Obstruction to integrability

A subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ assigns to each point $x$ of the manifold a subspace $E_x$ of the tangent space $T_xM$ . An integrable manifold of the subbundle $E$ is a submanifold $N$ of $M$ whose tangent space $T_xN$ at each point $x$ in $N$ is $E_x$ . The subbundle $E$ is said to be integrable if for each point $x$ in $M$ , there is an integrable manifold of $E$ passing through $x$ .

By the Frobenius theorem, often proven in a first-year graduate course, a subbundle $E$ of the tangent bundle $TM$ is integrable if and only if its space of sections $\Gamma (E)$ is closed under the Lie bracket.

The Pontryagin ring $\operatorname{Pont}(V)$ of a vector bundle $V$ over $M$ is defined to be the subring of the cohomology ring $H^*(M)$ generated by the Pontryagin classes of the bundle $V$ . In [51] Bott found an obstruction to the integrability of $E$ in terms of the Pontryagin ring of the quotient bundle $Q:=TM/E$ . More precisely, if a subbundle $E$ of the tangent bundle $TM$ is integrable, then the Pontryagin ring $\operatorname{Pont}(Q)$ vanishes in dimensions greater than twice the rank of $Q$ .

What is so striking about this theorem is not only the simplicity of the statement, but also the simplicity of its proof. It spawned tremendous developments in foliation theory in the Seventies, as recounted in [C] and [H1].