The total surgery obstruction by Andrew Ranicki, Princeton University Let n~ 5. According to the Browder-Novikov-Sullivan-Wall theory of surgery ([BI],[B2],[N],[Sul],[WI]) a finite n-dimensional Poincar6 complex X is homotopy equivalent to a compact topological manifold if and only if i) the Spivak normal flbration~x:X ~BG(k) (k~n) admits a topological ~x:X ~BTOP(k), in which case topological transversality applied reduction to a degree I map ~x:S n+k.o ~T(~ x) gives a topological manifold N n = ~xI(X)Csn+k and a map of topological bundles b:~M----~ X covering the degree I map f = ~X 1 : M ~X, and hence a surgery obstruction ~(f,b)~Ln(E1(X)) ii) there exists a topological reduction~ x such that 6(f,b) = O, in which case the normal map (f,b):M ~X is normal bordant to a homotopy equivalence. The theory was initially developed in the smooth and PL categories; the extension to the topological category is due to Kirby and Siebenmann ([KS]). We present here the preliminary account of a theory which replaces the two-stage obstruction with a single invariant, ~the total surgery obstruction'. We shall only consider the oriented case, but in principle there exists an unoriented version involving twisted coefficients. For the sake of the s-cobordism theorem we shall be working with simple homotopy types and the Wall LS-groups, but there is also an ordinary homotopy version which we discuss briefly • I at the end. Thus Polncare complexes will be finite, simple and oriented; manifolds will be compact, topological and oriented. The invariant lies in one of the groups ~.(X) (defined for any space X) appearing in an exact sequence of abelian groups • .. ~ Hn(X;_mO) ~. ~Ln(~I(X)) ~n (x) ~ Hn_I(X;_~O) ~... , where -~O is a 1-connective~-spectrum with Oth space homotopy equivalent to G/TOP and ~. is a universal assembly map. Both -~0 and G. were originally constructed by Proceedings 1978 Arhus Topology Conference, Springer Lecture Notes 763, 275-316 (1979)