## Characteristic classes of foliations

I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

I thought it would be a nice idea to discuss some pieces of the theory in a blog post (note that I have made no attempt to bring the material “up to date”). There are two starting points for the theory; the first is the work of Gelfand-Kazhdan on formal vector fields, which establishes the existence of a natural homomorphism $H^*(W_p) \to H^*(J^1M)$ where $J^1M$ is the frame bundle (i.e. the bundle of 1-jets) of a p-dimensional manifold $M$, and $W_p$ is the Lie algebra of formal vector fields on $\mathbb{R}^p$. The second is the work of Godbillon-Vey who discovered a 3-dimensional characteristic class associated to a codimension 1 foliation on a manifold, which is a kind of transgression of a characteristic class of the normal bundle $Q$. These ideas were synthesized by the work of Bernstein-Rosenfeld, who showed how to construct a homomorphism $H^*(W_p) \to H^*(J^1Q)$. Classes in the image can be integrated over the fiber to produce characteristic classes on $M$, of which the simplest is the Godbillon-Vey invariant.

The Godbillon-Vey invariant of a codimension 1 foliation $\mathcal{F}$ on a 3-manifold $M$ can be described using an auxiliary Riemannian metric. Under holonomy transport, leaves spread apart from each other infinitesimally; the logarithmic derivative of a transverse measure (i.e. the multiplicative rate of spreading apart of leaves) defines a vector field $X$ tangent to $\mathcal{F}$. The Godbillon-Vey form measures the infinitesimal rate at which $X$ spins as one moves transverse to $\mathcal{F}$; Thurston famously called this “helical wobble”. If one uses the metric to make $X$ dual to a 1-form $\alpha$ then the Godbillon-Vey form $\alpha \wedge d\alpha$ measures how non-integrable $\alpha$ is, and the Godbillon-Vey invariant is the integral of this form over $M$. For example, if $M=UTS$, the unit tangent bundle of a hyperbolic surface with its stable foliation (see this post), then $X$ is the geodesic flow itself (up to a change of orientation), and the Godbillon-Vey invariant is the volume of $UTS$ (up to a nonzero constant). The figure below (taken from Thurston’s paper) shows such an example of constant helical wobble locally.

The differential-geometric approach to constructing foliated characteristic classes goes via connections and curvature. Let’s fix a manifold $M$ and a codimension p foliation $\mathcal{F}$. The issue of smoothness is very important for foliations, so for simplicity assume that the tangent field to the foliation is a $C^\infty$ distribution $E$, and let $Q:=TM/E$ denote the normal bundle. Dual to $Q$ is $Q^*$, the collection of 1-forms on $M$ whose kernel contains $E$ (pointwise). Working with $Q^*$ in place of $Q$ makes it easier to use the language of differential algebra. The crucial property of the sections $\Gamma(Q^*)$ is that they generate a differential ideal; i.e. if $\omega_i$ are forms which locally give a basis for $Q^*$ at each point in an open neighborhood $U$, then each $d\omega_i$ can be expressed as a linear combination $d\omega_i = \sum_j \theta_{ij}\wedge \omega_j$ for certain 1-forms $\theta_{ij}$. This statement is equivalent (and dual) to Frobenius’s theorem, which characterizes the integrability of a distribution $E$ (i.e. the property that it should be tangent to a foliation) precisely by saying that sections $\Gamma(E)$ form a Lie algebra: i.e. for sections $X,Y \in \Gamma(E)$ we have $[X,Y] \in \Gamma(E)$. This property of $\Gamma(Q^*)$ enables one to construct a certain connection on $Q^*$ which is said to be torsion-free. Recall that a connection $\nabla$ on $T^*M$ defines a map

$\nabla: \Omega^1(M) \to \Omega^1(M) \otimes \Omega^1(M)$

and it is said to be torsion-free if the composition with the antisymmetrizing map $\wedge \circ \nabla: \Omega^1(M) \to \Omega^2(M)$ coincides with exterior d. In local coordinates therefore one can define a connection on $Q^*$ by the formula

