On the bordism of almost free 𝑍_{2𝑘} actions

Abstract

An “almost free” Z 2 k {Z_{{2^k}}} action on a manifold is one in which only the included Z 2 {Z_2} may possibly fix points of the manifold. For k = 2, these are the stationary-point free actions. It is shown that almost free Z 2 k {Z_{{2^k}}} bordism is generated by three subalgebras: the extension from Z 2 {Z_2} actions, a coset of Z 2 {Z_2} extensions being the restrictions of circle actions and a certain ideal of elements which annihilate the whole ring. The additive structure is determined. Free Z 2 k {Z_{{2^k}}} bordism is shown to split as an algebra. It is shown that the kernel of the extension homomorphism from Z 2 {Z_2} to Z 2 k {Z_{{2^k}}} bordism is equal to the image of the corresponding restriction homomorphism.