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 where
is the frame bundle (i.e. the bundle of 1-jets) of a p-dimensional manifold
, and
is the Lie algebra of formal vector fields on
. 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
. These ideas were synthesized by the work of Bernstein-Rosenfeld, who showed how to construct a homomorphism
. Classes in the image can be integrated over the fiber to produce characteristic classes on
, of which the simplest is the Godbillon-Vey invariant.
The Godbillon-Vey invariant of a codimension 1 foliation on a 3-manifold
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
tangent to
. The Godbillon-Vey form measures the infinitesimal rate at which
spins as one moves transverse to
; Thurston famously called this “helical wobble”. If one uses the metric to make
dual to a 1-form
then the Godbillon-Vey form
measures how non-integrable
is, and the Godbillon-Vey invariant is the integral of this form over
. For example, if
, the unit tangent bundle of a hyperbolic surface with its stable foliation (see this post), then
is the geodesic flow itself (up to a change of orientation), and the Godbillon-Vey invariant is the volume of
(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 and a codimension p foliation
. The issue of smoothness is very important for foliations, so for simplicity assume that the tangent field to the foliation is a
distribution
, and let
denote the normal bundle. Dual to
is
, the collection of 1-forms on
whose kernel contains
(pointwise). Working with
in place of
makes it easier to use the language of differential algebra. The crucial property of the sections
is that they generate a differential ideal; i.e. if
are forms which locally give a basis for
at each point in an open neighborhood
, then each
can be expressed as a linear combination
for certain 1-forms
. This statement is equivalent (and dual) to Frobenius’s theorem, which characterizes the integrability of a distribution
(i.e. the property that it should be tangent to a foliation) precisely by saying that sections
form a Lie algebra: i.e. for sections
we have
. This property of
enables one to construct a certain connection on
which is said to be torsion-free. Recall that a connection
on
defines a map
and it is said to be torsion-free if the composition with the antisymmetrizing map coincides with exterior d. In local coordinates therefore one can define a connection on
by the formula
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 and exterior d satisfy the Leibniz formula) and one thereby obtains a torsion-free connection on
. Differentiating the equation
gives
where is the
entry in the curvature of the connection
. This last equation implies that
is in the (differential) ideal generated by the
, and therefore any homogeneous polynomial in the
of degree
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
.
On the other hand, we can choose a Riemannian connection on
(this does not make any use of integrability at all), and then the associated curvature matrix
will be skew-symmetric. In particular, the invariant homogeneous polynomials in
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
and
denote the differential forms on
of dimension
representing the Chern classes associated to the connections
and
respectively (i.e. they are, up to a constant, pointwise the
th coefficients of the characteristic polynomial of the
and
respectively), then
is identically zero for i odd, and every polynomial in the
of total degree
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
be a connection on the pullback of
to
restricting to
on
, let
be the associated Chern class, and let
be the integral of
along the fibers point
. Then
is a form on
satisfying
.
We define to be the following differential graded algebra:
where , and where
has degree
, and
has degree
, and the differential is given by
and
. A choice of a pair of connections
determines a map of dgas from
to
, and the induced map on cohomology
is independent of all choices. The images are the characteristic classes of the foliation. For example, if
then the Godbillon-Vey class is the image of
.
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 on
, there is a submersion
for which the kernel is precisely
. So we can identify
with forms on
locally. Consider the principal
(frame) bundle
whose fiber at each point is a basis for
at that point. There is a canonical trivialization of the pullback
; for each
, a point
is a frame for
, and the fiber of
over
is itself a copy of
, so one can trivialize it by the tautological frame
. Dualizing, we obtain p canonical sections
of
. Since exterior d commutes with projection, these generate a differential ideal in
so there are forms
with
. The form
is not unique, but there is a canonical choice if we first pull back to a further bundle
over
, namely the “bundle of 2-jets”. In fact, one can reinterpret
as
, the bundle of 1-jets, and consider it as the first step in a tower of bundles
where the fiber of over
keeps track of the derivatives of order
of a local submersion to
sending
to
. The conclusion is that we obtain canonical 1-forms
on
satisfying
, canonical 1-forms
on
satisfying
and so on (each form on
pulls back to a form of the same name on all
with
which is where these formulae hold). Let
denote the Lie algebra, which is a module on p generators
over the ring of formal power series on
, with Lie bracket defined (formally) in the obvious way. We can think of
as the Lie algebra of formal vector fields on
. The continuous dual
(with respect to the obvious topology) has a basis consisting of the forms
, and there is a differential graded algebra
obtained by dualizing the Lie bracket. From the discussion above, there is a map of dgas
and thereby a map on cohomology
. Now, topologically, the fiber of each fibration
is contractible for
, so at the level of cohomology we may identify
with
. Dual to projection there is a map
identifying the (de Rham) cohomology of
with the cohomology of the complex of
-invariant forms on
. Up to homotopy we can replace
by
; the Lie algebra
of
sits inside
in an obvious way (by thinking of elements of
as vector fields on
and thence as formal vector fields), and we obtain a map
. The relation to the discussion above is that there is a canonical isomorphism of
with
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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