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Suspension Foliations: Interesting Examples of Topology and Geometry

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Suspension Foliations: Interesting Examples of Topology and Geometry SUSPENSION FOLIATIONS: INTERESTING EXAMPLES OF TOPOLOGY AND GEOMETRY KEN RICHARDSON Abstract. We give examples of foliations on suspensions and comment on their topological and geometric properties 1. Idea of foliation by suspension Here is the simplest example of foliation by suspension. Let X be a manifold of dimension 1 q, and let f : X ! X be a bijection. Then we define the suspension M = S ×f X as the quotient of [0; 1] × X by the equivalence relation (1; x) ∼ (0; f (x)). 1 M = S ×f X = [0; 1] × X ∼ Then automatically M carries two foliations: F2 consisting of sets of the form F2;t = f(t; x)∼ : x 2 Xg and F1 consisting of sets of the form F2;x0 = f(t; x): t 2 [0; 1] ; x 2 Ox0 g, where the orbit Ox0 is defined as −2 −1 2 Ox0 = :::; f (x0) ; f (x0) ; x0; f (x0) ; f (x0) ; ::: ; where the exponent refers to the number of times the function f is composed with itself. −2 Note that Ox0 = Of(x0) = Of (x0), etc., so the same is true for F1;x0 . Understanding the foliation F1 is equivalent to understanding the dynamics of the map f. If the manifold X is already foliated, you can use the construction to increase the codimension of the foliation, as long as f maps leaves to leaves. The first set of examples concerns foliations of a map from the circle to itself. Example A: Let X = S1, let α be a fixed real number, and let f : S1 ! S1 be defined iα 1 1 by f (z) = e z. The S ×f S is topologically the 2-torus. It is a cylinder with the two ends identified with a twist. Note that if α is a rational multiple of 2π, then all of the leaves are closed. If α is irrational, then all of the leaves are dense. This is called a Kronecker foliation. Note that all leaves have no holonomy. 1 1 1 1 1 Example B: Let X = S , let f : S ! S be defined by f (z) = z. The S ×f S is topologically the Klein bottle. It is a cylinder with the two ends identified with a reflection. Observe that all leaves are closed | two of them have z2 holonomy, and the others have trivial holonomy. The next example is a codimension-2 foliation on a 3-manifold. Example C: (This one is from [8] and [9].) Consider the one-dimensional foliation ob- tained by suspending an irrational rotation on the standard unit sphere S2. On S2 we use the cylindrical coordinates (z; θ), related to the standard rectangular coordinates by x0 = p(1 − z2) cos θ, y0 = p(1 − z2) sin θ, z0 = z. Let α be an irrational multiple Date: August, 2009. 1991 Mathematics Subject Classification. 53C12, 58G11, 58G18, 58G25. 1 2 KEN RICHARDSON of 2π, and let the three-manifold M = [0; 1] × S2= ∼, where (1; z; θ) ∼ (0; z; θ + α). Here the function f : S2 ! S2 is f (z; θ) = (z; θ + α) : ∼ 2 Endow M with the product metric on Tz,θ;tM = Tz,θS × TtR. Let the foliation F = F1 be defined by the immersed submanifolds Lz,θ = [n2Z [0; 1] × fzg × fθ + αg (not unique in θ). The leaf closures Lz for jzj < 1 are two-dimensional, and the closures corresponding to the poles (z = ±1) are one-dimensional. The basic functions are functions of z alone. Example D: This foliation is a suspension of an irrational rotation of S1 composed with an irrational rotation of S2 on the manifold S1 × S2. As in Example ??, on S2 we use the cylindrical coordinates (z; θ), related to the standard rectangu- lar coordinates by x0 = p(1 − z2) cos θ, y0 = p(1 − z2) sin θ, z0 = z. Let α be an irrational multiple of 2π, and let β be any irrational number. We con- sider the four-manifold M = [0; 1] × S2 × [0; 1] = ∼, where (0; z; θ; t) ∼ (1; z; θ; t), (1; z; θ; s) ∼ (0; z; θ + α; s + β mod 1). Let the foliation F = F1 be defined by the immersed submanifolds Lz,θ;s = [n2Z [0; 1] × fzg × fθ + αg × fs + βg (not unique in θ or s). The leaf closures Lz for jzj < 1 are three-dimensional, and the closures corresponding to the poles (z = ±1) are two-dimensional. The following two examples are related to an example in [2]. The first is a codimension two foliation that does not admit a Riemannian foliation structure, and the