The Wirthmüller isomorphism revisited. (English) Zbl 1025.55004

May, J. Peter

Theory Appl. Categ. 11, 132-142 (2003).

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.

Reviewer: Rainer Vogt (Osnabrück)

MSC:

55P42	Stable homotopy theory, spectra
55P91	Equivariant homotopy theory in algebraic topology

Keywords:

equivariant stable homotopy category; \(G\)-spectra; Wirthmüller isomorphism; Grothendieck isomorphism; Adams isomorphism

Citations:

Zbl 1042.18008