimensional solvable Lie groups G2 and G3 by 80et 0 0 x1 9 ˆ kt ˇ > k : 0 0 0 1 ; and 80eˇt cos t eˇt sin t 0 x1 9 ˆ ˇt ˇt ˇ > h : 0 0 0 1 ; where k R, 0 < h2 < 4, and d e 2ˇ R. We construct homogeneous Lie g–flows 2 D 2 on the homogeneous spaces Gk and Gh . n 2 n 3 0 Example 3.3 Let be a uniform lattice of G2 . Define a homomorphism 0et 0 0 x1 0et 0 x1 B 0 1 0 yC D0 G A.3/ by B C @ 0 1 yA : W ! @ 0 0 e t zA 7! 0 0 1 0 0 0 1

Then D0 defines a homogeneous Lie a.2/–flow on G. n k Example 3.4 Assume that the Lie group G2 has a uniform lattice . Define a k k homomorphism D0 G G by W 2 ! 2 0 t 1 e 0 0 x 0 t 1 kt e 0 x B 0 e 0 yC kt D0 B C @ 0 e yA : W @ 0 0 e .1 k/t zA 7! C 0 0 1 0 0 0 1

k Then D0 defines a homogeneous Lie g –flow on G. 2 n h Example 3.5 We assume that G3 has a uniform lattice . Define a homomorphism h h D0 G G by W 3 ! 3 0 ˇt ˇt 1 e cos t e sin t 0 x 0 ˇt ˇt 1 ˇt ˇt e cos t e sin t x Be sin t e cos t 0 yC ˇt ˇt D0 B C @e sin t e cos t yA : W @ 0 0 d t zA 7! 0 0 1 0 0 0 1

h 0 h Then D0 deﬁnes a homogeneous Lie g 6D –ﬂow on G . 3 n 3

Algebraic & Geometric Topology, Volume 16 (2016) 2760 Naoki Kato

0 0 0 Example 3.6 Let be a uniform lattice of G3 R. Let D0 G3 R G3 be the W 0 ! 0 natural homomorphism. Then D0 defines a homogeneous Lie g –flow on G . 3 n 3 Remark We can extend the definition of Gh to the case when h 0. Then G0 3 D 3 coincides with SO.2/ Ë R2 R, which is not simply connected. The homomorphism 0 0 D0 G3 R G3 defined in Example 3.6 coincides with the lifted homomorphism W !2 2 D0 SO.2/ Ë R R SO.2/ Ë R defined in Example 3.5. W ! 4 Proof of Theorem A

Let F be a Lie g–ﬂow on a closed manifold M . Assume that F has a closed orbit. Then any orbit of F is closed.

Let D M G be the developing map and h 1.M / G be the holonomy homo- W ! W ! morphism. Since any orbit of F is closed, the holonomy group is discrete in G and the basic ﬁbration D M G is an oriented S 1–bundle over the homogeneous W ! n space G . n Let g be the dual of g, which is naturally identiﬁed with the set of left-invariant 1–forms on G . Consider the inclusion map V g A . G/ W ! n and the induced map H .g/ H . G/; W ! dR n where H .g/ is the cohomology of the Lie algebra g, A . G/ is the de Rham n complex of G , and H . G/ is the de Rham cohomology of G . We call n dR n n a k –form ! Ak . G/ algebraic if ! is in .Ak .g//. 2 n Let e.D/ H 2 . G/ be the real Euler class of the S 1–bundle D. We use the 2 dR n following lemma, which is a special case of[12, Theorem 5.1].

Lemma 4.1 If e.D/ is represented by an algebraic 2–form, then F is homogeneous.

Suppose the Euler class e.D/ is represented by an algebraic 2–form .ˇ/ A2. G/. 2 n Then there exists a homogeneous Lie g–ﬂow . G; F0/ which is diffeomorphic n to .M; F/. By the proof of[12, Theorem 5.1], the Lie algebra g of G coincides with z the central extension 0 R g g 0 ! ! z ! ! of g by R with the Euler class 2Œˇ H 2.g/. Hence G is a central extension of G 2 by R. Therefore we have the following proposition.

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2761

Proposition 4.2 If e.D/ is represented by an algebraic 2–form, then F is diffeomorphic to the Lie g–ﬂow in Example 3.1.

Proof of Theorem A First, we assume that g is a solvable Lie algebra of type (R). Let e.D/ H 2 . G/ be the real Euler class of the oriented S 1–bundle. 2 dR n Since g is of type (R), by Hattori[9, Theorem 4.1], the homomorphism

 H .g/ H . G/ W ! dR n is an isomorphism. Therefore e.D/ is represented by an algebraic 2–form. Hence, by Proposition 4.2, F is diffeomorphic to the Lie g–ﬂow in Example 3.1.

