Formal groups and the Atiyah–Hirzebruch formula

In this article, manifolds with actions of compact Lie groups are considered. For each rational Hirzebruch genus $h\colon\Omega_*\to Q$, an “equivariant genus” $h^G$, a homomorphism from the bordism ring of $G$-manifolds to the ring $K(BG)\otimes Q$, is constructed. With the aid of the language of formal groups, for some genera it is proved that for a connected compact Lie group $G$, the image of $h^G$ belongs to the subring $Q\subset K(BG)\otimes Q$. As a consequence, extremely simple relations between the values of these genera on bordism classes of $S^1$-manifolds and submanifolds of its fixed points are found. In particular, a new proof of the Atiyah–Hirzebruch formula is obtained.