Equivariant formal group laws (Q2766369)

This paper is devoted to developing the correct definition of an equivariant formal group law for use in equivariant stable homotopy theory. In the nonequivariant situation, a complex oriented cohomology theory \(E^*(\cdot)\) has associated to it a formal group law over \(\pi_*(E)\) coming from the coproduct on \(E^*({\mathbb C}P^\infty)\). A fundamental theorem, due to D. Quillen, is that for \(E=MU\), complex cobordism, this formal group law is universal and \(\pi_*(MU)\) can be identified with Lazard's universal ring.NEWLINENEWLINENEWLINEThe authors' goal is to develop an analogue of this construction in the \(A\)-equivariant case when \(A\) is a compact abelian Lie group. The paper is divided into 3 parts with the first and second providing topological motivation for the algebraic definition given in the third.NEWLINENEWLINENEWLINEThe first section summarizes information about \({\mathbb C}P^\infty\) in the equivariant situation, including an account of the first author's splitting theorem for \(E^*_A({\mathbb C} P^\infty)\).NEWLINENEWLINENEWLINEThe second part examines \(E^*_A({\mathbb C}P^\infty)\) in several special cases including equivariant \(K\) theory, bordism theories and Borel cohomology.NEWLINENEWLINENEWLINEIn the third part an \(A\)-equivariant formal group law over a ring \(k\) is defined to be a topological Hopf \(k\)-algebra, \(R\), together with a co-action of \(A^*\) on \(R\) and special regular elements \(y(\alpha)\in R\) so that \(R/y(\alpha) =k\) for all \(\alpha\) and \(R\) is complete with respect to the ideal \((\Pi_\alpha y(\alpha))\). The existence of the representing ring for such objects, \(L_A(F)\), is established and some of its structure is described.