0 Next, we assume that g is isomorphic to g3 . By the classification of uniform lattices of G0 , the homogeneous space G0 is isomorphic to the mapping torus 3 n 3 T 3 T 2 R=.x; t 1/ .Ax; t/; A D C where A SL.2 Z/ such that Ap I for some p Z. If A I , then the mapping 3 2 I D 3 2 D torus TA is the three-dimensional torus T . 0 2 Define left-invariant 1–forms T , X and Y on G R Ë R by 3 D T dt; X cos t dx sin t dy and Y sin t dx cos t dy: D D C D C 2 0 0 Then the second cohomology H .g3/ of the Lie algebra g3 is generated by the cohomology class ŒX Y Œdx dy. ^ D ^ On the other hand, we have RŒdt dx RŒdt dy RŒdx dy if A I, H 2 .T 3/ ^ ˚ ^ ˚ ^ D dR A D RŒdx dy if A I. ^ 6D 2 3 2 0 Therefore if A I , then HdR.TA / is isomorphic to H .g3/ and the Euler class e.D/ 6D 0 is represented by an algebraic 2–form. Hence F is diffeomorphic to the Lie g3–flow in Example 3.1.

In the case where A I , by the following lemma, there exists a diffeomorphism 3 3 D f T T such that the pullback f e.D/ is represented by an algebraic 2–form. W ! 1 1 Then the S –bundle M is diffeomorphic to the S –bundle f M , which is diffeomorphic to the Lie g–ﬂow in Example 3.1.

Lemma 4.3 For any Œ! aŒdt dx bŒdt dy cŒdx dy H 2.T 3 Z/, there D ^ C ^ C ^ 2 I exists an integer matrix A SL.3 Z/ Diff.T 3/ such that A Œ! ZŒdx dy. 2 I 2 ^

Algebraic & Geometric Topology, Volume 16 (2016) 2762 Naoki Kato

Proof By the Smith normal form, we can show that there exist an integer d and an invertible 3 3 integer matrix B such that 0a1 0d1 B @bA @0A : D c 0 Therefore there exists A SL.3 Z/ and an integer n such that A Œ! nŒdx dy 2 I D ^ for some n Z. 2 5 A construction of a diffeomorphism of ﬂows

Let F1 and F2 be Lie g–ﬂows on closed manifolds M1 and M2 and let 1 and 2 be the holonomy groups of F1 and F2 , respectively. Suppose that F1 and F2 have no closed orbits and 1 is conjugate to 2 in G .

By replacing the developing map D1 M1 G and the holonomy homomorphism W ! h1 1.M1/ G of F1 by W ! 1 g D1 M1 G and g h1 g 1.M1/ G W ! W ! for some g G , we may assume that F1 and F2 have the same holonomy group . The 2 aim of this section is to construct a diffeomorphism between .M1; F1/ and .M2; F2/ according to Ghys’s method; see[7; 6; 14]. By results of Haefliger[8, Section 3], a Lie g–foliation of a closed manifold M is a classifying space for .G; / if every leaf of F is contractible. Thus .M1; F1/ and .M2; F2/ are classifying spaces for .G; /. By the uniqueness of classifying spaces, there exists a homotopy equivalence f M1 M2 , which we may assume is smooth, such that W ! f F2 F1 . In general, this map f is not a diffeomorphism. However, by using the D averaging technique (see[7; 6]), we can modify f to a diffeomorphism from .M1; F1/ to .M2; F2/. t t Parametrize F1 and F2 by 1 and 2 , respectively. Then we can define a smooth function u M1 R R W ! by the equation f .t .x// u.x;t/.f .x//: 1 D 2 The function u satisfies the cocycle condition u.x; s t/ u.x; t/ u.t .x/; s/: C D C 1 By this equation and by the compactness of M , we obtain the following lemma.

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2763

Lemma 5.1 There exists a constant C > 0 such that ˇ ˇ ˇ @ ˇ ˇ u.x; t/ˇ < C ˇ @t ˇ for any x M and t R. 2 2

Let 1 D1 M1 Q1 G be the basic fibration of F1 . Fix a fiber F1 D W ! D nk k of 1 M1 Q1 . Let .Ui; gi/ i 1 be a local trivialization of 1 , where Ui i 1 W ! f g D 1 f g D is an open covering of Q1 and gi .Ui/ Ui F1 is a diffeomorphism as fiber W 1 ! bundles. Let V1;:::; V be a refinement of U1;:::; U such that the closure V i f k g f k g is contained in Ui for each i 1;:::; k . D Since f is transverse to the flow F2 , by replacing V1;:::; Vk with smaller ones if necessary, for each i there exists a codimension-one open ball Di F1 such that 1 (1) Ni g .Vi Di/ is transverse to F1 , WD i (2) f .Ni/ is transverse to F2 , and

