A thickening, in the topological category, of a complex K is an equivalence class of simple homotopy equivalences gO:K--M, where M is a topological manifold with boundary. Here it is shown that for stable thickenings (dim M>>dim K), the set 3(K) of stable thickenings is in 1-1 correspondence with homotopy classes of maps of K into BTop. Wall [1] and Mazur [2] have studied a "functor" of complexes called a thickening. Given a complex K, a thickening of K is essentially an m-manifold M which is homotopy equivalent to K. The set of these, under a suitable equivalence relation, forms a set 3m(K). It is clear that this kind of construction can be done in the differentiable, piecewise-linear, or topological categories. In [I ] and [2 ] it is shown that the stable thickenings of a complex K are a representable functor, i.e. if we denote the stable thickenings of K by 3(K), then we have 3(K)[K, BO] in the smooth category and 3(K) : [K, BPL] in the piecewise-linear category. In this note we establish the analogous result for the topological category. In a subsequent paper, we will give an analogous result for the homotopy category. 1. Definition. We are able to use the same definition as Wall. Let K be a finite complex of dimension k with basepoint *, and q:KM, a simple homotopy equivalence of K into a compact topological manifold-with-boundary of dimension m, m_k+3. The notion of simple homotopy equivalence is well defined in the topological category since, by Kirby-Siebenmann [3], every compact topological manifold has the homotopy type of a finite complex. We require that the basepoint * of M lie in dM and that the inclusion i:dMCM induce an isomorphism i*:7r1(OM)-7r1(M), and that the tangent space of M at * be oriented. Then we say that the pair (M, 4) defines a pre-m-thickening of K. Define two pre-thickenings (M1, q1), (M2, qt2) of K to be equivalent, if there is a (topological) homeomorphism h: Ml-*M2, preserving * and the given orientations of the tangent space there, such that Received by the editors October 22, 1969. AMS 1970 subject classifications. Primary 57A15; Secondary 55D15, 57CI0.