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Transchromatic generalized character maps. (English) Zbl 1300.55011
Given a finite group \(G\) one of the most classical results in representation theory (and thus \(K\)-theory) says that associating to a complex representation of the group its character induces an isomorphism \(L \otimes R(G) \to Cl(G,L)\); here \(L\) denotes a minimal field of characteristic zero containing all roots of unity, \(R(G)\) denotes the complex representation ring, and \(Cl(G,L)\) denotes the class functions with values in \(L\). On the other hand there is the classical Atiyah-Segal completion theorem stating that the Borel \(K\)-cohomology group \(K^0(EG\times_G X)\) is isomorphic to \(K_G^0(X)\otimes_{R(G)} R(G)\widehat{_I}\), the tensor product of the equivariant \(K\)-cohomology of \(X\) and the completion of the represention ring with respect to the augmentation ideal. The two statements in fact can be combined, and in [J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)], M. J. Hopkins et al. proved a generalization of the combined statement for a large class of complex oriented cohomology theories \(E\), e.g. like \(K\)-theory, Morava \(E\)-theories \(E_n\) or the Lubin-Tate-theories \(E\widehat{_n}\) (see Theorem C, ibid.), namely that there is an isomorphism \(L(E^{*}) \otimes_{E^{*}} E^{*}(EG \times_G X) \cong Cl(G,X; L(E^*))\) where \(L(E^*)\) denotes a suitable ring constructed from the cohomology theory \(E^{*}\), and \(Cl(G,X; L(E^*))\) is a ring of generalized class functions with values in \(L(E^*)\) built from \(X\). The result can be viewed as a generalization of the classical statements for \(K\)-theory, which is a complex oriented cohomology theory of height 1, to corresponding statements for complex oriented cohomology theories of higher height.
In his paper Stapleton now constructs for Morava \(E_n\)-theory and any \(0\leq t <n\) a complex oriented cohomology theory \(C_{n,t}^{*}\) of height \(t\) together with an according character isomorphism \(C_{n,t} \otimes_{E_n^{*}} E_{n}^{*}(EG \times_G X) \cong C_{n,t}^*(X)\) where \(C_{n,t}\) is a suitable ring over \(E_n^*\). And indeed, the isomorphism of Kuhn, Hopkins and Ravenel coincides with the special case of \(t=0\). One major input for getting this result is an investigation of the formal group \(\mathbb{G}\) obtained from the formal group \(\mathbb{G}_{E_{n}}\) of \(E_n\) by extending scalars along the homomorphism \(\pi_0 E_n \to \pi_0 L_{K(t)}E_n\) induced by the Bousfield localization with respect to Morava \(K(t)\)-theory. The author shows that \(\mathbb{G}\) fits into a short exact sequence of \(p\)-divisible groups \(\mathbb{G}_0 \to \mathbb{G} \to \mathbb{G}_{\acute{e}t}\), where \(\mathbb{G}_0\) is formal and \(\mathbb{G}_{\acute{e}t}\) is étale, and that the short exact sequence splits into a formal group of height \(t\) and a constant height \(n-t\) étale \(p\)-divisible group after extending scalars from \(E_{n}^{*}\) to \(C_{n,t}\). In fact, \(C_{n,t}\) is essentially characterized by the property that it is initial with that property. After the splitting has been established the author uses decomposition techniques for \(G\)-spaces similar to those used in [ibid.] to finally obtain his desired generalized character isomorphism.

55N91 Equivariant homology and cohomology in algebraic topology
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
Full Text: DOI arXiv
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