where p is the natural map onto the unit interval. We show that the f.pdf

Waldhausens Nil Groups andContinuously Controlled K-Theory.Hans J. Munkholmy Stratos PrassidiszxDecember 18, 1997AbstractLet = 1 G 2 be the pushout of two groups i, i = 1; 2; over a commonsubgroup G, and H be the double mapping cylinder of the corresponding diagram ofclassifying spaces B1 BG ! B2. Denote by  the diagram I p H 1! X = H ,where p is the natural map onto the unit interval. We show that the fNil groupswhich occur in Waldhausens description of K(Z) coincide with the continuouslycontrolled groups eKcc (), dened by Anderson and Munkholm. This also allowsus to identify the continuously controlled groups eKcc (+) which are known to forma homology theory in the variable , with the \homology part in Waldhausensdescription of K1(Z). A similar result is also obtained for HNN extensions.1 RecollectionsLet the group be a pushout as in the abstract. By rst passing to the correspondingpushout of integral group rings, next applying Theorem 1 of [8], and nally taking homo-topy groups, we arrive at the following version of Waldhausens result concerning K(Z)Theorem 1.1 (Waldhausen [8]) For a group as above, there is a chain complex ofabelian groups


! Kj(ZG)! Kj(Z1)Kj(Z2)! Kj(Z)! Kj1(ZG)!


;which is exact except that at each Kj(Z) the homology is fNilj1(ZG;B1; B2) where Bi =Z[iG] as a ZG bimodule (i = 1; 2).IMADA, Odense University, Campusvej 55, DK{5230 Odense M, Denmark (hjm@imada.ou.dk).yPartially supported by SNF (Denmark) under contract no. 9502188.zDept. of Math., Vanderbilt University, Nashville, TN 37240, USA (prassie@).xPartially supported by SNF (Denmark) under contract no. 9502188, by NSF (USA) under grant DMS-9504479, and by a Vanderbilt University Summer Research Fellowship.1 H BpXq HXcyl(q) c(B) BFigure 1The continuously controlled K-theory of Anderson and Munkholm is dened in section7 of [4] as a (spectrum valued) functor eKcc : T OP=CM ! SPEC. Here T OP=CM isthe category of \diagrams of holink type, i.e