Godbillon-Vey class and secondary characteristic classes of foliations

Intuition Beginner

Picture a thick stack of sheets of paper filling a room — not flat sheets, but sheets that bend and twist so that they fit together with no gaps and no crossings. Each sheet is a leaf, and the whole stack is a foliation of the room. A foliation is just a way of slicing a space into smooth layers, the way the pages of a book slice the book, or the way surfaces of constant pressure slice the atmosphere.

Now imagine you walk along one leaf while the neighbouring leaves slowly rotate relative to you, the way the pages of a fanned-out book splay apart. Some stacks are calm: the leaves stay lined up, like a fresh ream of paper. Other stacks are restless: the leaves keep shearing and turning as you move, never settling. We want a single number that measures how much built-in twisting a stack carries that cannot be combed away by re-laying the sheets.

The Godbillon-Vey number is exactly such a measure. It is a quantity you compute from the way the leaves turn, and its remarkable feature is that no matter how you smooth or re-describe the layering, the number does not change. It is a fingerprint of the twisting, baked into the layered space itself.

Visual Beginner

Alt text: Two stacks of curved sheets fill side-by-side boxes. In the left box the sheets are nested parallel layers that never rotate, a calm foliation whose twisting measure is zero. In the right box the sheets fan and shear so that a small frame carried across the stack steadily rotates, a restless foliation. A curved arrow marks the rate at which neighbouring leaves turn relative to one another. The picture shows that the Godbillon-Vey number reads off the accumulated turning of the leaves, the part of the layering that cannot be undone by relabelling the sheets.

Worked example Beginner

Take a solid doughnut — a bagel-shaped region. There is a famous way to fill it with leaves called the Reeb foliation. Near the central core, one leaf wraps around as a tube. Moving outward, the leaves are bowl-shaped surfaces that spiral and flatten, pressing closer and closer to the boundary torus, which is itself one big leaf forming the skin of the doughnut.

The leaves near the boundary spiral in a way that keeps turning, and that turning does not cancel out as you sweep around the doughnut. When you glue two such solid doughnuts together to make a 3-sphere, the turning of one of the pieces can be dialled up. The Godbillon-Vey number of the result comes out nonzero, and you can make it as large as you like by spinning the leaves faster near the gluing surface.

What this shows: a layered 3-sphere is not always the calm flat ream. There are layerings whose accumulated leaf-twisting is genuinely there, recorded by a number you cannot reduce to zero by any honest re-slicing. This single example is the seed of the whole subject.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $M$ is a smooth manifold and $\mathcal{F}$ is a transversely orientable codimension-1 foliation of $M$: a decomposition of $M$ into immersed hypersurface leaves, locally the level sets of a submersion. Such an $\mathcal{F}$ is given by a nowhere-vanishing defining 1-form $\omega \in {\Omega }^1(M)$ whose kernel at each point is the tangent space to the leaf; the leaves are the integral hypersurfaces of the distribution $\ker \omega$. By the Frobenius theorem 03.02.04, the distribution $\ker \omega$ is integrable — that is, admits leaves — if and only if $$ \omega \wedge d\omega = 0 . $$ This is the integrability condition in its 1-form guise, equivalent to $d\omega$ lying in the ideal generated by $\omega$.

The auxiliary form $\eta$. The condition $\omega \wedge d\omega =0$ says precisely that $d\omega$ is a multiple of $\omega$ at the level of forms: there exists a 1-form $\eta \in {\Omega }^1(M)$ with $$ d\omega = \eta \wedge \omega . $$ Such an $\eta$ exists because $\omega$ is nowhere zero, so $\omega$ generates a line subbundle of the cotangent bundle and any 2-form annihilated by wedging with $\omega$ factors through $\omega$. The form $\eta$ is not unique: it may be altered by adding any multiple of $\omega$, since $(\eta +g\omega )\wedge \omega =\eta \wedge \omega$ for any function $g$.

