XX THE SEGAL CONJECTURE explain in the next section we can analyze the singular terms in in terms of these sub quotient theories Assume that k is einvariant for every prop er sub quotient J of Theorem J and let Y E P G i If G is not elementary Ab elian then k Y EG G r r r ii If G Zp then k Y EG is the direct sum of p copies of G r k S GG arning the nonequivariant theory k is usually quite dierent from the W GG underlying nonequivariant theory k k e As we shall explain in Section we can use Adams sp ectral sequences to analyze the free terms in Theorem Assume that k is split and k is b ounded b elow and let Y G E P Y EG i If G is not elementary Ab elian then k G r k is nite Y EG is the ii If G Zp and H dimensional then k G r r r direct sum of p copies of k S The hyp othesis that H k b e nite dimensional in ii is extremely restrictive although it is satised trivially when k is the sphere sp ectrum The hyp othesis are is actually necessary We shall see in Section that the theories B G G einvariant for nite groups They satisfy all other hyp otheses of our theorems are dierent In such cases the calculation of k Y EG but here k and k G GG falls out from the einvariance which must b e proven dierently and reduction is now the case of the following immediate inductive Carlssons G consequence of the rst parts of Theorems and Theorem Supp ose that G is not elementary Ab elian Assume i k is einvariant for all elementary Ab elian sub quotients J J ii k is split and k is b ounded b elow for all nonelementary Ab elian sub J KK quotients J H K Then k is einvariant for all sub quotients J including J G J to cohomotopy and the pro of of the Segal conjecture it only remains Returning r to prove that the map in is an isomorphism when G Zp We assume that the result has b een proven for q r Comparing Theorems and we see that the map in is a map b etween free mo dules on the same APPROXIMATIONS OF SINGULAR SUBSPACES OF GSPACES numb er of generators It suces to show that is a bijection on generators which means that it is an isomorphism in degree r Here is a map b etween free mo dules on the same numb er of generators over the padic integers Z so that it p will b e an isomorphism if it is a monomorphism when reduced mo d p EG H F where H F is the Eilenb ergMacLane G To prove this let k F p G p sp ectrum asso ciated to the constant Mackey functor at F that we obtain from p This theory like any other theory represented by a function sp ectrum IX G F EG is einvariant Since H F F we have a unit map S H F p G p p and we comp ose with H F S k There is an k to obtain G G G p induced map S S k and a little calculation shows that it sends the GG GG unit in S to an element that is nonzero mo d p We can also check that the are all einvariant By sub quotient theories k the naturality of we have the J commutative diagram r r Y EG Y EG G G r r k Y EG Y EG k G G Y The left map is The b ottom map is an isomorphism since k G r r r the sum of p copies of S k and is therefore a GG monomorphism mo d p Thus the top map is a monomorphism mo d p and this concludes the pro of J H Gunawardena and H Miller The Segal conjecture for elementary Ab elian J F Adams pgroupsI Top ology G Carlsson Equivariant stable homotopy and Segals Burnside ring conjecture Annals Math J Caruso J P May and S B Priddy The Segal conjecture for elementary Ab elian pgroupsI I Top ology Approximations of singular subspaces of Gspaces Let SX denote the singular set of a Gspace X namely the set of p oints with nontrivial isotropy group The starting p oint of the pro of of Theorem is the space level observation that the inclusions SX X and S EG XX THE SEGAL CONJECTURE induce bijections S X EG X X EG X S X X G G G We may represent theories on nite GCW complexes via colimits of space level homotopy classes of maps The precise formula is not so imp ortant What is imp ortant is that when calculating k X EG we get a colimit of terms of the G general form SW Z We can replace S here by other functors T on spaces that G satisfy appropriate axioms and still get a cohomology theory in X called k X T G transformations T T induce Such functors are called S functors Natural maps of theories contravariantly We have a notion of a cobration of S functors and cobrations give rise to long exact sequences In sum we have something like a cohomology theory on S functors T that approximates the singular functor S We construct a ltered S functor A Let A A G b e the partially ordered set of nontrivial elementary Ab elian subgroups of G thought of as a Gcategory with