Andrew H Wallace's An introduction to algebraic topology PDF

By Andrew H Wallace

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Chapter Generalized Equivariant Cohomology In this chapter we show how to construct equivariant cohomology how any generalized to the "classical" I. theories, theory generalized using G-spectra. is connected by a spectral theory of Chapter I. Equivariant cohomology via G-spectra We work with the category in this section. We then show sequence of spaces with base points Let Y be a G-spectrum. 2) locally compact = -n CECX,Y)) --Note invariant that under ] = limk[SkX~Ym+k] ]. then this is same as = l i m k = k - n (E(X'Yk))" [[skX~Yn+k ] ] = [[x~kYn+k]].

Discussion. 4} (coinciding with an isomorphism if each Hq(K,L;Z) For example stationary points Hurewicz h o m o m o r p h i s m this the for q < n. = 0 for q < n. (in (obvious} eG) § --qH(K;Z) for 0 < q < n. Thus case. g. k0) and that theorem, is an isomorphism L = ~). 5} when now j u s t i f y projectives denote assignment the in set of f + f(K) our earlier e G. For contention any equivariant clearly yields G-sets maps that S and S § T. a one-one there T let For K C G, correspondence E(GIK,S) % SK.

Proof. characteristic = c9 = c~ . F is constant, Let with = Bnx(o) usn-lxI is we p u t of d 0 = d F, O. 3) If maps Kn UL § Y an e q u i v a r i a n t FO)(x,1) d~,F,@(o) is equivariant § Y by The d e f o r m a t i o n It and O a r e § Y h e an e q u i v a r i a n t L and O[K n - I U L . ~r denote that K, n o t map such that in L, and d~, 0 = d. and choose : ( B n , S n - l ) § (Kn,K n - l ) f o r a . Let o G C B n x l a n d d e f i n e u j n § y o by u a = ~(fo(x)). As s h o w n i n n o n - e q u i v a r i a n t obstruction theory, u may G b e e x t e n d e d t o a map u ~(Bn• § y o representing the element G ( o r a n y e l e m e n t ) d ( o ) ~ ~ (Y o ) .

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An introduction to algebraic topology by Andrew H Wallace


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