Definition (Godbillon-Vey 3-form). Given the data $(\omega ,\eta )$ above, the Godbillon-Vey 3-form is $$ \mathrm{gv} ;=; \eta \wedge d\eta ;\in; \Omega^3(M). $$

Definition (Godbillon-Vey class). The de Rham class $$ \mathrm{GV}(\mathcal F) ;=; [,\eta \wedge d\eta,] ;\in; H^3(M;\mathbb R) $$ is the Godbillon-Vey class of the transversely orientable codimension-1 foliation $\mathcal{F}$. The Key theorem below shows that $\eta \wedge d\eta$ is closed and that its class is independent of the choices of $\omega$ and $\eta$, so $\mathrm{GV}(\mathcal{F})$ depends only on $\mathcal{F}$.

This is a secondary, or exotic, characteristic class. The primary Chern-Weil classes 03.06.06 of the normal line bundle $\nu =TM/T\mathcal{F}$ of a codimension-1 foliation vanish in the relevant range — this is the simplest instance of Bott vanishing — and the Godbillon-Vey class lives precisely in the room that the vanishing of the primary classes opens up. It is built by a transgression, in the same spirit as the Chern-Simons form 03.06.07: the secondary class is the transgression of a primary class that has been forced to vanish.

Counterexamples to common slips

• $\mathrm{gv}$ is not the volume of anything. Although $\eta \wedge d\eta$ is a 3-form one integrates over a closed 3-manifold, it is not the Riemannian volume nor a curvature integrand; it is built from the leaf turning recorded in $\eta$, and on a product foliation $\eta$ can be taken to be $0$, giving $\mathrm{gv}=0$ even when the volume is positive.
• The class needs codimension 1 transverse orientability. Without transverse orientability there is no global defining 1-form $\omega$; the construction must then be carried out on a double cover or recast through the normal bundle, and the naive $\eta \wedge d\eta$ recipe does not directly apply.
• Nonzero $\mathrm{GV}$ does not force a closed leaf or any leaf topology. The invariant measures transverse turning, not the topology of an individual leaf; the Roussarie example has nonzero $\mathrm{GV}$ while its leaves are ordinary planes and cylinders.

Key theorem with proof Intermediate+

Theorem (Godbillon-Vey). Let $\mathcal{F}$ be a transversely orientable codimension-1 foliation of $M$, defined by a 1-form $\omega$ with $\omega \wedge d\omega =0$, and let $\eta$ satisfy $d\omega =\eta \wedge \omega$. Then:

1. The 3-form $\mathrm{gv}=\eta \wedge d\eta$ is closed.
2. The de Rham class $[\eta \wedge d\eta ]\in H^3(M;\mathbb{R})$ is independent of the choice of $\eta$ for a fixed $\omega$, and independent of the choice of defining form $\omega$.

Hence $\mathrm{GV}(\mathcal{F})=[\eta \wedge d\eta ]$ is a well-defined invariant of $\mathcal{F}$.

Proof.

Step 1: a derived identity. Differentiate $d\omega =\eta \wedge \omega$. Since $d^2=0$, $$ 0 = d(d\omega) = d\eta\wedge\omega - \eta\wedge d\omega = d\eta\wedge\omega - \eta\wedge(\eta\wedge\omega). $$ The last term vanishes because $\eta \wedge \eta =0$. Therefore $$ d\eta\wedge\omega = 0, $$ so $d\eta$, like $d\omega$, lies in the ideal generated by $\omega$: there is a 1-form $\xi$ with $d\eta =\xi \wedge \omega$.

Step 2: $\mathrm{gv}$ is closed. Compute $$ d(\eta\wedge d\eta) = d\eta\wedge d\eta - \eta\wedge d(d\eta) = d\eta\wedge d\eta . $$ Now $d\eta \wedge d\eta =(\xi \wedge \omega )\wedge (\xi \wedge \omega )=-\xi \wedge \xi \wedge \omega \wedge \omega =0$, since $\omega \wedge \omega =0$. Hence $d(\eta \wedge d\eta )=0$ and $\mathrm{gv}$ defines a class in $H^3(M;\mathbb{R})$. This proves (1).

