The Euler class of a foliation and Thurston's Euler class one conjecture #
The tangent field of a foliation F of rank k of a real manifold M is a real vector bundle of
rank k (TauCeti.IsContMDiffDistribution.vectorBundle). An orientation of it is a Thom class
u ∈ Hᵏ(TF, TF₀) (TauCeti.Foliation.IsThomClass), and its Euler class is the class
e(u) ∈ Hᵏ(M) of TauCeti.eulerClass (TauCeti.Foliation.eulerClass). A class e ∈ Hᵏ(M) is
an Euler class of F (TauCeti.Foliation.IsEulerClass) when it is e(u) for some orientation
u of the tangent field. Reversing the orientation negates the Euler class, and a tangent field
with a nowhere-vanishing continuous section has Euler class zero.
For a codimension-one foliation of a closed 3-manifold the tangent field is a plane field, with
an Euler class in H²(M; ℤ), and evaluating it on embedded surfaces bounds the topology of the
leaves. A class e ∈ H²(X; ℤ) satisfies the Euler class inequality
(TauCeti.EulerClassBounded) when |⟨e, f_*[S]⟩| ≤ -χ(S) for every closed connected oriented
surface S with χ(S) ≤ 0 and every embedding f : S → X; here ⟨e, f_*[S]⟩ is the evaluation
TauCeti.HomologicalOrientation.evalFundamentalClass of e on the image of the fundamental
class. The inequality is sharp for e (TauCeti.EulerClassNormOne) when moreover equality
holds on some such surface with χ(S) < 0. On a closed hyperbolic 3-manifold these two
conditions say that e has dual Thurston norm at most one, respectively exactly one: the Thurston
norm of a class is the least -χ of an embedded surface without sphere components representing
it, and by Gabai it is also the least -χ of a singular surface representing it, so merely
topological embeddings may be allowed. That equivalence is not formalized here; the two
predicates are the surface conditions themselves.
Thurston proved that the Euler class of a taut foliation of a closed hyperbolic 3-manifold
satisfies the Euler class inequality (TauCeti.ThurstonEulerClassInequality), and conjectured
conversely that every even integral class for which the inequality is sharp is the Euler class of
a taut foliation (TauCeti.EulerClassOneConjecture). Evenness is necessary: the Euler class of an
oriented plane field ξ on a closed orientable 3-manifold reduces mod 2 to
w₂(ξ) = w₂(M) = 0, so it is divisible by 2. Yazdi found even counterexamples
(TauCeti.YazdiEulerClassOneTheorem), refuting the conjecture
(TauCeti.YazdiEulerClassOneTheorem.not_eulerClassOneConjecture). These statements are
recorded, not proved; foliations in them are smooth, so the counterexample statement is the
weakest form of Yazdi's theorem.
Main definitions #
TauCeti.Foliation.thomPair: the Thom pair(TF, TF₀)of the tangent field of a foliation.TauCeti.Foliation.IsThomClass: an orientation of the tangent field, as a Thom class.TauCeti.Foliation.eulerClass: the Euler classHᵏ(TF, TF₀) ⟶ Hᵏ(M).TauCeti.Foliation.IsEulerClass: a class is the Euler class of some orientation of the tangent field.TauCeti.EulerClassBounded,TauCeti.EulerClassNormOne: the Euler class inequality on embedded surfaces, and its sharpness.TauCeti.ThurstonEulerClassInequality,TauCeti.EulerClassOneConjecture,TauCeti.YazdiEulerClassOneTheorem: Thurston's inequality, his Euler class one conjecture, and Yazdi's counterexample.
Main results #
TauCeti.Foliation.IsEulerClass.neg: the negative of an Euler class is an Euler class.TauCeti.Foliation.eulerClass_eq_zero_of_forall_ne_zero: a nowhere-vanishing continuous tangent vector field kills the Euler class.TauCeti.not_eulerClassNormOne_zero: the inequality is never sharp for the zero class.TauCeti.YazdiEulerClassOneTheorem.not_eulerClassOneConjecture: Yazdi's theorem refutes the Euler class one conjecture.
References #
- W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339 (the Thurston norm, the Euler class inequality and the Euler class one conjecture).
- D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), 445–503, Section 6 (the Thurston norm is computed by singular surfaces).
- M. Yazdi, On Thurston's Euler class-one conjecture, Acta Math. 225 (2020), 313–368.
The Thom pair (TF, TF₀) of the tangent field TF of a foliation: the total space of the
tangent field, as a subspace of the tangent bundle, relative to its nonzero vectors.
Instances For
An orientation of the tangent field of a foliation of rank k, recorded as a Thom class
u ∈ Hᵏ(TF, TF₀) (TauCeti.IsThomClass): a relative class restricting to a generator of the
cohomology of every fibre pair.
