Cohomology for groups of \(\text{rank}_ pG=2\) and Brown-Peterson cohomology
For a finite group \(G\) and \(p\) a prime let \(BP^*(BG)\) be the \(BP\)- theory and \(H^*(BG)\) the ordinary cohomology theory of the classifying space \(BG\) with coefficients in \(\mathbb{Z}(p)\). The Chern subring \(Ch(G)\) (resp. \(Ch_{BP}(G)\)) is the subring of \(H^{\text{even}}(BG)\) (resp. the \(BP^*\)-subalgebra of \(BP^*(BG)\)) generated by the Chern classes of complex representations. For a metacyclic group \(G\) and some other \(\text{rank}_ p G = 2\) groups (\(\text{rank}_ pG\) is the maximal rank of an elementary abelian \(p\)-subgroup of \(G\)) \(BP^*(BG) =Ch_{BP}(G)\) and \(BP^*(BG) \otimes_{BP^*} \mathbb{Z} (p) = H^{\text{even}} (BG)\) and hence \(Ch(G) = H^{\text{even}} (BG)\). However, \textit{I. Leary} and \textit{N. Yagita} [in Bull. Lond. Math. Soc. 24, 165-168 (1992; Zbl 0723.20029)] show that there are \(\text{rank}_ p(G) = 2\) groups for \(p \geq 5\) such that \(BP^* (BG) \neq Ch_{BP} (G)\) and \(H^{\text{even}} (BG) \neq Ch(G)\). These groups also give counter examples to the Atiyah conjecture about filtrations on the complex representation ring \(R(G)\). As in \textit{M. Hopkins}, \textit{N. Kuhn} and \textit{D. Ravenel} [Generalized group characters and complex oriented cohomology theories, preprint] let \(\text{Tre}(G)\) (resp. \(\text{Tre}_{BP} (G)\) and \(\text{Tre}_{K(n)}(G)\)) be the subring of \(H^* (BG)\) (the \(BP^*\)- submodule of \(BP^* (BG)\) and the \(K(n)^*\)-submodule of \(K(n)^* (BG)\)) generated by the transfer (corestriction) of the Euler classes of complex representations (where \(K(n)^* (-)\) is the Morava \(K\)- tgeory). In this very interesting paper the author proves: Theorem. Let \(G\) be a \(p\)-group with \(\text{rank}_ p G= 2\) and \(p \geq 5\). Then \(BP^*(BG) = \text{Tre}_ BP (G)\), \[ BP^* (BG) \otimes_{BP^*} K(n)^* \simeq K(n)^* (BG), \quad BP^*(BG) \otimes_{BP^*} \mathbb{Z}(p) \cong H^{\text{even}}(BG) \] and hence \(K(n)^* (BG) = \text{Tre}_{K(n)} (G)\), \(H^{\text{even}} (BG) = \text{Tre}(G)\). Moreover, he finds explicit generators of \(BP^* (BG)\) as a \(BP^*\)-algebra.
Chern classes of complex representations
scientific article; zbMATH DE number 562715