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5. Obstruction Theory I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY Our results on GCW approximation of Gspaces and on cellular approximation that these are welldened functors on the category hGU Sim of Gmaps imply ilarly we can approximate any pair X A by a GCW pair X A Less obvi ously if X A B is an excisive triad so that X is the union of the interiors of A and B we can approximate X A B by a triad X A B where X is the union of its sub complexes A and B That is all there is to the construction of ordinary equivariant homology and b erg cohomology groups satisfying the evident equivariant versions of the Eilen Steenro d axioms G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol S Illman Equivariant singular homology and cohomology Memoirs Amer Math So c No S J Willson Equivariant homology theories on Gcomplexes Trans Amer Math So c Obstruction theory Obstruction theory works exactly as it do es nonequivariantly and Ill just give n Recall that a connected space X is said to b e nsimple a quick sketch Fix if X is Ab elian and acts trivially on X for q n Let X A b e a relative q H b e a Gspace such that Y is nonempty connected GCW complex and let Y n and nsimple if H o ccurs as an isotropy subgroup of X n A Let f X A Y n We ask when f can b e extended to X Comp osing the attaching b e a Gmap n H S maps GH X of cells of X n A with f gives elements of Y These n elements sp ecify a welldened co cycle n C X A Y c f n G n n if and only if c A Y and f extends to X If f and f are maps X f n and h is a homotopy rel A of the restrictions of f and f to X A then f f and h together dene a map n Y hf f X I 0 Applying c to cells j I we obtain a deformation co chain f f h n 0 Y d C X A f f h n G 0 0 such that Moreo d c c ver given f and d there exists f that coin f f h f f n 0 d where the constant homotopy h is cides with f on X and satises d f f UNIVERSAL COEFFICIENT SPECTRAL SEQUENCES understo o d This gives the rst part of the following result and the second part is similar n n For f X A Y the restriction of f to X A extends Theorem n n to a map X A Y if and only if c in H X A Y Given maps f n G n n f f X Y and a homotopy rel A of their restrictions to X A there is n Y that vanishes if and only if the restriction of an obstruction in H X A n G n the given homotopy to X A extends to a homotopy f f rel A G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol Universal co ecient sp ectral sequences While easy to dene Bredon cohomology is hard to compute However we do t sp ectral sequences which we describ e next have universal co ecien Let W H b e the comp onent of the identity element of WH and dene a co e cient system J X by H J X GH H X W H Z X coincides with the obvious co ecient system H X if G is discrete Thus J We claim that if G is a compact Lie group then J X is the co ecient system X The p oint is that a Lie theoretic that is obtained by taking the homology of C argument shows that H H GK GK W H H We deduce that the ltration of X W H induced by the ltration of X gives rise X GH to the chain complex C We can construct an injective resolution Q of the co ecient system M and C X Q tial the sum of the This is a bicomplex with total dieren form Hom G X and of Q It admits two ltrations Using dierentials induced by those of C pq comes from the dierential on Q and E one of them the dierential on E pq X M Since C X is pro jective the higher Ext groups are zero C is Ext G and E reduces to C X M Thus E E H X M and our bicomplex G G computes Bredon cohomology Filtering the other way the dierential on E comes from the dierential on C X and we can identify E Using a pro jective we obtain an analogous homology sp ectral sequence resolution of N I EQUIVARIANT CELLULAR AND HOMOLOGY THEORY Theorem Let G b e either discrete or a compact Lie group and let X b e a GCW complex There are universal co ecient sp ectral sequences pq pq n H X M X M Ext J E G G and G G E Tor J X N H X N pq pq n We should say something ab out change of groups and ab out pro ducts in coho mology but it would take us to o far aeld to go into detail For the rst we simply note that for H G we can obtain H co ecient systems from Gco ecient sys G that sends H K to GK G H K For the tems via the functor H H second we note that for groups H and G pro jections give a functor from the orbit category of H G to the pro duct of the orbit categories of H and of G so that we can tensor an H co ecient system and a Gco ecient system to obtain an H Gco ecient system When H G we can then apply change of groups to the diagonal inclusion G G G The resulting