Title: Equivariant infinite loop space theory, I. The space level story,
Abstract: 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
naive version which gives naive G-spectra for any topological group G, but our
focus is on the construction of genuine G-spectra when G is finite.
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 naive
G-spectra for general G but fails hopelessly for genuine G-spectra when G is
finite. Even in the nonequivariant case, our comparison theorem is considerably
more precise, giving a 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 some
corrections to results in the literature.