Bordism of semi-free \(S^1\)-actions (Q1771982)

The author describes both the geometric bordism ring \(\Omega^{SF}_*\) and the homotopical bordism ring \(MU^{SF}_*\) associated to stably complex semi-free \(S^1\)-manifolds. He presents sets of generators of \(MU^{SF}_*\) and relations among those generators. Using this result and the Pontryagin-Thom map \(\Omega^{SF}_* \to MU^{SF}_*\), he deduces the ring structure of \(\Omega^{SF}_*\) by giving sets of explicit generators and relations. He also shows that if \(M\) is a stably complex semi-free \(S^1\)-manifold with isolated fixed points, then \(M\) is equivariantly bordant to a disjoint union of products of \(P(\mathbb{C} \oplus \rho)\), where \(P(\mathbb{C} \oplus \rho)\) is the space of complex lines in \(\mathbb{C} \oplus \rho\) and \(\rho\) is the standard one-dimensional representation of \(S^1\).