$\nabla(\omega_i) = \sum_j \theta_{ij}\otimes \omega_j \in \Omega^1(M)\otimes \Gamma(Q^*)$

and observe that integrability implies that this connection is torsion-free. Taking convex combinations of connections defined on open neighborhoods (by using a partition of unity) preserves the torsion-free property (since both $\nabla$ and exterior d satisfy the Leibniz formula) and one thereby obtains a torsion-free connection on $Q^*$. Differentiating the equation $d\omega_i = \sum_j \theta_{ij}\wedge \omega_j$ gives

$0 = \sum_j d\theta_{ij}\wedge \omega_j - \sum_j\theta_{ij}\wedge (\sum_k \theta_{jk}\wedge \omega_k)$

$=\sum_k(d\theta_{ik} - \sum_j \theta_{ij}\wedge\theta_{jk}) \wedge \omega_k = \sum_k K_{ik}\wedge\omega_k$

where $K_{ik}$ is the $i,k$ entry in the curvature of the connection $\nabla$. This last equation implies that $K_{ik}$ is in the (differential) ideal generated by the $\omega_j$, and therefore any homogeneous polynomial in the $K_{ij}$ of degree $>p$ is identically zero. This observation is due to Bott, and implies (for example) that the (rational) Pontriagin classes of the normal bundle of a smooth foliation of codimension p vanish in degrees $>2p$.

On the other hand, we can choose a Riemannian connection $\nabla'$ on $Q^*$ (this does not make any use of integrability at all), and then the associated curvature matrix $K_{ij}'$ will be skew-symmetric. In particular, the invariant homogeneous polynomials in $K_{ij}'$ of odd degree will vanish identically (this is just the usual observation that the odd rational Chern classes of a real vector bundle vanish). If we let $c_i$ and $c_i'$ denote the differential forms on $M$ of dimension $2i$ representing the Chern classes associated to the connections $\nabla$ and $\nabla'$ respectively (i.e. they are, up to a constant, pointwise the $i$th coefficients of the characteristic polynomial of the $K_{ij}$ and $K_{ij}'$ respectively), then $c_i'$ is identically zero for i odd, and every polynomial in the $c_i$ of total degree $> p$ is also identically zero. Now, Chern showed that for any two connections on a bundle, the difference of the associated Chern forms is exact, and is exterior d of a canonical form of one dimension lower. To see this in our context, let $\tilde{\nabla}$ be a connection on the pullback of $Q^*$ to $M \times[0,1]$ restricting to $\nabla,\nabla'$ on $M\times\lbrace 0,1\rbrace$, let $\tilde{c}_i$ be the associated Chern class, and let $u_i$ be the integral of $\tilde{c}_i$ along the fibers point $\times [0,1]$. Then $u_i$ is a form on $M$ satisfying $du_i=c_i-c_i'$.

We define $WO_p$ to be the following differential graded algebra:

$WO_p = \Lambda(u_1,u_3,\cdots,u_{2\ell+1})\otimes \mathbb{R}(c_1,c_2,\cdots,c_p)/\text{ideal of degree}>2p$

where $\ell=\lfloor p/2\rfloor$, and where $u_{2i-1}$ has degree $4i-3$, and $c_i$ has degree $2i$, and the differential is given by $du_i=c_i$ and $dc_i=0$. A choice of a pair of connections $\nabla,\nabla'$ determines a map of dgas from $WO_p$ to $\Omega^*(M)$, and the induced map on cohomology $H^*(WO_p) \to H^*(M)$ is independent of all choices. The images are the characteristic classes of the foliation. For example, if $p=1$ then the Godbillon-Vey class is the image of $u_1c_1$.

