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By Lowell Jones

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Note {g-} k+ is contained in the image of the map k lim [S *,T (BSG) A (e/8e)] - lim [S k + *,T k (BSG)A(R/R)] which is induced k-*» . k-*» by the inclusion e/3e c R/R, because the latter inclusion induces an isomorphism on Z -homology groups. Moreover, if BSG * denotes the k k n*-primary component of BSG, the inclusion of spectra T (BSG *) -> T (BSG) is a homotopy equivalence mod n-torsion; so {g-} * is contained in the image of lim [S k+ *,T k (BSG * ) A ( R / R ) ] - lim [S k + *,T k (BSG) A (R/R)].

Outline of Step 1. Let TT denote a finitely presented group. The surgery classifying spaces ^ P (TT) have been defined by F. Quinn [24]. A good reference for these spaces is [[5], pg. 90-94]. 5]. These spaces are useful in this paper because of the following lemmas. Let C. e L f(Z ) denote the subset over which surgery has been completed on the universal surgery problem. space. Note CQ is a contractible For any space X, with base point x , [(X,x ) , (L (Z ),C )] will denote the homotopy classes of maps from (X,x ) to (L (7L ) ,C ) .

F ). _)*£_. where B ( f ) is the base space of ? defined on [[9], pg. 490], and all the block spaces R>£0>fl have K** ^ for base space. 14(a) do make sense. # # Now we continue with the detailed description of (c (t),6c (t)): (c # (Ip,6c # (^)) - (c#(f0),6c#(50)). 14 (a) we get a definition o£ # c (C 0 ),6c (^). (c) The pull-back c # (t): c#(f^) -> c #(f Q) is defined to be the — — # — # restriction of 1 *t: n*£' + n*^ to c (£0) , and 6c (t) is the restriction of l 6n *t: 6n*ro - n*? 0 to 6 c # ( ^ ) .

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