2017/04/11 by J. P. May, May, J. Peter, Mona Merling +3 · 1 citation
Mathematics · Physics and Astronomy · #18E30 #55P42 #55P43 #55P48 #55P91 (Primary) 18A25 #55U35 (Secondary) #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.1704.03413
openalex publication_date 2017/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We rework and generalize equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. There is a classical version which gives classical Ω-G-spectra for any topological group G, but our focus is on the construction of genuine Ω-G-spectra when G is finite. We also show what is and is not true when G is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical G-spectra for general G but fails for genuine G-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.