By Glen E. Bredon
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Additional resources for Equivariant Cohomology Theories
Ffi 0 f o r L § Y has K' without general Theorem H -Gr § that k: = k *(Tn§ variant then ~n+l(f-k) (This If on t h e cochain level). 5) to and equivariant) defined. 11) also ~ f[L = g[L, Let then argument Theorem If we a l s o and if f,g: have that K § Y are f and g are e q u i v a r i a n t l y to L) iff n ( f , g ) homo- = 0. now p r o v e s the following (Classification). Assume result: that (1) holds that ~ H (K,L;&r(Y)) ~Pl (1,S,4) = 0 for n < r < dim(K-L) A standard and (Homotop~v~. r-1 = 0 = HG ( K , L ; ~ r ( Y ) ) (K,L;~r(u r+l HG ( K , L ; ~ r ( Y ) ) 0 f: K § Y be an e q u i v a r i a n t classes (relative one-one correspondence to L) map.
F T ~ ~ (SqT +) ~ ~ (Y)}. q q Proof. of . 5). ). ~ (via F T - K n Uf C S n R +. 6) we have ~ = 0 for r < n. r If K q has b e e n constructed (q > n + 1) such of d i m e n s i o n 0 < r n(K q) ~ "'lKqj = r let V be a G-set such that for r there FV 2C+ ~ q Kq~vsqV + . Then, by by q + I. 6), q 7. The m e t h o d We shall in the use following type K q+l This of k i l l i n g it h e r e n < q (Kq). 7) is c l e a r l y y and with let K q+l = q replaced of type a G-complex G-complexes the section map satisfies n-connected last and groups is, in a r a t h e r ~ q used of course, in the construction an i m p o r t a n t straightforward way tool.
A G-complex of Chap. shall always to c o n s t r u c t n > i. (~,i) on the w map ~ * : HnG ( K ' ; ~ ) ~ Proposition how addition. 1) of ~ e ~ G and n ~ convenience [[fur,]]. we c a n ~ for any First we a r e as well of = cellular) way, We shall (~,n) if additivity in [[f,]] correspondence case, necessarily obvious + This when following our is not K has two K of type attention, much type loss (5,2). lemmas which use 10. with some a disjoint of T + (that member further is, of T), the and notation.