Step 3: independence of $\eta$ for fixed $\omega$. Any two choices differ by ${\eta }^{\prime }=\eta +g\omega$ for a smooth function $g$, because both satisfy $d\omega =\eta \wedge \omega ={\eta }^{\prime }\wedge \omega$, forcing $({\eta }^{\prime }-\eta )\wedge \omega =0$, so ${\eta }^{\prime }-\eta$ is a multiple of $\omega$. Then $$ \eta'\wedge d\eta' = (\eta + g\omega)\wedge d(\eta + g\omega) = (\eta+g\omega)\wedge(d\eta + dg\wedge\omega + g,d\omega). $$ Expand, using $\omega \wedge \omega =0$, $\eta \wedge \eta =0$, and $d\omega =\eta \wedge \omega$: $$ \eta'\wedge d\eta' = \eta\wedge d\eta + \eta\wedge dg\wedge\omega + g,\omega\wedge d\eta + g,\eta\wedge(\eta\wedge\omega) . $$ The final term vanishes. Using $d\eta =\xi \wedge \omega$ from Step 1 gives $\omega \wedge d\eta =\omega \wedge \xi \wedge \omega =0$, killing the third term. For the second term, observe $$ d\big(g,\eta\wedge\omega\big) = dg\wedge\eta\wedge\omega + g,d\eta\wedge\omega - g,\eta\wedge d\omega = dg\wedge\eta\wedge\omega $$ because $d\eta \wedge \omega =0$ (Step 1) and $\eta \wedge d\omega =\eta \wedge \eta \wedge \omega =0$. Hence $\eta \wedge dg\wedge \omega =dg\wedge \eta \wedge \omega \cdot (-1)\cdot (-1)=d(g\eta \wedge \omega )$ up to sign bookkeeping, an exact 3-form. Therefore $$ \eta'\wedge d\eta' = \eta\wedge d\eta + d\big(\pm, g,\eta\wedge\omega\big), $$ so the two 3-forms differ by an exact form and define the same class.

Step 4: independence of $\omega$. Any other defining form for the same $\mathcal{F}$ is $\tilde{\omega }=u\omega$ for a nowhere-zero function $u$. Then $$ d\widetilde\omega = du\wedge\omega + u,d\omega = du\wedge\omega + u,\eta\wedge\omega = \Big(\tfrac{du}{u} + \eta\Big)\wedge\widetilde\omega, $$ so a valid auxiliary form for $\tilde{\omega }$ is $\tilde{\eta }=\eta +d(\log |u|)$. Since $d(\log |u|)$ is closed and exact, $$ \widetilde\eta\wedge d\widetilde\eta = (\eta + d\log|u|)\wedge d\eta = \eta\wedge d\eta + d\log|u|\wedge d\eta . $$ The extra term is exact: $d\log |u|\wedge d\eta =d(\log |u|d\eta )$ because $d(d\eta )=0$. Combining with Step 3, every admissible pair $(\tilde{\omega },\tilde{\eta })$ yields the same de Rham class. This proves (2). $■$

The class $\mathrm{GV}(\mathcal{F})$ is natural: if $f:N\to M$ is transverse to $\mathcal{F}$, the pulled-back foliation $f^{\ast }\mathcal{F}$ is defined by $f^{\ast }\omega$ with auxiliary form $f^{\ast }\eta$, so $\mathrm{GV}(f^{\ast }\mathcal{F})=f^{\ast }\mathrm{GV}(\mathcal{F})$. In particular $\mathrm{GV}$ is a cobordism-type invariant of foliations, which is how it first entered Thurston's work below.

