I would like to talk on c_1-spherical bordism theory W - an intermediate theory between complex and special unitary cobordism, introduced and firstly studied in the work of E. Conner and P. Floyd on the torsion in the SU-bordism ring. This bordism theory is based on stably complex manifolds with the first Chern class induced from the sphere CP^1. This theory does not have a natural multiplication (since a Cartesian product of two c_1-spherical manifolds is not necessarily c_1-spherical) but it has the natural module structure over the SU-bordism. R. Stong defined the SU-bilinear multiplication on W via the SU-linear projection from complex cobordism onto W and described the torsion-free part of the SU-bordism ring in terms of the obtained ring W. But there are a lot of other SU-linear projections and multiplications on W, as well as complex orientations (unlike the SU-bordism).

I would like to talk about SU-linear operations in complex cobordism in general, about SU-linear projections and multiplications on W, and about complex orientations of W, corresponding formal group laws and Landweber exactness of W