0. The Main Results, Applications, and some History and Preview s Let G be a compact Lie group and £ the category of smooth G mani- folds. If Y € &, Y denotes its underlying homotopy type. Much of the i m - portant work in the subject of transformation groups deals with the basic project of realizing invariants within a homotopy type. This means an invariant I( . ) defined on £ i s given and sought i s a description of the values values obtained in the set {l(Y) | Y c Jfr 9 Y= M} . Here M is some fixed homotopy type, e . g . , a sphere or a disk. As examples of l(Y), consider: Y the G fixed set, {TY | x c Y } the isotropy representations at fixed G points, and dim Y the dimension of the fixed set. Henceforth G i s f i n i t e . The main result of this paper is Theorem 7A. It provides a tool for treating the basic project. Theorem 7A gives the exact G surgery sequence 0.1 h S G ( Y ' X J ~^~* N G ( Y ' X } ~ ~ ^ I ( G X )# The sequence reduces to the exact Wall sequence [Wl] when G acts freely on Y and Y i s 1-connected. Roughly, the element s of hS (Y,X) consist of equivalence c l a s s e s of pairs (X,f) with X € £ and f:X -* Y a pseudoequivalence i. e. , a G map whose underlying map is a homotopy equivalence. Then {I(X) | (X,f) € hS G (Y,X) , Y= M} contributes to the values of l( . ) on manifolds homotopy equivalent to M. Here are three specific applications which fit into this context. v i i

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