Computations of complex equivariant bordism rings (Q2747074)

Unlike non-equivariant versions, equivariant bordism tends not to be fully susceptible to purely homotopy theoretic investigation because of the lack of equivariant transversality results. Instead, the bordism of \(G\)-manifolds is only approximated by the homotopy of suitable equivariant Thom spectra. The present paper focuses on the algebraic structure of the homotopy theoretic complex bordism groups \(MU^G_*\) for nilpotent groups and makes a significant advance in the computation of equivariant bordism groups. It is worth remarking that recently a considerable amount of attention has been directed at equivariant generalizations of Quillen's non-equivariant theory of complex orientations and formal group laws for complex cobordism and this work may well be related to some aspects of this.NEWLINENEWLINENEWLINEAs is often the case in equivariant situations, localization at fixed points plays a central part. To each representation of the compact Lie group \(G\) is assigned an Euler class \(e_V\in MU^G_{-m}\), where \(m=\dim_{\mathbb{R}}V\). If \(V\) is irreducible, \(e_V\neq 0\); thus \(MU^G_*\) can be non-zero in negative degrees, in sharp contrast to the non-equivariant situation. Let \(R_0\subset MU^G_*\) is the subring generated by the Euler classes \(e_V\) and \(Z_{n,V}=[\mathbb{P}(\underline{n}\oplus V)]\) for irreducible \(V\), and \(S\) is the multiplicative set generated by these same \(e_V\). Here are some of the basic results proved. NEWLINENEWLINENEWLINETheorem 1.1. Let \(G\) be nilpotent. The inclusion of \(R_0\) into \(MU^G_*\) is an isomorphism after inverting \(S\). If \(T\) is a torus and \(V\) a non-trivial irreducible of \(T\), let \(K(V)\) be the kernel of the representation \(V\). NEWLINENEWLINENEWLINETheorem 1.2. The following sequence is exact: NEWLINE\[NEWLINE 0\rightarrow MU^T_*@>\cdot e_V>> MU^T_* @>\text{res}^T_{K(V)}>>MU^{K(V)}_*\rightarrow 0. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINETheorem 1.3. There are inclusions of \(MU_*\)-algebras NEWLINE\[NEWLINE R_0=MU_*[e_V,Z_{n,V}]\subset MU^T_*\subset S^{-1}MU^T_*, =MU_*[e_V^{\pm 1},Z_{n,V}] NEWLINE\]NEWLINE where \(V\) ranges over the irreducibles and \(n\) over the natural numbers. NEWLINENEWLINENEWLINEUsing these results, the main theorem 1.5 is deduced, giving a description of algebra generators for \(MU^T_*\) and families of relations amongst them. When \(T=S^1\), these relations are known to be complete.