(3) the restriction f N Ni f .Ni/ is a diffeomorphism. j i W ! 1 Lemma 5.2 There exists T0 > 0 such that, for any i 1;:::; k and any x 1 .V i/, t D 2 the orbit O .x .0; T0// .x/ 0 < t < T0 intersects Ni . 1 I WD f 1 j g 1 Proof Define a function ri .Ui/ R by W 1 ! t ri.x/ inf t > 0 .x/ Ni : D f j 1 2 g Since F1 is minimal on each fiber, the function ri is well-defined and upper semi- 1 continuous. Since .V i/ is compact, for each i 1;:::; k , there exists an upper 1 2 f g bound Ti . Then we should take T0 max T1;:::; T 1. D f k g C

For any x Ni , we deﬁne si.x/ Z to be the number of times that the orbit 2 2 O f .x/ . 2C T0 ı; 2C T0 ı/ intersects f .Ni/, where ı is a small positive 2 I C number and C and T0 are the constants in Lemmas 5.1 and 5.2, respectively. By the choice of Ni , the function s Ni Z is bounded. W ! For any x Ni , we consider the set 2 ˚ ˇ t Ti.x/ t R ˇ .x/ Ni and u.x; t/ < 2C T ı : D 2 1 2 j j C t t 0 If t and t 0 satisfy u.x; t/ u.x; t 0/ and 1.x/ and 1 .x/ are in Ni , then we have t t D t t f . .x// f . 0 .x//. Since f N is a diffeomorphism, we have .x/ 0 .x/. 1 D 1 j i 1 D 1 Since F1 has no closed orbits, this implies that t t . Hence, if t and t are distinct D 0 0

Algebraic & Geometric Topology, Volume 16 (2016) 2764 Naoki Kato

points of Ti.x/, then u.x; t/ u.x; t /. Therefore the number of elements of Ti.x/ 6D 0 is less than si.x/ and hence bounded.

For an arbitrary point x Ni , we can take a sufﬁciently small connected neighbor- 2 hood Ax of x in Ni and a sufﬁciently large number tx > 0 such that tx is an upper bound of Ti.y/ for any y Ax . Then we have 2 u.y; t/ 2C T0 ı > 2C T0 j j C t if t > tx , y Ax and .y/ Ni . Since the basic manifold Q1 G is compact, 2 1 2 D n we can choose connected open subsets A ;:::; A of N and large numbers t such i1 iki i ij that

(1) 1.Ai / i 1;:::; k; j 1;:::; ki is a reﬁnement of V1;:::; V , and f j j D D g f k g (2) ti is an upper bound of Ti.y/ for any y Ai . j 2 j

Let t0 > max ti i 1;:::; k; j 1;:::; ki be an arbitrary number. Then, for f j j D D g any x Ai , we have 2 j u.x; t/ > 2C T0 j j t if t t0 and .x/ Ni . 1 2

Lemma 5.3 For any ij , one of the following holds:

(a) u.x; t/ > CT0 for any x Ai and any t t0 . 2 j (b) u.x; t/ < CT0 for any x Ai and any t t0 . 2 j 

Proof Let x Aij be an arbitrary point. Let t0 s0 < s1 < s2 < be the maximal 2 sl D sequence such that 1 .x/ Ni for l 1. By Lemma 5.2, we obtain sl 1 sl < T0 2 C for any l 0. On the other hand, we have u.x; s / > 2C T0 for any l 1. j l j Lemma 5.1 implies that

u.x; sl 1/ u.x; sl / < C.sl 1 sl / < CT0: j C j C Hence we have either u.x; s / > 2C T0 for any l 1 or u.x; s / < 2C T0 for l l any l 1. For any t t0 , there exists l 0 such that sl t sl 1 . By Lemma 5.1, we have Ä Ä C u.x; sl 1/ u.x; t/ C.sl 1 t/ < C.sl 1 sl / < CT0: j C j Ä C C Therefore we have either u.x; t/ > CT0 for any t t0 or u.x; t/ < CT0 for any t t0 . By the continuity of u, we have either u.x; t/ > CT0 for any x Ai and any t t0 2 j or u.x; t/ < CT0 for any x Ai and any t t0 . 2 j

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2765

1 Let Wi 1.Ai / and Ei .Wi /. j D j j D 1 j

Lemma 5.4 There exists 0 such that, for each Eij , one of the following holds:

(a) u.x; t/ > 0 for any x Ei and any t 0 . 2 j (b) u.x; t/ < 0 for any x Ei and any t 0 . 2 j Proof By Lemma 5.2, we have t Sij sup inf t > 0 1.x/ Aij < D x Ei f j 2 g 1 2 j for any ij . Let S max Si 1 i k; 1 j ki ; D f j j Ä Ä Ä Ä g ˛ max u.x; t/ x M; 0 t S ; C D f j 2 Ä Ä g ˛ max u.x; t/ x M; 0 t S ; D f j 2 Ä Ä g and ˛ max ˛ ; ˛ : D f C g Take an integer n and a constant 0 satisfying

