Equivariant stable homotopy theory by L. Gaunce Jr. Lewis, J. Peter May, Mark Steinberger, J.E.

By L. Gaunce Jr. Lewis, J. Peter May, Mark Steinberger, J.E. McClure

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Unless t h e indexing universe i s G - t r i v i a l , t h e component spaces of U'. J -isotropict1. 8. 3 -objects a r e l i t e r a t u r e . 10). a?. It reads a s follows on t h e spectrum l e v e l . 2. and F i s a weak 3 -equivalence of G-spectra, e: E + F If [ D , E ] +~ [D,FIG i s an isomorphism f o r every 3 -CW e,: a r e themselves 3 C W s p e c t r a , then e Examples of f a m i l i e s a r e legion. spectrum then If D. E and i s a G-homotopy equivalence. We introduce n o t a t i o n s f o r those of i n t e r e s t t o us here.

Complex $1. Change of universe functors Let U and U' be G-universes and l e t f : U + U' have observed t h a t t h e r e i s a change of universe functor be a G-linear isometry. We f*: GdU' s p e c i f i e d by l e t t i n g 4 . i s an isomorphism; f o r It i s v i t a l t o our work t h a t E' ( ~ v ) * fwifV s f* E' ( fw) -%a . 5. 3. f , w-l(kf ) This gives t h e adjunction, and t h e l a s t statement i s c l e a r . , even when k ' : f*D + D' ( f*E1 ) ( v ) = E' ( f v ) , with s t r u c t u r a l maps ( f v ) = E' (fv)nsW-' Z"~E' w Then GLU The prespectrum l e v e l functor f a i l s t o be an isomorphism.

The pullback The intersection of subobjects Dl and Dl1 of D may be described as Appendix. 1. D1l - Dl1 1 -i A cofibration of spectra is a spacewise closed inclusion. Dl This applies in particular to the inclusion of a subcomplex in a CW-spectrum. D To answer our questions, we need the following kinds of prespectra. 2. (i) A prespectrum D o :CWmV~v + DW is a closed inclusion. prespectra, but not conversely as the (li) A prespectrum D is an injection is a C-inclusion prespectrum if each Note that C-inclusion prespectra are inclusion A map f:D + Dl is a closed inclusion if and only if there is a pair of maps Dl *DD" such that the diagram example of actual spectra makes clear.

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