second is a codimension two Riemannian foliation that is not taut. Example E: Consider the flat torus T 2 = R2Z2. Consider the map F : T 2 ! T 2 defined by x 2 1 x F = mod 1 y 1 1 y 2 Let M = [0; 1] × T ∼, where (1; a) ∼ (0;F (a)). Let F1 be the foliation whose leaves are of the form La = f(t; p)∼ 2 M : t 2 [0; 1] ; p 2 Oag. This is an example of an Anosov foliation. Example F: Consider the flat torus T 2 = R2Z2. Consider the map F : T 2 ! T 2 defined by x 2 1 x F = mod 1 y 1 1 y 2 0 Let M = [0; 1] × T ∼, where (1; a) ∼ (0;F (a)). Let v, v bep orthonormalp eigenvec- 3+ 5 3− 5 tors of the matrix above, corresponding to the eigenvalues 2 , 2 , respectively. Let the linear foliation F be defined by the vector v0 on each copy of T 2. 2. More general suspensions The most general type of suspension is as follows. Let Y be a manifold with fundamental group π1 (Y ) and universal cover Ye, let X be another manifold, and let φ : π1 (Y ) ! Maps (X), where by Maps we mean some group of bijective maps from X to itself, such as continous maps, smooth maps, analytic maps, isometries, etc. Then we define the suspension M = Y ×φ X by M = Y ×φ X = Ye × Xπ1 (Y ) ; SUSPENSION FOLIATIONS 3 −1 −1 where g 2 π1 (Y ) acts on Ye × X by g (y; x) = (y · g ; φ (g)(x)), where y is mapped to y · g e e−1 e e by the deck transformation corresponding to g 2 π1 (Y ). Note that this quotient is the same as the quotient by the equivalence relation (ye · g; x) ∼ (y;e φ (g)(x)). The foliations of the suspension M come from choosing the immersed Y -parameter submanifolds or the immersed X-parameter submanifolds, or other submanifolds of these submanifolds. The standard foliation to choose is the foliation F1 of Y -parameter submanifolds, that is sets of n 0 0 o the form Lx = [(y;e x )] : ye 2 Ye ; x 2 Ox , where Ox = fφ (g)(x): g 2 π1 (Y )g ⊂ X. Note that this generalizes the previous section, where in that section Y = S1 = [0; 1] (0 = 1), n π1 (Y ) = Z, and the homomorphism is φ (n)(x) = f (x). Remark 2.1. If you have any discrete, finitely-generated group G of bijective maps on X, there always exists a closed manifold Y and a homomorphism φ : π1 (Y ) ! G. This follows from the fact that every such group can be realized as the fundamental group of a closed 4- manifold, where φ can be then taken to be the identity. In general one may usually take Y to be simpler. For example if fg1; g2; g3g generates G, then one could take Y to be the connected sum of three copies of S2 × S1, which has fundamental group the free product Z ∗ Z ∗ Z, and nj the homomorphism could be generated by φ (nj) = gj in the group, for j = 1; 2; 3. Remark 2.2. The choice of group of maps determines the transverse type of foliation F1. If the homomorphism φ maps to isometries of X, then F1 is a Riemannian foliation. If φ maps to morphisms of a K¨ahlermanifold X, then F1 is a transversely K¨ahlerfoliation. We now give two examples of these more general suspensions. The following example is a codimension two transversally oriented Riemannian foliation in which all the leaf closures have codimension one, and the leaf closure foliation is not transversally orientable. There are two leaf closures with Z2 holonomy. Example G: This foliation is the suspension of an irrational rotation of the flat torus and a Z2-action. Let X be any closed Riemannian manifold such that π1(X) = Z ∗ Z | the free group on two generators fα; βg. We normalize the volume of X to be 1 1 1. Let Xe be the universal cover. We define M = Xe × S × S π1(X), where π1(X) acts by deckp transformations on Xe and by α (θ; φ) = (2π − θ; 2π − φ) and β (θ; φ) = θ; φ + 2π on S1 × S1. We use the standard product-type metric. The n o leaves of F are defined to be sets of the form (x; θ; φ)∼ j x 2 Xe . Note that the foliation is transversally oriented. The leaf closures are sets of the form n o [ n o Lθ = (x; θ; φ)∼ j x 2 X;e φ 2 [0; 2π] (x; 2π − θ; φ)∼ j x 2 X;e φ 2 [0; 2π] : The next example is a codimension two Riemannian foliation with dense leaves, such that some leaves have holonomy but most do not. Example H: This Riemannian foliation is a suspension of a pair of rotations of the 2 sphere S .
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