Swan-Tate cohomology of meromorphic circle actions

We propose a toy model for symmetry-breaking or bubbling, in terms of cobordism of manifolds with circle actions free on a possible boundary. The Swan-Tate cohomology $t_{\text{T}}E$ of a complex-oriented $E_{\mathrm{\infty }}$ ring-spectrum $E$ is the extension of a Hopf algebra by its dual, which provides an algebraic rigidification of geometric interest. This note reviews the cases $E=H,K$ and $MU$, with special attention to $\lambda$-ring structures.