Contact structures, equivariant spin bordism, and periodic fundamental groups (Q5947241)

Let G be a finite group of odd order and finite cohomological period. The authors prove that every closed 5-dimensional spin manifold with fundamental group isomorphic to G admits a contact structure if the order of G is not divisible by 9. The idea of the proof goes as follows. The first step is a bordism theorem, which transforms the problem of constructing a contact structure to that of finding in the same equivariant spin bordism class a contact manifold with vanishing first Chern class (of its contact bundle). The second step is a reduction theorem, which allows to reduce the problem to studying the p-Sylow subgroups of the fundamental group. Finally, one shows that the relevant equivariant bordism groups are generated by contact manifolds with vanishing first Chern class.