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The Wirthmüller isomorphism revisited. (English) Zbl 1025.55004
In [ibid. 11, 107-131 (2003; Zbl 1042.18008)] H. Fausk, P. Hu and J. P. May proved a formal Wirthmüller and a formal Grothendieck isomorphism theorem. In the present paper the author shows that the Wirthmüller isomorphism theorem in equivariant stable homotopy theory is a fairly simple consequence of the formal one.
While the Wirthmüller isomorphism relates the categories of \(G\)-spectra and \(H\)-spectra for \(H\subset G\), the Adams isomorphism relates the categories of \(G\)-spectra and \(G/N\)-spectra. The setting of the Adams isomorphism satisfies the “formal hypotheses” of the Wirthmüller isomorphism theorem, but the conclusion fails, because an additional hypothesis does not hold.
Finally, the author notes that a change of universes leads into a setting where the formal Grothendieck isomorphism theorem may apply, but that again one of the hypotheses is violated.