Invariants of contact structures and transversally oriented foliations (Q1126455)

Let \(S\) be a contact structure on a manifold \(M\) defined by a global contact 1-form \(\omega\). Denote by \(\text{Diff}_S(M)\) the group of automorphisms of \((M, S): \phi\in \text{Diff}_S(M)\) iff \(\phi\in \text{Diff}(M)\) and \(\phi^\ast\omega =u^\omega_\phi\cdot\omega\) for a positive scalar function \(u^\omega_\phi\) on \(M\). Also, denote by \(h_S(M)\) the Lie algebra of the group \(\text{Diff}_S(M)\): a field \(X\) on \(M\) lies in \(h_S(M)\) iff \(\mathcal L_X\omega= v^\omega_\phi\cdot\omega\) for a scalar function \(v^\omega_\phi\). In the paper under review, the author shows that the maps \(D_\omega: \text{Diff}_S(M)\ni \phi\mapsto \log u^\omega_{\phi^{-1}}\) and \(\Delta_\omega: h_S(M)\ni X\mapsto v^\omega_{\phi^{-1}}\) are suitable 1-cocycles and define cohomology classes in \(H^1(\text{Diff}_S(M), C^\infty (M))\) and \(H^1(h_S(M), C^\infty (M))\), respectively. These classes are shown to be nontrivial and independent of the choice of the form \(\omega\) defining \(S\), and thus become invariants of the structure \(S\). Similar constructions are performed for flows (i.e., one-dimensional transversely oriented foliations) and transverse symplectic structures on contact manifolds.