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By Glen E. Bredon

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Example text

Be homotopy 69 type cone obtain pp. to shown type be here. we c o m b i n e a long exact § z [[sny~z]] In (or this G-spaces) collection section with Y = {YnlneZ} with points. of that the SX The 299-544). 1) and the C. has with the this pro- sequence ~ § [[snx;z]] -,.... G-spectra we work base sequence as does [[S n-I 3. of f canonical Z, the for Y - ~ Cf § C 39 ~ SX and c o n t i n u e sequence § mapping two [[X~Z]] exact. homotopy to f i n a l l y w i t h base map b e t w e e n reduced f# , give the d e t a i l s As in ...

If 9 dim(K-L) extension we a r e f[L and equi- we h a v e not to two These we a l s o then an K iff equivariant 7n+l(f) equivariant induce have ffi 0 . maps an e q u i v a r i a n t map Q = (KX~I) U (LXI). ) an e q u i v a r i a n t Now s u p p o s e § K is (Extension). 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.

2) taking F(XAY,Z) ~-+ F(X,F(Y,Z)) f into f defined by (f(x))(y) = f(XAy). 2) is e q u i v a r i a n t . 63 clearly X = X G. SX = S ^ X is a G - s p a c e , trivial. Similarly, Thus as above. 53. we h a v e loop m u l t i p l i c a t i o n (see Chap. II, w and e l e m e n t a r y , that In p a r t i c u l a r , the is homotopies. 6) preserves equivariant before, equivariant homotopy we h a v e the on on F(SX,Y) of H o p f - s p a c e s . It pro- F(X,flY). 5) ~ reduced on the on X, so that In p a r t i c u l a r , E(X,F(Y,Z)) Now the sets means compact.

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Equivariant Cohomology Theories by Glen E. Bredon


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