Instances For
The Euler class Hᵏ(TF, TF₀) ⟶ Hᵏ(M) of the tangent field of a foliation of rank k:
TauCeti.eulerClass, restriction to the total space followed by pullback along the zero
section, read in degree k.
Instances For
A class e ∈ Hᵏ(M) is an Euler class of the foliation F of rank k when it is the Euler
class of some orientation of the tangent field of F.
Instances For
The defining property of an orientation of the tangent field.
The Euler class of a foliation is that of its tangent field, read in degree k.
The defining property of an Euler class of a foliation.
The Euler class of an orientation of the tangent field is an Euler class of the foliation.
The negative of a Thom class of the tangent field is a Thom class: the reversed orientation.
Reversing the orientation of the tangent field negates its Euler class.
A nowhere-vanishing tangent vector field kills the Euler class. If the tangent field of
F has a section s that is continuous as a vector field on M and vanishes nowhere, then the
Euler class map of F is zero, so 0 is the only possible Euler class of F. The instance
IsManifold I 1 M, implied by the C^(n + 1) structure, gives the tangent bundle its
topology.
If the tangent field of F has a continuous nowhere-vanishing section, then every Euler
class of F is zero.
The Euler class inequality #
The Euler class inequality for an integral class e ∈ H²(X; ℤ): for every closed
connected oriented surface S of Euler characteristic χ(S) ≤ 0 and every embedding
f : S → X, the evaluation of e on f_*[S] satisfies |⟨e, f_*[S]⟩| ≤ -χ(S). Closed
surfaces have finite CW type, which makes χ(S) defined. On a closed hyperbolic 3-manifold
this says that e has dual Thurston norm at most one.
Instances For
The Euler class inequality is sharp for an integral class e ∈ H²(X; ℤ) when e
satisfies it and some closed connected oriented surface S with χ(S) < 0 embeds in X by an
f with |⟨e, f_*[S]⟩| = -χ(S). On a closed hyperbolic 3-manifold this says that e has
dual Thurston norm exactly one.
Instances For
A class satisfying the Euler class inequality obeys the bound |⟨e, f_*[S]⟩| ≤ -χ(S) on
every embedded closed connected oriented surface S with χ(S) ≤ 0.
The zero class satisfies the Euler class inequality.
A class satisfies the Euler class inequality exactly when its negative does.
Alias of the reverse direction of TauCeti.eulerClassBounded_neg_iff.
A class satisfies the Euler class inequality exactly when its negative does.
A class satisfying the Euler class inequality vanishes on every embedded closed connected oriented surface of Euler characteristic zero, such as an embedded torus.
A class satisfying the Euler class inequality is sharp for it once some embedded closed
connected oriented surface S with χ(S) < 0 has |⟨e, f_*[S]⟩| = -χ(S).
A class for which the Euler class inequality is sharp satisfies it.
A class for which the Euler class inequality is sharp has a sharpness witness: an embedded
closed connected oriented surface S with χ(S) < 0 and |⟨e, f_*[S]⟩| = -χ(S).
The Euler class inequality is sharp for a class exactly when it is sharp for its negative.
Alias of the reverse direction of TauCeti.eulerClassNormOne_neg_iff.
The Euler class inequality is sharp for a class exactly when it is sharp for its negative.
The Euler class inequality is never sharp for the zero class.
Thurston's Euler class one conjecture #
Thurston's Euler class inequality: on a closed orientable hyperbolic 3-manifold M,
that is a compact connected orientable smooth 3-manifold without boundary admitting a complete
Riemannian metric of constant curvature -1, every Euler class of a smooth taut codimension-one
foliation satisfies the Euler class inequality.
Instances For
Thurston's Euler class one conjecture: on a closed orientable hyperbolic 3-manifold M,
every even integral class e ∈ H²(M; ℤ) for which the Euler class inequality is sharp is an
Euler class of a smooth taut codimension-one foliation of M. Evenness is necessary, since the
Euler class of an oriented plane field on M reduces mod 2 to w₂(M) = 0.
The conjecture is false, by Yazdi
(TauCeti.YazdiEulerClassOneTheorem.not_eulerClassOneConjecture).
Instances For
Under Thurston's Euler class inequality, every Euler class of a smooth taut codimension-one
foliation of a closed orientable hyperbolic 3-manifold satisfies the Euler class
inequality.
Under the Euler class one conjecture, every even integral class of a closed orientable
hyperbolic 3-manifold for which the Euler class inequality is sharp is an Euler class of a
smooth taut codimension-one foliation.
Yazdi's theorem: some closed orientable hyperbolic 3-manifold M carries an even
integral class e ∈ H²(M; ℤ) for which the Euler class inequality is sharp, but which is not an
Euler class of any smooth taut codimension-one foliation of M.
Instances For
Yazdi's theorem refutes the Euler class one conjecture.