Abstract:
In this paper generators are found for the rings $U^{S^1}_*$ (the unitary $S^1$-bordism ring) and $U_*(S^1,\{\mathbf Z_s\})$ (the unitary bordism ring with actions of the group $S^1$ without fixed points). The generators found are $S^1$-manifolds of the form $(S^3)^k\times\mathbf CP^n/(S^1)^k$. By an obvious construction the ring $U^{S^1}_*$ allows one to establish a relation between numerical invariants of manifolds with unitary actions of $S^1$ and the set of fixed points, without using a theorem of the type of an integrality theorem. In particular, we obtain a new proof of the Atiyah–Hirzebruch formula for the generalized Todd genus of $S^1$-manifolds.
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