Bridge. This construction builds toward the entire theory of exotic characteristic classes of foliations, and the foundational reason it works is the same transgression mechanism that produced the Chern-Simons form: a primary curvature class is forced to vanish, and its vanishing is witnessed by a lower-degree form whose own derivative is the (vanished) primary class. Here the vanishing is Bott vanishing, and the witness is $\eta$. This is exactly the pattern of 03.06.07, where a flat connection makes a Pontryagin form exact and the transgressing Chern-Simons form carries secondary information; the Godbillon-Vey class generalises that to the normal bundle of a foliation. The integrability identity $\omega \wedge d\omega =0$ of 03.02.04 is the central insight that makes $\eta$ exist at all, so the Frobenius theorem and the secondary-class theory are dual sides of one fact. Putting these together, the Godbillon-Vey class appears again in the cohomology of the classifying space $B{\Gamma }_1$, where it becomes the first universal secondary class, a thread the Master tier takes up.

Exercises Intermediate+

Advanced results Master

Bott vanishing as the source of secondary classes. Let $\mathcal{F}$ be a codimension-$q$ foliation with normal bundle $\nu =TM/T\mathcal{F}$. The Bott vanishing theorem ^{[Bott LNM 279 §3]} states that the Pontryagin ring of $\nu$, computed via Chern-Weil 03.06.06 from any connection adapted to $\mathcal{F}$ (a Bott connection, flat along the leaves), vanishes in degrees greater than $2q$: $$ \mathrm{Pont}^{k}(\nu) = 0 \quad\text{for } k > 2q . $$ The proof is a one-line miracle of Chern-Weil theory: a Bott connection has curvature with no purely-transverse components of the forbidden type, so any invariant polynomial of degree $>q$ in the curvature is forced to be zero as a form, not merely as a class. The vanishing of a primary class as a form is exactly the hypothesis that lets one transgress: the secondary classes are the transgression forms that witness this vanishing, living in the cohomology of a truncated Weil algebra $W{O}_q$ (the Gelfand-Fuks / Bott-Haefliger algebra). For $q=1$ the algebra $W{O}_1$ has a single generator in degree $3$, and its class is the Godbillon-Vey class; the codimension-1 theory is the first and simplest stratum of this picture, with $\eta \wedge d\eta$ the explicit transgression of the vanished first Pontryagin form of the normal line bundle.

The Roussarie-Reeb example: $\mathrm{GV}\ne 0$. Godbillon and Vey exhibited their class as nonzero on an explicit foliation built from the geometry of $\mathrm{PSL}(2,\mathbb{R})$ ^{[Godbillon-Vey 1971]}. On a compact quotient $\Gamma \backslash \mathrm{PSL}(2,\mathbb{R})$ of the unit tangent bundle of a hyperbolic surface, the weak-stable horocycle foliation is defined by a left-invariant 1-form $\omega$ with $d\omega =\eta \wedge \omega$ where $\eta$ is another invariant form, and the Maurer-Cartan structure equations give $\eta \wedge d\eta$ equal to a nonzero multiple of the bi-invariant volume form. Integrating over the closed $3$-manifold yields $\int \mathrm{gv}\ne 0$, so $\mathrm{GV}(\mathcal{F})\ne 0$ in $H^3$. The Roussarie construction (reported in the same note) recasts this on a Reeb-type foliation, and the modern statement is that the Godbillon-Vey number of the weak-stable foliation of a hyperbolic surface bundle is a nonzero multiple of the Euler characteristic — the first concrete value of a secondary class.

Thurston's continuity theorem. Reeb's foliation and the Roussarie example might have suggested that a topological invariant of foliations would take only discrete values. Thurston proved the opposite ^{[Thurston 1972]}: there is a smooth one-parameter family of foliations ${\mathcal{F}}_t$ of the $3$-sphere whose Godbillon-Vey numbers $\mathrm{GV}({\mathcal{F}}_t)={\int }_{S^3}{\mathrm{gv}}_t$ vary continuously and surjectively onto an interval of $\mathbb{R}$. Concretely, by spinning the leaves of a Reeb component faster one dials the transverse turning, and $\eta \wedge d\eta$ integrates to a value that moves continuously with the spin rate. This is a structural surprise: the moduli of foliations are not rigid, and a foliation invariant can take a continuum of real values, unlike the integer-valued primary characteristic numbers. The Godbillon-Vey number thereby distinguishes uncountably many non-cobordant foliations of $S^3$, the title result of Thurston's note.

