A Comparison of Norm Maps

We present a spectrum-level version of the norm map in equivariant homotopy theory based on the algebraic construction in the 1997 paper by Greenlees and May. We show that this new norm map is the same as the construction in the 2009 paper by Hill, Hopkins and Ravenel. Our comparison of the two norm maps gives a conceptual understanding of the choices inherent in the definition of the multiplicative norm map. This paper serves to contextualize and explain the construction of the norm maps in equivariant stable homotopy theory. The norm map should be thought of as a sort of multiplicative induction functor that takes H–equivariant spectra to G–equivariant spectra, where H is a finite index subgroup of a compact Lie group G. We compare the definition of the norm map used in the recent solution to the Kervaire invariant one problem [3] to the norm map implicit in earlier work of Greenlees and May [2]. Greenlees and May define an algebraic norm map on homotopy groups, whereas Hill, Hopkins and Ravenel have a spectrum-level construction that agrees with the Greenlees–May construction on the algebraic level. We give a spectrum-level version of the Greenlees–May construction and compare the two constructions at the spectrum level. Our comparison gives a new conceptual understanding of the choices inherent in the definition of the multiplicative norm map. First, note that we will use orthogonal spectra and push all of the choice of universe issues into choice of model structure. It follows from [7, Theorem V.1.7] that we can thus get away with thinking of G–spectra as simply G–objects in the category of spectra. In turn, we will think of these as covariant functors from the one-object category G into the category S of spectra. Let G be a compact Lie group and H < G be a subgroup such that [G : H] = n. Let BG/HG be the translation groupoid of G acting on G/H. That is, BG/HG has objects x ∈ G/H and morphisms x g −→ gx for all g ∈ G. If we think of H as a one-object category, we have an inclusion of groupoids ι : H ↪→ BG/HG given by sending the object ∗H of H to the identity coset. Since BG/HG is connected and the endomorphisms of eH ∈ G/H are just H, this is an equivalence of categories. Received by the editors January 27, 2012 and, in revised form, April 30, 2012. 2010 Mathematics Subject Classification. Primary 55P91, 55P42; Secondary 18D30. The first author thanks MSRI for hosting the Hot Topics: Kervaire Invariant workshop of October 2010, which inspired this research. The second author was supported by an NSF graduate research fellowship and by an NSF Mathematical Sciences Postdoctoral Research Fellowship. c ©2014 American Mathematical Society 1413 Licensed to Mathematical Sciences Research Institute. Prepared on Fri Apr 18 19:09:46 EDT 2014 for download from IP 198.129.65.19. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1414 ANNA MARIE BOHMANN Definition 1. The Hill–Hopkins–Ravenel norm map is the following composite: