Calculation of contact and symplectic cobordism groups (Q1209384)

Two compact symplectic \(2n\)-manifolds \((M_ i,\omega_ i)\) are called \(S\)-cobordant if there is an oriented cobordism \(W\) between \(M_ 1\) and \(M_ 2\) such that the forms \(\omega_ i\) extend to a closed 2-form \(\Omega\) on \(W\) which has maximal rank (i.e. \(\Omega^ n\neq 0)\). Similarly, two cooriented contact \((2n+1)\)-manifolds are \(C\)-cobordant if there is a cobordism between them which is equipped with a 1-form \(\alpha\) of maximal rank (i.e. \(\alpha\wedge(d\alpha)^ n\neq 0)\) which defines the given contact structures on the boundary. By using Gromov's method of convex integration, it is easy to see that the \(S\)- and \(C\)- cobordism groups are determined by topological data. The author calculates these groups. He shows, for example, that two integral symplectic manifolds are \(S\)-cobordant if and only if their mixed characteristic numbers \(c_ I(M,\omega)\) are equal, where, given a multi-index \(I\), \(c_ I\) denotes the approximate product of Chern classes of the symplectic manifold \((M,\omega)\) and \(c_ I(M,\omega)=\bigl\langle[\omega]^{n-| I|}\cup c_ I,[M]\bigr\rangle\). He also gives an explicit proof that every 3- dimensional contact manifold is null \(C\)-cobordant using contact surgery.