nC T0 > ˛ and 0> n.t0 S/: C Fix ij . By Lemma 5.3, we have either u.y; t/ > CT0 for any x Ai and any t t0 2 j or u.x; t/ < CT0 for any x Ai and any t t0 . 2 j First, we suppose that u.x; t/ > CT0 for any x Ai and any t t0 . Fix an arbitrary 2 j point x Eij and any t 0 . Define a sequence 0 v1 < v2 < < vn inductively 2 Ä v1 as follows: Let v1 be the first arrival time of x to Ai . Thus we have .x/ Ai j 1 2 j and 0 v1 S . For l 1, let vl 1 be the first arrival time to Aij of x after the Ä Ä vl 1 C time vl t0 . Thus we have 1 C Aij and vl t0 vl 1 vl t0 S . C 2 C Ä C Ä C C Since v1 S and vl 1 vl t0 S , we have Ä C Ä C vn S .n 1/.t0 S/ < 0 t0 t t0: Ä C C Ä Since vl 1 vl t0 and t vn > t0 , we have C n 1 P u.x; t/ u x; v1 .vl 1 vl / t vn D C l 1 C C D n 1 P vl vn u.x; v1/ u.1 .x/; vl 1 vl / u.1 .x/; t vn/ D C l 1 C C D > ˛ .n 1/CT0 CT0 C C > ˛ nC T0 > 0: C

Algebraic & Geometric Topology, Volume 16 (2016) 2766 Naoki Kato

In the case where u.x; t/ < CT0 for any x Ai and any t t0 , by the same 2 j argument, we have

n 1 P u.x; t/ u x; v1 .vl 1 vl / t vn D C l 1 C C D n 1 P vl vn u.x; v1/ u.1 .x/; vl 1 vl / u.1 .x/; t vn/ D C l 1 C C D < ˛ .n 1/CT0 CT0 C < ˛ nC T0 < 0: Finally, we prove the following lemma.

Lemma 5.5 One of the following holds:

(a) u.x; t/ > 0 for any x M1 and any t 0 . 2 (b) u.x; t/ < 0 for any x M1 and any t 0 . 2 Proof By the continuity of u and the connectedness of M , if there exists ij such that u.x; t/ > 0 for any x Ei and any t 0 , then u.x; t/ > 0 for any x M and 2 j 2 any t 0 . Similarly, if there exists ij such that u.x; t/ < 0 for any x Ei and 2 j any t 0 , then u.x; t/ < 0 for any x M and any t 0 . 2 We construct a diffeomorphism from .M1; F1/ to .M2; F2/. Let T R be a positive 2 constant such that T 0 . Deﬁne T M1 R and fT M1 M2 by W ! W ! Z T 1 .x/ T .x/ u.x; / d and fT .x/ 2 .f .x//: D T 0 D By the equation Z T t 1 t T .1.x// u.1.x/; / d D T 0 1 Z T u.x; t / u.x; t/ d; D T 0 f C g we have 1 R T t T 0 u.x;t / d fT . .x// C .f .x//: 1 D 2 Therefore, for any x M1 , we have 2 d 1 Z T 1 u.x; t / d .u.x; t T / u.x; t// dt T 0 C D T C 1 u.t .x/; T /: D T 1

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2767

By Lemma 5.5, u.x; T / 0 for any x M1 . Therefore fT M1 M2 is a local 6D 2 W ! diffeomorphism. Since M1 is closed, fT is a covering map. Since fT is homotopic to f via t.x/ F M1 Œ0; 1 M2 deﬁned by F.x; t/ .f .x// W ! D 2 and f is a homotopy equivalence, the map fT is a diffeomorphism. Therefore we obtain the following theorem.

Theorem 5.6 [7] Let F1 and F2 be Lie g–ﬂows on closed manifolds M1 and M2 , respectively. Suppose F1 and F2 have no closed orbits and the holonomy group of F1 is conjugate to the holonomy group of F2 in G . Then F1 and F2 are diffeomorphic.

6 Proof of Theorem B

Let g be a 3–dimensional solvable Lie algebra and let F be a Lie g–ﬂow on a closed manifold M which has no closed orbits. Let be the holonomy group of F . Since F has no closed orbits, the holonomy homomorphism h 1.M / G is injective. W ! Hence the fundamental group 1.M / is isomorphic to the holonomy group . If the Lie algebra g is nilpotent, then the Lie algebra g is isomorphic to either R3 or n.3/. By the theorem of Ghys[7, Section 2], .M; F/ is diffeomorphic to a homogeneous Lie g–ﬂow . G; F0/, where G is a simply connected nilpotent Lie group. n Since any 1–dimensional ideal of a nilpotent Lie algebra is contained in its center, the kernel of the induced homomorphism

dD0 g g W z ! is contained in the center of g. Hence G is a central extension of G by R and F is z diffeomorphic to the Lie g–ﬂow in Example 3.1. We suppose that g is not nilpotent. First, we consider the case where g is isomorphic to a.3/.