The universal class on $B{\Gamma }_1$. The natural home of the Godbillon-Vey class is the Haefliger classifying space $B{\Gamma }_q$, which classifies codimension-$q$ foliations up to integrable homotopy ^{[Lawson 1974]}. The truncated Weil algebra $W{O}_q$ maps to $H^{\ast }(B{\Gamma }_q;\mathbb{R})$, and for $q=1$ the generator transgresses to a universal class $\mathrm{gv}\in H^3(B{\Gamma }_1;\mathbb{R})$ — the first secondary characteristic class of foliations. Thurston's continuity result is equivalent to the statement that this universal class is nonzero and that the induced map detects a real-valued summand of $H^3(B{\Gamma }_1;\mathbb{R})$, so $B{\Gamma }_1$ has cohomology beyond what the normal bundle's primary classes can produce.

Synthesis. The Godbillon-Vey class is the foundational reason the secondary-class program exists, and it ties together at least four strands into one object. It is the transgression image of the Chern-Simons construction 03.06.07: a primary Pontryagin form, forced to vanish by Bott vanishing, leaves behind a witness whose class is the secondary invariant, exactly as a flat connection forces a curvature form to vanish and leaves the Chern-Simons form behind. It consumes the Frobenius integrability theorem 03.02.04 as its enabling hypothesis — the central insight is that $\omega \wedge d\omega =0$ is what lets the auxiliary form $\eta$ exist, so foliation theory and secondary characteristic classes are dual to one another. It generalises the rigid integer-valued world of primary characteristic numbers 03.06.06 to a continuously-varying real invariant, and putting these together with Thurston's theorem shows the moduli of foliations are soft where ordinary characteristic numbers are rigid. Finally it is the first nonzero class of $H^{\ast }(B{\Gamma }_1;\mathbb{R})$, so the explicit $3$-form $\eta \wedge d\eta$ on a single manifold and a universal cohomology class on a classifying space are two faces of the same invariant — the bridge from a hands-on differential-form computation to the homotopy theory of the Haefliger space.

Full proof set Master

Proposition (the auxiliary form exists and is unique modulo $\omega$). Let $\omega$ be a nowhere-zero $1$-form with $\omega \wedge d\omega =0$. Then there is a $1$-form $\eta$ with $d\omega =\eta \wedge \omega$, and any two such differ by a multiple of $\omega$.

Proof. Locally choose a frame in which $\omega =d{u}_1$ after a change of coordinates is not assumed; instead argue pointwise. At a point $p$, the condition $\omega \wedge d\omega =0$ says $d\omega \in \omega \wedge {\Lambda }^1{T}_p^{\ast }M$, because wedging with the nonzero covector $\omega (p)$ is injective on the quotient ${\Lambda }^2/(\omega \wedge {\Lambda }^1)$ in the relevant degree, so its kernel on $2$-forms is exactly the image of $\omega \wedge (-)$. Hence $d\omega (p)=\eta (p)\wedge \omega (p)$ for some covector $\eta (p)$. A smooth global $\eta$ is obtained by a partition of unity, or by fixing a Riemannian metric and taking $\eta ={\iota }_{V}d\omega$ for $V$ the dual vector field of $\omega /|\omega {|}^2$; one checks $\eta \wedge \omega =d\omega$ directly. If $\eta \wedge \omega ={\eta }^{\prime }\wedge \omega$ then $(\eta -{\eta }^{\prime })\wedge \omega =0$, and since $\omega$ is nowhere zero this forces $\eta -{\eta }^{\prime }=g\omega$ for a smooth function $g$. $■$

Proposition ($\mathrm{gv}$ is closed). $d(\eta \wedge d\eta )=0$.