a map A B when B A e the classifying space B A is Gcontractible with G acting by conjugation If G In fact if C is a central subgroup of order p then the diagram A AC C displays the values on an ob ject A of three Gequivariant functors on A together with two equivariant natural transformations b etween them these induce a G homotopy from the identity to the constant Gmap at the vertex C we construct a top ological We can parametrize A by p oints of SX Precisely A Gcategory A X whose ob jects are pairs A x such that x X there is a mor if B A and y x and G acts by g A x g Ag g x phism A x B y Pro jection on the X co ordinate gives a functor A X SX where SX is a cate gory in the trivial way and B A X BSX SX is a Ghomotopy equivalence of B A X is Gcontractible Let AX B A X B A We The subspace B A still have a Ghomotopy equivalence AX SX but now A is an S functor and our equivalences give a map of S functors F or any space Y we have k Y A k Y S k Y EG G G G The functor A arises from geometric realizations of simplicial spaces and carries the simplicial ltration F A here F A and F A A where r rank G q r Insp ection of denitions shows that the successive sub quotients satisfy q A F AF AX G X q q H AN INVERSE LIMIT OF ADAMS SPECTRAL SEQUENCES Here runs over the Gconjugacy classes of strictly ascending chains A A q of nontrivial elementary Ab elian subgroups of G H is the isotropy group of namely fg jg A g A i q g and A A For each normal subgroup i i q K of a subgroup H of G there is an S functor C K H whose value on X is K G X and as S functors H q F AF A C A H q q By direct insp ection of denitions we nd that for any space Y K k Y k Y C K H H K G This is why the xed p oint functors enter into the picture To prove Theorem we restrict attention to Y E P If G is not elementary K Ab elian then Y is contractible and the sub quotients H K are prop er for all r pairs K H that app ear in If G Zp and q r this is still true r All these terms vanish by hyp othesis If G Zp we are left with the case q r Here A H G for all chains there are pp chains G and Y S Using Theorem follows G Carlsson Equivariant stable homotopy and Segals Burnside ring conjecture Annals Math J Caruso J P May and S B Priddy The Segal conjecture for elementary Ab elian pgroupsI I Top ology An inverse limit of Adams sp ectral sequences We turn to the pro of of Theorem Its hyp othesis that k is split allows G us to reduce the problem to a nonequivariant one and the hyp othesis that the underlying nonequivariant sp ectrum k is b ounded b elow ensures the convergence of the relevant Adams sp ectral sequences We prove Theorem by use of a particularly convenient mo del Y for E P namely the union of the Gspheres nV S where V is the reduced regular complex representation of G It is a mo del G H since V fg and V for H P V In general for any representation V there is a Thom sp ectrum BG Here we V BG may think of V as the negative of the representation bundle EG G regarded as a map V BG BO Z If V is suitably oriented for example V if V is complex there is a Thom isomorphism showing that H BG is a free H BGmo dule on one generator of degree n where n is the real dimension v of V We take cohomology with mo d p co ecients For V W there is a W V V W map f BG BG such that f H BG H BG carries XX THE SEGAL CONJECTURE to W V Here V H BG is the Euler class of V which is the v w Euler class of its representation bundle For a split Gsp ectrum k we have an G isomorphism V G V k BG k S EG W V For V W the map f k BG k BG corresp onds under the V W isomorphisms to the map induced by e S S The pap er of mine cited at the end gives details on all of this With our mo del Y for E P we now see that q nV G lim k BG Y EG k Y EG k q G q Rememb er that we are working padically we complete sp ectra at p without change of notation The inverse limit E of Adams sp ectral sequences of an inverse r sequence fX g of b ounded b elow sp ectra of nite typ e over the padic integers Z n p converges from E Ext colim H X F A n p nV to lim X With X k BG this gives an inverse limit of Adams sp ectral n n sequences that converges from nV colim H BG F E Ext H k p A to k Y EG The colimit is taken with resp ect to the maps G nV nV V H BG H BG H Since V fg V restricts to zero in H BH for all H P A theorem of implies that V must b e nilp otent if G is not elementary Ab elian and Quillen this implies that E This proves part i of Theorem r r p Now assume that G Zp Let L BG Then V H nV colim H BG H BGL It is easy to write L down explicitly and the heart of part ii is the