Global group laws and equivariant bordism rings

Abstract

For every abelian compact Lie group $A$, we prove that the homotopical $A$-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the $A$-equivariant Lazard ring, introduced by Cole--Greenlees--Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A=C_2$ due to Hanke--Wiemeler (2018).

We work in the framework of global homotopy theory, which is essential for our proof. In addition to the statements for a fixed group $A$, we also prove a global algebraic universal property that characterizes the collection of all equivariant complex bordism rings simultaneously. We show that they form the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate; the coordinate is given by the universal Euler class at the circle group. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among functors equipped with a strict $n$-tuple of coordinates.