pairings of co ecient systems allow us to dene cup pro ducts exactly as in ordinary cohomology using cellular approximations of the diagonal maps of GCW complexes G Bredon Equivariant cohomology theories Springer Lecture Notes in Mathematics Vol S J Willson Equivariant homology theories on Gcomplexes Trans Amer Math So c CHAPTER II ostnikov Systems Lo calization and Completion P Eilenb ergMacLane Gspaces and Postnikov systems Let M b e a co ecient system An Eilenb ergMac Lane Gspace K M n is a Gspace of the homotopy typ e of a GCW complex such that M if q n K M n q if q n While our interest is in Ab elian groupvalued co ecient systems we can allow M to b e setvalued if n and groupvalued if n I will give an explicit construc tion later Ordinary cohomology theories are characterized by the usual axioms and by checking the axioms it is easily veried that the reduced cohomology of based Gspaces X is represented in the form n X K M n X M H G G are understo o d where homotopy classes of based maps in hGT Recall that a connected space X is said to b e simple if A is Ab elian and acts trivially on X for n More generally a connected space X is said to n b e nilp otent if X is nilp otent and acts nilp otently on X for n A n H Gconnected Gspace X is said to b e simple if each X is simple A Gconnected H Gspace X is said to b e nilp otent if each X is nilp otent and for each n H H the orders of nilp otency of the X groups X have a common b ound n We shall restrict our sketch pro ofs to simple Gspaces for simplicity in the next few sections but everything that we shall say ab out their Postnikov towers and ab out lo calization and completion generalizes readily to the case of nilp otent G spaces The only dierence is that each homotopy group system must b e built up I I POSTNIKOV SYSTEMS LOCALIZATION AND COMPLETION in nitely many stages rather than all at once APostnikov system for a based simple Gspace X consists of based Gmaps X X and p X X n n n n n X X for n such that X is a p oint induces an isomorphism n n q q for q n p and p is the Gbration induced from the path space n n n n n bration over a K X K by a map k X n X n It n n n n Our requirement X and that X for q follows that X K n q that Eilenb ergMac Lane Gspaces have the homotopy typ es of GCW complexes ensures that each X has the homotopy typ e of a GCW complex The maps n induce a weak equivalence X lim X but the inverse limit generally will n n not have the homotopy typ e of a GCW complex Just as nonequivariantly the k invariants that sp ecify the tower are to b e regarded as cohomology classes n n k H X X n n G These classes together with the homotopy group systems X sp ecify the weak n homotopy typ e of X On passage to H xed p oints a Postnikov system for X H gives a Postnikov system for X We outline the pro of of the following standard result since there is no complete published pro of and my favorite nonequivariant pro of has also not b een published The result generalizes to nilp otent Gspaces Theorem A simple Gspace X of the homotopy typ e of a GCW complex has a Postnikov tower Pr oof Assume inductively that X X has b een constructed By the n n homotopy excision theorem applied to xed p oint spaces we see that the cob er C is n connected and satises n C X n n n More precisely the canonical map F C induces an isomorphism on n n for q n We construct q j C K X n n n by inductively attaching cells to C to kill its higher homotopy groups We n n take the comp osite of j and the inclusion X C to b e the k invariant k n n n By our denition of a Postnikov tower X must b e the homotopy b er of k n x consisting of a path I K Its p oints are pairs X n and a n SUMMARY LOCALIZATIONS OF SPACES n p oint x X such that and k x The map p X n n n X is given by p x x and the map X X is given by n n n n j x t x t b eing a p oint on x x x where xt n n the cone CX C Clearly p It is evident that induces an n n n n n isomorphism on for q n and a diagram chase shows that this also holds for q q n Summary lo calizations of spaces ariantly lo calization at a prime p or at a set of primes T is a standard Nonequiv rst step in homotopy theory We quickly review some of the basic theory Say that a map f X Y is a T cohomology isomorphism if f H Y A H X A is an isomorphism for all T lo cal Ab elian groups A are equivalent Theorem The following prop erties of a nilp otent space Z When they hold Z is said to b e T lo cal a Each Z is T lo cal n b If f X Y is a T cohomology isomorphism then f Y Z X Z is a bijection c The integral homology of Z is T lo cal Theorem Let X b e a nilp otent space
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