The algebro-geometric approach goes via formal vector fields, thought of as living on the local “space of leaves”. In every sufficiently small open ball $U$ on $M$, there is a submersion $f:U \to \mathbb{R}^p$ for which the kernel is precisely $E$. So we can identify $\Gamma(Q^*)$ with forms on $\mathbb{R}^p$ locally. Consider the principal $GL_p$ (frame) bundle $\pi:P(Q) \to M$ whose fiber at each point is a basis for $Q$ at that point. There is a canonical trivialization of the pullback $\pi^*(Q)$; for each $x\in M$, a point $y\in \pi^{-1}(x)$ is a frame for $Q_x$, and the fiber of $\pi^*(Q)$ over $y$ is itself a copy of $Q_x$, so one can trivialize it by the tautological frame $y$. Dualizing, we obtain p canonical sections $\omega_i$ of $\pi^*(Q^*)\subset T^*P(Q)$. Since exterior d commutes with projection, these generate a differential ideal in $\Omega^*P(Q)$ so there are forms $\theta_{ij} \in \Omega^1P(Q)$ with $d\omega_i = \sum \theta_{ij}\wedge \omega_j$. The form $\theta_{ij}$ is not unique, but there is a canonical choice if we first pull back to a further bundle $J^2(Q)$ over $P(Q)$, namely the “bundle of 2-jets”. In fact, one can reinterpret $P(Q)$ as $J^1(Q)$, the bundle of 1-jets, and consider it as the first step in a tower of bundles

$\cdots \to J^{i+1}(Q) \to J^i(Q) \to J^{i-1}(Q) \to \cdots$

where the fiber of $J^i(Q)$ over $x\in M$ keeps track of the derivatives of order $\le i$ of a local submersion to $\mathbb{R}^p$ sending $x$ to $0$. The conclusion is that we obtain canonical 1-forms $\theta_{ij}$ on $J^2(Q)$ satisfying $d\omega_i = \sum \theta_{ij}\wedge \omega_j$, canonical 1-forms $\theta_{ijk}$ on $J^3(Q)$ satisfying $d\theta_{ij} = \sum \theta_{ijk}\wedge \omega_k + \sum \theta_{ik} \wedge \theta_{kj}$ and so on (each form on $J^i$ pulls back to a form of the same name on all $J^j$ with $j>i$ which is where these formulae hold). Let $L_p$ denote the Lie algebra, which is a module on p generators $\partial/\partial x_i$ over the ring of formal power series on $\mathbb{R}^p$, with Lie bracket defined (formally) in the obvious way. We can think of $L_p$ as the Lie algebra of formal vector fields on $\mathbb{R}^p$. The continuous dual $L_p^*$ (with respect to the obvious topology) has a basis consisting of the forms $\omega_i,\theta_{ij},\theta_{ijk},\cdots$, and there is a differential graded algebra $\Lambda^*(L_p^*)$ obtained by dualizing the Lie bracket. From the discussion above, there is a map of dgas $\Phi:\Lambda^*(L_p^*) \to \lim_{n\to\infty} \Omega^*(J^n(Q))$ and thereby a map on cohomology $\Phi^*:H^*(L_p^*) \to H^*(J^{\infty}(Q))$. Now, topologically, the fiber of each fibration $J^{i+1}(Q) \to J^i(Q)$ is contractible for $i\ge 1$, so at the level of cohomology we may identify $H^*(J^{\infty}(Q))$ with $H^*(J^1(Q))$. Dual to projection there is a map $H^*(M) \to H^*(J^1(Q))$ identifying the (de Rham) cohomology of $M$ with the cohomology of the complex of $GL_p$-invariant forms on $H^*(J^1(Q))$. Up to homotopy we can replace $GL_p$ by $O_p$; the Lie algebra $o_p$ of $O_p$ sits inside $L_p$ in an obvious way (by thinking of elements of $o_p$ as vector fields on $\mathbb{R}^p$ and thence as formal vector fields), and we obtain a map $H^*(L_p^*,o_p^*) \to H^*(M)$. The relation to the discussion above is that there is a canonical isomorphism of $H^*(L_p^*,o_p^*)$ with $H^*(WO_p)$ defined above.

This (highly abbreviated) discussion brings us roughly to the end of the third chapter of Harsh’s book. A fourth chapter discusses how to measure the variation of the characteristic classes in families of foliations. There is also an appendix, giving a short exposition of the Chern-Weil theory of (ordinary) characteristic classes, and another appendix on the cohomology of Lie algebras. Composing this blog post gave me an excuse to read Harsh’s book again (for the first time in quite a few years), and I must say it was every bit as good as I remember. Mathematics is a conversation in which the participants might be separated by unbridgeable distances in space or in time, but it is some consolation to know that we will still have the opportunity — through our work — to take part in this conversation once we are gone.

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