2014/04/30 by Holger Reich, Marco Varisco
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Equivalence (formal languages) #Equivariant map #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Isomorphism (crystallography) #Suspension (topology) #Weak equivalence #math.AT #msc:55P42 #msc:55P91 #n-connected
paper · pdf · doi:10.2140/agt.2016.16.1493
published as Algebr. Geom. Topol. 16 (2016), no. 3, 1493-1566 · Final version, to appear in Algebraic & Geometric Topology. 58 pages
arxiv created 2015/09/22 · openalex created_date 2016/06/24 · openalex publication_date 2016/07/01 · arxiv updated 2016/07/05 · openalex updated_date 2026/08/05
We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group G , any normal subgroup N of G , and any orthogonal G -spectrum X , we construct a natural map A of orthogonal G=N -spectra from the homotopy N -orbits of X to the derived N -fixed points of X , and we show that A is a stable weak equivalence if X is cofibrant and N -free. This recovers a theorem of Lewis, May and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize that our Adams map A is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by--spectra construction with good functorial properties, which we believe is of independent interest.