6A a.3/ case

Let F be a Lie a.3/–ﬂow on a closed manifold M without closed orbits, and ﬁx a nonsingular a.3/–valued Maurer–Cartan form ! of F . The Lie algebra a.3/ has the explicit description 80t 0 x1 ˇ 9 < ˇ = ˇ a.3/ @0 0 yA ˇ t; x; y R : D : 0 0 0 ˇ 2 ;

Algebraic & Geometric Topology, Volume 16 (2016) 2768 Naoki Kato

Then there exist nonsingular 1–forms !T , !X and !Y on M such that 0 1 !T 0 !X ! @ 0 0 !Y A : D 0 0 0 Since ! is a Maurer–Cartan form, we have the following equations: 1 d!T 0 and d!X !T !X and d!Y 0: D D 2 ^ D Therefore !T and !Y are nonsingular closed 1–forms and ! ! ! T X 0 D 0 0 is a nonsingular a.2/–valued Maurer–Cartan form of M . The nonsingular closed 1–form !T and the nonsingular Maurer–Cartan form !0 define two foliations G and F0 of M whose codimensions are one and two, respectively. Since !0 is a a.2/–valued Maurer–Cartan form, the foliation F0 is a Lie a.2/–foliation. By an observation of Matsumoto and Tsuchiya[13, Section 7], we can see that the closed 1–form !T is a rational form. Therefore each leaf of G is compact and the leaf space M=G is diffeomorphic to S 1 . Let M S 1 M=G be the natural projection and fix a fiber N of . Since the W ! D tangent bundle T F coincides with Ker.!/ and Ker.!T / includes Ker.!/, each orbit of the Lie a.3/–flow F is tangent to the fibers of .

Let F N be the foliation deﬁned by the restriction of F to the ﬁber N . j

Lemma 6.1 The fiber N is diffeomorphic to the 3–dimensional torus and the flow F N j is diffeomorphic to a linear flow.

Proof The tangent bundle TN coincides with Ker.!T N /. By the equation j 1 d!X !T !X ; D 2 ^ the 1–form !X N on N is a nonsingular closed 1–form. Since T F coincides with j Ker.!/ Ker.!T / Ker.!X / Ker.!Y /; D \ \ T F N coincides with Ker.!X N / Ker.!Y N /. Since the 1–forms !X N and !Y N j j \ j j j are closed, the nonsingular R2–valued 1–form !X N N j D !Y N j is a Maurer–Cartan form. Hence the ﬂow F N is a Lie R2–ﬂow on N . By the j

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie flows of codimension 3 2769 theorem of Caron and Carrière[2, Theorem 1], the manifold N is diffeomorphic to T 3 and the flow F N is diffeomorphic to a linear flow. j By Lemma 6.1, the manifold M is a T 3–bundle over S 1 . Let F Diff .T 3/ be 3 1 2 C the monodromy map of the T –bundle M S . Fix generators ˛1 , ˛2 and ˛3 3 3 W ! 1 of 1.T / Z and an element ˇ of 1.M / such that .ˇ/ 1.S / is a generator. ' 3 3 2 Then the induced map F 1.T / 1.T / defines an integer matrix A SL.3 Z/ W ! 3 2 I and the fundamental group 1.M / is isomorphic to Z ËA Z .

Set A .aij /. Then we have D 1 a1j a2j a3j ˇ˛j ˇ ˛ ˛ ˛ for j 1; 2; 3: D 1 2 3 D Since F has no closed orbits, the holonomy homomorphism h 1.M / is an iso- W ! morphism. Let 0 be the abelian subgroup of generated by h.˛1/, h.˛2/ and h.˛3/. Since 1.N / is a normal subgroup of 1.M /, 0 is a normal subgroup of .

Lemma 6.2 Let H be an abelian subgroup of A.3/. Then H is contained in either

80 t 1 9 801 0 x1 ˇ 9 et 0 1 e x ˇ < ˇ = <ˆ 1 et0 0 ˇ => 2 R @0 1 yA ˇ x; y R or H.t ;x / B 0 1 y C ˇ t; y R D ˇ 2 0 0 D @ A ˇ 2 : 0 0 1 ˇ ; :ˆ 0 0 1 ˇ ;> for some x0 R and t0 0. 2 6D Proof Suppose that H is not contained in R2 . Then there exists

0 t0 1 e 0 x0 g0 @ 0 1 y0A H D 2 0 0 1 such that t0 0. Let 6D 0et 0 x1 g @ 0 1 yA H D 2 0 0 1 be an arbitrary element. Since H is abelian, we have g0g gg0 . Then we obtain the D equation 1 et x x0: D 1 et0 2 Lemma 6.3 The abelian subgroup 0 of is contained in R .

Algebraic & Geometric Topology, Volume 16 (2016) 2770 Naoki Kato

2 Proof Suppose that 0 is not contained in R . By Lemma 6.2, 0 is contained 2 in H for some x0 R and t0 0. Since R , there exists .t0;x0/ 2 6D 0 6 0 t 1 et 0 1 e x 1 et0 0 g B 0 1 y C D @ A 2 0 0 0 1 such that t 0. 6D Set 0 t 1 e 0 0 x0 h.ˇ/ @ 0 1 y A A.3/: D 0 2 0 0 1

Since 0 is normal in , we have h.ˇ/gh.ˇ/ 1 H : 2 0 .t0;x0/ Then we obtain the equation

t ˚ t0 t .1 e / .1 e /x .1 e 0 /x0 0: 0 D Since t 0, this equation implies that h.ˇ/ H . Thus is contained in H . 6D 2 .t0;x0/ .t0;x0/ However, this contradicts the fact that the holonomy group is uniform in A.3/. 2 Therefore 0 is contained in R .

Set

0 1 0 t0 1 1 0 xi e 0 x0 h.˛i/ @0 1 yiA for i 1; 2; 3 and h.ˇ/ @ 0 1 y0A : D D D 0 0 1 0 0 1

Since is uniform in A.3/, we have t0 0. Moreover, by conjugating in A.3/, we 6D may assume that x0 0. D

Lemma 6.4 A is conjugate to the matrix

0et0 0 01 t0 @ 0 e 0A : 0 0 1

Proof By the equation 1 a1j a2j a3j ˇ˛j ˇ ˛ ˛ ˛ ; D 1 2 3

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2771 we have 0 t0 1 0 1 1 0 e xj 1 0 a1j x1 a2j x2 a3j x3 C C @0 1 yj A @0 1 a1j y1 a2j y2 a3j y3A : D C C 0 0 1 0 0 1

Thus we have the equations 0 1 0 1 0 1 0 1 x1 x1 y1 y1 t t0 t A @x2A e @x2A and A @y2A @y2A : D D x3 x3 y3 y3

Since is uniform in A.3/, we can show that 0 1 0 1 x1 y1 @x2A 0 and @y2A 0: 6D 6D x3 y3

t0 t0 Therefore e , 1 and e are the eigenvalues of A.

1 Deﬁne elements ˛i and ˇ of G by y y 2

0 1 0 t0 1 1 0 0 xi e 0 0 0 B0 1 0 yiC B 0 1 0 y0C ˛i B C and ˇ B t0 C ; y D @0 0 1 zi A y D @ 0 0 e 0 A 0 0 0 1 0 0 0 1 where 0 1 z1 @z2A z3 is an eigenvector of A corresponding to the eigenvalue et0 . Let be the subgroup 1 of G generated by ˛1 , ˛2 , ˛3 and ˇ. Since 2 y y y y 0 1 0 1 0 1 x1 y1 z1 @x2A ; @y2A and @z2A x3 y3 z3

t0 t0 are eigenvectors of A SL.3 Z/ corresponding to the eigenvalues e , 1 and e , 2 I 1 respectively, the subgroup is discrete in G2 . Therefore is a uniform lattice 1 of G2 .

Algebraic & Geometric Topology, Volume 16 (2016) 2772 Naoki Kato

1 Deﬁne an submersion homomorphism D0 G A.3/ by W 2 ! 0et 0 0 x1 0et 0 x1 B 0 1 0 yC D0 B C @ 0 1 yA : W @ 0 0 e t zA 7! 0 0 1 0 0 0 1

1 Then D0 defines a Lie a.3/–flow F0 on G whose holonomy group coincides n 2 with . Therefore, by Theorem 5.6, the Lie a.3/–flow F is diffeomorphic to F0 . Hence F is diffeomorphic to the flow in Example 3.3.

k 6B g2 case k We consider the case where g is isomorphic g2 . In this case, the basic dimension of F is one; see[5; 11]. Hence the manifold M is diffeomorphic to a T 3–bundle over S 1 .

Let ˛1 , ˛2 , ˛3 and ˇ be the same as deﬁned in Section 6A. Then there exists an integer matrix A .aij / SL.3 Z/ such that the fundamental group 1.M / is isomorphic D3 2 I to Z ËA Z .

Let 0 be the normal abelian subgroup of generated by h.˛1/, h.˛2/ and h.˛3/. k Lemma 6.5 Let H be an abelian subgroup of G2 . Then H is contained in either

80 t 1 9 8 9 ˆ et 0 1 e x > 01 0 x1 ˆ 1 et0 0 ˇ > < = <ˆB C ˇ => 2 B kt 1 ekt C ˇ R @0 1 yA or H.t0;x0;y0/ 0 e y0 t R D D B 1 ekt0 C ˇ 2 : 0 0 1 ; ˆ@ A ˇ > :ˆ 0 0 1 ;> for some x0; y0 R and t0 0. 2 6D Proof Suppose that H is not contained in R2 . Then there exists

0 t0 1 e 0 x0 kt0 g0 @ 0 e y0A H D 2 0 0 1 such that t0 0. Let 6D 0et 0 x1 g @ 0 ekt yA H D 2 0 0 1 be an arbitrary element. Then gg0 g0g implies the equations D t0 t t0 t .1 e /x .1 e /x0 and .1 e /x .1 e /y0: D D Since t0 0, these equations imply that g H . 6D 2 .t0;x0;y0/