Proof. From $d(d\omega )=0$ and $d\omega =\eta \wedge \omega$ we get $d\eta \wedge \omega -\eta \wedge d\omega =0$, and $\eta \wedge d\omega =\eta \wedge \eta \wedge \omega =0$, so $d\eta \wedge \omega =0$. Thus $d\eta =\xi \wedge \omega$ for some $1$-form $\xi$. Then $d(\eta \wedge d\eta )=d\eta \wedge d\eta =(\xi \wedge \omega )\wedge (\xi \wedge \omega )=-\xi \wedge \xi \wedge \omega \wedge \omega =0$. $■$

Proposition (well-definedness of the class). The class $[\eta \wedge d\eta ]\in H^3(M;\mathbb{R})$ is independent of $\eta$ and of $\omega$.

Proof. For fixed $\omega$, replacing $\eta$ by $\eta +g\omega$ changes $\eta \wedge d\eta$ by $d(\pm g\eta \wedge \omega )$, an exact form, by the computation of Step 3 of the Key theorem, which uses only $d\eta \wedge \omega =0$ and $\eta \wedge d\omega =0$. Replacing $\omega$ by $u\omega$ ($u$ nowhere zero) admits $\eta +d\log |u|$ as auxiliary form, and $\eta \wedge d\eta$ changes by $d\log |u|\wedge d\eta =d(\log |u|d\eta )$, again exact. Composing the two reductions, any admissible $(\omega ,\eta )$ gives a class differing from the reference one only by exact forms. $■$

Proposition (naturality under transverse maps). If $f:N\to M$ is transverse to $\mathcal{F}$, then $\mathrm{GV}(f^{}\mathcal F) = f^{}\mathrm{GV}(\mathcal F)$.

Proof. Transversality guarantees $f^{\ast }\omega$ is a nowhere-zero defining form for the pulled-back foliation $f^{\ast }\mathcal{F}$. Since pullback commutes with $d$ and $\wedge$, applying $f^{\ast }$ to $d\omega =\eta \wedge \omega$ gives $d(f^{\ast }\omega )=(f^{\ast }\eta )\wedge (f^{\ast }\omega )$, so $f^{\ast }\eta$ is an admissible auxiliary form, and $f^{\ast }(\eta \wedge d\eta )=(f^{\ast }\eta )\wedge d(f^{\ast }\eta )$ represents $\mathrm{GV}(f^{\ast }\mathcal{F})$. Passing to cohomology and using $f^{\ast }[\alpha ]=[f^{\ast }\alpha ]$ yields the claim. $■$

The Bott vanishing theorem, the Roussarie computation on $\Gamma \backslash \mathrm{PSL}(2,\mathbb{R})$, and Thurston's continuity construction are stated above without full proof; see Bott ^{[Bott LNM 279 §3]} for vanishing, Godbillon-Vey ^{[Godbillon-Vey 1971]} for the example, and Thurston ^{[Thurston 1972]} for continuity.

Connections Master

• Frobenius theorem and integrable distributions 03.02.04. The entire construction rests on the integrability condition $\omega \wedge d\omega =0$ proved there; it is exactly this condition that produces the auxiliary form $\eta$ via $d\omega =\eta \wedge \omega$. Without integrability there are no leaves and no defining $1$-form to transgress, so the Godbillon-Vey class is the characteristic-class layer sitting directly on top of the Frobenius integrability structure.

• Chern-Simons and transgression 03.06.07. The Godbillon-Vey class is a secondary class built by the same transgression mechanism: a primary characteristic form of the normal bundle is forced to vanish (Bott vanishing), and the secondary class is the transgression form witnessing the vanishing, exactly as the Chern-Simons form transgresses a Pontryagin form that a flat connection makes exact.

