Equivariant orientations and G-bordism theory (Q1112388)

This paper outlines a new geometric theory of orientations under the action of a group G and formulates bordism theories for G-oriented manifolds. These theories extend the classical G-bordism theories (graded on \({\mathbb{Z}})\), as well as the RO(G)-graded oriented G-bordism theories which describe bordism of G-manifolds with restricted local representation structure. The theories obtained account for oriented and unoriented bordism of G-manifolds with and without restricted local representation structure. It is shown further that there are spectral sequences covering to these theories through adjacent family constructions.