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2773

2 Lemma 6.6 The normal abelian subgroup 0 of is contained in R .

2 Proof Suppose that 0 is not contained in R . By Lemma 6.5, there exist x0; y0 R 2 2 and t0 0 such that is contained in H . Since R , there exists 6D 0 .t0;x0;y0/ 0 6 0et 0 x1 kt g @ 0 e yA 0 D 2 0 0 1 such that t 0. 6D Set 0 t 1 e 0 0 x0 h.ˇ/ @ 0 ekt 0 y A : D 0 0 0 1

Since 0 is a normal subgroup of , we have

h.ˇ/gh.ˇ/ 1 H : 2 0 .t0;x0;y0/ Thus we obtain the equations

t ˚ t t kt ˚ kt kt .1 e / .1 e /x0 .1 e 0 /x0 0 and .1 e / .1 e /x .1 e 0 /x0 0: 0 D 0 D

Since t 0, these equations imply that h.ˇ/ H.t0;x0;y0/ . This contradicts the fact 6D k 2 2 that is uniform in G2 . Therefore we have that 0 is contained in R .

Set 0 1 0 t0 1 1 0 xj e 0 x0 kt0 h.˛j / @0 1 yj A and h.ˇ/ @ 0 e y0A : D D 0 0 1 0 0 1

k k Since is uniform in G , we have t0 0. Moreover, by conjugating in G , we may 2 6D 2 assume that x0 0 and y0 0. D D By the same argument as the proof of Lemma 6.4, we can prove the following lemma.

Lemma 6.7 A SL.3 Z/ is conjugate to the matrix 2 I 0 1 et0 0 0 B 0 ekt0 0 C @ A : . 1 k/t0 0 0 e

Algebraic & Geometric Topology, Volume 16 (2016) 2774 Naoki Kato

k Define elements ˛i and ˇ of G by y y 2 0 1 0 t0 1 1 0 0 xi e 0 0 0 0 1 0 y 0 ekt0 0 0 ˛ B iC and ˇ B C ; i B C B .1 k/t0 C y D @0 0 1 zi A y D @ 0 0 e C 0A 0 0 0 1 0 0 0 1 where 0 1 z1 @z2A z3 is an eigenvector of A corresponding to the eigenvalue et0 . Let be the subgroup k k of G2 generated by ˛1 , ˛2 , ˛3 and ˇ. Then is a uniform lattice of G2 . Since y y y y k k coincides with via the homomorphism D G2 G2 defined in Example 3.4, the k Wk ! Lie g2–flow F is diffeomorphic to the Lie g2–flow in Example 3.4.

h 0 6D 6C g3 case

h 0 In the case in which g is isomorphic to g36D , the basic dimension of F is one; see[5; 11]. Hence the manifold M is diffeomorphic to a T 3–bundle over S 1 and 3 1.M / Z ËA Z for some A SL.3 Z/. D 2 I By the same argument as in Section 6B, we can prove the following lemma.

2 Lemma 6.8 The normal subgroup 0 of is contained in R .

By Lemma 6.8, we have

0 1 0 ˇt0 ˇt0 1 1 0 xi e cos t0 e sin t0 x0 ˇt0 ˇt0 h.˛i/ @0 1 yiA and @e sin t0 e cos t0 y0A : D 0 0 1 0 0 1

h h Since is uniform in G , we have t0 0. By conjugating in G , we may assume 3 6D 3 that x0 0 and y0 0. D D By the equation 1 a1j a2j a3j ˇ˛j ˇ ˛ ˛ ˛ ; D 1 2 3 we have

ˇt0 ˇt0 Ax e .cos t0x sin t0y/ and Ay e .sin t0x cos t0y/; D D C

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2775 where 0 1 0 1 x1 y1 x @x2A and y @y2A : D D x3 y3 By an easy calculation, we can show that x 0 and y 0. Hence x iy and x iy 6D .ˇ 6Di/t0 .ˇ Ci/t0 are eigenvectors corresponding to the eigenvalues e C and e , respectively.

Let d t0 e 2ˇt0 be the other eigenvalue of A, and let D 0 1 z1 z @z2A D z3 be an eigenvector of A corresponding to the eigenvalue d t0 . Let

0 1 0 ˇt0 ˇt0 1 1 0 0 xi e cos t0 e sin t0 0 0 ˇt0 ˇt0 B0 1 0 yiC Be sin t0 e cos t0 0 0C ˛i B C and ˇ B t C y D @0 0 1 zi A y D @ 0 0 d 0 0A 0 0 0 1 0 0 0 1

h h be elements of G3 and be the subgroup of G3 generated by ˛1 , ˛2 , ˛3 and ˇ. h y y y y Then the subgroup is a uniform lattice of G3 h h Since coincides with via the homomorphism D G3 G3 defined in Example 3.5, h h W ! the Lie g3–flow F is diffeomorphic to the Lie g3–flow in Example 3.5.