• Chern-Weil homomorphism 03.06.06. Bott vanishing — the engine that makes secondary classes exist — is a statement inside Chern-Weil theory: an invariant polynomial of the curvature of a Bott connection vanishes as a form in high degree. The Godbillon-Vey class lives in the room opened up when these primary Chern-Weil classes of the normal bundle vanish.

• Exterior derivative and de Rham cohomology 03.04.04, 03.04.06. The closedness of $\eta \wedge d\eta$ and the exactness of the correction terms are pure de Rham-complex computations; the invariant is by construction a de Rham class, and its independence of choices is the statement that the ambiguities are coboundaries in the de Rham complex.

Historical & philosophical context Master

Claude Godbillon and Jacques Vey announced their invariant in a 1971 Comptes Rendus note of barely four pages, Un invariant des feuilletages de codimension 1, defining $\mathrm{gv}=\eta \wedge d\eta$ and proving its de Rham class independent of choices ^{[Godbillon-Vey 1971]}. The note arrived in the wake of Raoul Bott's foundational program on the topology of foliations: Bott's vanishing theorem (in his Lectures, published as Lecture Notes in Mathematics 279 in 1972) had just shown that the Pontryagin classes of the normal bundle of a foliation vanish above twice the codimension, opening the structural question of what secondary invariants could live in the gap ^{[Bott LNM 279 §3]}. The Godbillon-Vey class was the first explicit answer, and it crystallised the Gelfand-Fuks cohomology of formal vector fields and the Bott-Haefliger theory of the classifying space $B{\Gamma }_q$ into a concrete, computable $3$-form.

The philosophical jolt came from William Thurston in 1972 ^{[Thurston 1972]}. Where characteristic numbers had always been integers — counting, rigid, discrete — Thurston showed the Godbillon-Vey number of foliations of the $3$-sphere varies continuously, taking every value in an interval, so that the moduli space of foliations is soft and a topological invariant can be real-valued and non-locally-constant. This dissolved the expectation that invariants of geometric structures must be quantised, and it reframed foliation theory as a subject with genuine continuous moduli. Lawson's 1974 survey Foliations gathered the resulting landscape ^{[Lawson 1974]}, and the Godbillon-Vey class remains the canonical example of a secondary characteristic class and the entry point to the cohomology of $B{\Gamma }_1$.

Bibliography Master

@article{GodbillonVey1971,
  author  = {Godbillon, Claude and Vey, Jacques},
  title   = {Un invariant des feuilletages de codimension 1},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences, Paris, S{\'e}rie A-B},
  volume  = {273},
  pages   = {A92--A95},
  year    = {1971}
}

@incollection{Bott1972Lectures,
  author    = {Bott, Raoul},
  title     = {Lectures on characteristic classes and foliations},
  booktitle = {Lectures on Algebraic and Differential Topology},
  series    = {Lecture Notes in Mathematics},
  volume    = {279},
  pages     = {1--94},
  publisher = {Springer},
  year      = {1972},
  note      = {Notes by Lawrence Conlon; contains the Bott vanishing theorem}
}

@article{Thurston1972Noncobordant,
  author  = {Thurston, William P.},
  title   = {Noncobordant foliations of $S^3$},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {78},
  pages   = {511--514},
  year    = {1972}
}

@book{Tondeur1997,
  author    = {Tondeur, Philippe},
  title     = {Geometry of Foliations},
  series    = {Monographs in Mathematics},
  volume    = {90},
  publisher = {Birkh{\"a}user},
  year      = {1997}
}

@article{Lawson1974Foliations,
  author  = {Lawson, H. Blaine},
  title   = {Foliations},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {80},
  pages   = {369--418},
  year    = {1974}
}

@book{CandelConlon2000,
  author    = {Candel, Alberto and Conlon, Lawrence},
  title     = {Foliations I},
  series    = {Graduate Studies in Mathematics},
  volume    = {23},
  publisher = {American Mathematical Society},
  year      = {2000}
}