Stable equivariant complex cobordism of the symmetric group on three elements (Q6039223)

For abelian groups \(G\), much progress has been done recently in the calculation of the homotopy cobordism rings \((MU_G)_\ast\), that is, the homotopy rings of the \(G\)-equivariant Thom spectra \(MU_G\) for (\(G\)-equivariant) complex cobordism. In particular, a theory of \(G\)-equivariant formal group laws has been constructed and the (\(G\)-equivariant) formal group law on \((MU_G)_\ast\) has been seen to be the universal one in certain cases (for example, if \(G = \mathbb Z_2\)), and conjectured to also be universal for \textit{any} abelian \(G\). No such progress has been achieved whenever \(G\) is not abelian, and the paper at hand aspires to start for this case a project similar to the one for abelian groups. In this direction, the authors calculate the equivariant cobordism ring \((MU_{\Sigma_3})_*\) for \(G=\Sigma_3\), the symmetric group on three elements. This is presented as Theorem 5.4, and the method used is referred to as \textit{isotropy separation}: one uses the properties of the orbit category of \(G\) to investigate certain spectra related to a given equivariant spectrum, and these ``building-block'' spectra can be used to construct the desired homotopy rings. For the computation in this paper, these building blocks are the geometric fixed point spectrum \(\Phi^G E\), the Borel cohomology spectrum \(F(EG_+,E)\) (both for \(G=\Sigma_3\) and \(E = MU_{\Sigma_3}\)), the spectrum \(S^{\infty \alpha} \wedge MU_{\Sigma_3}\) (where \(\alpha\) is the sign representation for \(\Sigma_3\)), and the \textit{intermediate Borel cohomology} \(F(S(\infty \alpha)_+, MU_{\Sigma_3})\). The geometric fixed point spectrum \(\Phi^{\Sigma_3} MU_{\Sigma_3}\) was described by tom Dieck, and this description is used here. The Borel cohomology spectrum \(MU^*B\Sigma_3\) is calculated in Theorem 2.1. The coefficients of \(S^{\infty \alpha} \wedge MU_{\Sigma_3}\) appear in Theorem 3.1. The coefficients of the intermediate Borel cohomology \(F(S(\infty \alpha)_+, MU_{\Sigma_3})\) are shown in Theorem 4.3, and are obtained from recent explicit computations of \((MU_{\mathbb Z_p})_\ast\) that generalize Strickland's previous result for \(p=2\) (these results are reproduced as Theorem 4.3). Section 5 presents the main result of the paper, the above-mentioned Theorem 5.4, where all the ingredients from the previous sections are combined in order to expose \((MU_{\Sigma_3})_*\), obtained as the limit of a certain diagram of rings. This and the previous partial results depend on the process of finding generators for rings that appear as pullbacks of Tate diagrams. This process is discussed in more general terms in Section 6, culminating in Example 6.2, which attempts a geometric description of the rings \((MU_{{\mathbb Z}_p})_\ast\) for a prime \(p\).