0 6D g3 case

0 Suppose that g is isomorphic to g3 . By[10, Corollaries 2.4 and 2.7] and the theorem of Caron and Carrière[2, Theorem 1], the manifold M is diffeomorphic to the 4– dimensional torus T 4 .

4 Fix generators ˛1 , ˛2 , ˛3 and ˛4 of 1.M / Z and set ' xi 0 Ë 2 h.˛i/ ti; G3 R R : D yi 2 D

Since is uniform in G0 , is not contained in 0 R2 . Hence we may assume that 3 f g t1 0. 6D

Lemma 6.9 ti 2Z, for each i . 2

Algebraic & Geometric Topology, Volume 16 (2016) 2776 Naoki Kato

Proof Suppose that there exists i such that ti 2Z. We may assume that t1 2Z. 62 62 Since is abelian, we can show that is contained in H , where ( ˇ ) 1 cos t sin t x0 ˇ H t; 1 ˇ t R D sin t 1 cos t y ˇ 2 10 0 is a simply connected 1–dimensional closed subgroup of G3 and 1 x 1 cos t1 sin t1 x1 10 : y D sin t1 1 cos t1 y1 10 Since is uniform in G0 , the homogeneous space H G0 is compact. 3 n 3 0 2 On the other hand, H G3 is homeomorphic to R , since H is a simply connected n 0 1–dimensional closed subgroup of G3 . This is a contradiction.

0 3 By Lemma 6.9, we have ti 2ni . Deﬁne a diffeomorphism F G R by D W 3 ! 0t 1 x F t; @xA : W y 7! y

3 Then F R is a homomorphism and F is F –equivariant, that is, j W ! j F. g/ F . / F.g/ D j for any and any g G0 . Therefore the rank of the matrix 2 2 3 0 1 2n1 2n2 2n3 2n4 @ x1 x2 x3 x4 A y1 y2 y3 y4 is three. We may assume that 0 1 0 1 0 1 2n1 2n2 2n3 @ x1 A ; @ x2 A ; @ x3 A y1 y2 y3

0 2 are linearly independent. Consider the subgroup of G R RË R R generated 3 D by ˛1 , ˛2 , ˛3 and ˛4 , where y y y y xi x4 ˛i 2ni; ; 0 for i 1; 2; 3 and ˛4 2n4; ; 1 : y D yi D y D y4

Algebraic & Geometric Topology, Volume 16 (2016) Solvable Lie ﬂows of codimension 3 2777

Then is a uniform lattice of G0 R and coincides with via the homomorphism 3 D G0 R G0 in Example 3.6. Therefore the Lie g0–ﬂow is diffeomorphic to the W 3 ! 3 3 ﬂow in Example 3.6.

References

[1] A Andrada, M L Barberis, I G Dotti, G P Ovando, Product structures on four dimensional solvable Lie algebras, Homology Homotopy Appl. 7 (2005) 9–37 MR [2] P Caron, Y Carrière, Flots transversalement de Lie Rn , flots transversalement de Lie minimaux, C. R. Acad. Sci. Paris Sér. A-B 291 (1980) A477–A478 MR [3] Y Carrière, Flots riemanniens, from “Transversal structure of foliations”, Astérisque 116, Société Mathématique de France, Paris (1984) 31–52 MR [4] E Fedida, Sur les feuilletages de Lie, C. R. Acad. Sci. Paris Sér. A-B 272 (1971) A999–A1001 MR [5] E Gallego, A Reventós, Lie flows of codimension 3, Trans. Amer. Math. Soc. 326 (1991) 529–541 MR [6] É Ghys, Flots d’Anosov sur les 3–variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984) 67–80 MR [7] É Ghys, Riemannian foliations: examples and problems (1988) MR Appendix E to Riemannian foliations, Progress in Mathematics 73, Birkhäuser, Boston (1988) [8] A Haefliger, Groupoïdes d’holonomie et classifiants, from “Transversal structure of foliations”, Astérisque 116, Société Mathématique de France, Paris (1984) 70–97 MR [9] A Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 289–331 MR [10] B Herrera, M Llabrés, A Reventós, Transverse structure of Lie foliations, J. Math. Soc. Japan 48 (1996) 769–795 MR [11] B Herrera, A Reventós, The transverse structure of Lie flows of codimension 3, J. Math. Kyoto Univ. 37 (1997) 455–476 MR [12] M Llabrés, A Reventós, Unimodular Lie foliations, Ann. Fac. Sci. Toulouse Math. 9 (1988) 243–255 MR [13] S Matsumoto, N Tsuchiya, The Lie affine foliations on 4–manifolds, Invent. Math. 109 (1992) 1–16 MR [14] H Tsunoda, Complete affine flows with nilpotent holonomy group, Hokkaido Math. J. 28 (1999) 515–533 MR 2778 Naoki Kato

Graduate School of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-9814, Japan [email protected]

Received: 11 February 2015 Revised: 13 November 2015

Geometry & Topology Publications, an imprint of mathematical sciences publishers msp