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Let K be a simplicial complex. The realization IKI of K admits the metric di (x, y) = x(v) -y(v) I, vEKO where x(v) and y(v), v E KO, are the barycentric coordinates of x and y respectively. The completion of the metric space (IKI,di) is called the 11completion and is denoted by IKI 1. In this paper, we prove that IKI 1 is an 12-manifold if and only if K is a combinatorial oo-manifold. 0. Introduction. Let K be a simplicial complex. The realization IKI of K admits the metric d1(x,y)= S Ix(v)-y(v)I, vEKO where x(v) and y(v), v E KO, are the barycentric coordinates of x and y respectively. The topology induced by d1 is the metric topology of IKI and the space IKI with this topology is denoted by IKim. The completion of the metric space (iKI, d4) is called the 11-completion of IKim and is denoted by IKI . In [Sall, the author proved that Kil 1 is an ANR and the inclusion i: IKim C IKl is a fine homotopy equivalence, that is, for each open cover U of IK1 there is a map f: LKI 1 IKim such that i o f is U-homotopic to id and f o i is i(I U)-homotopic to id, and conjectured that IKI 1 is an 12-manifold if K is a combinatorial oo-manifold. Here a combinatorial oo-manifold is a countable simplicial complex such that the star of each vertex is combinatorially equivalent to the countable-infinite full simplicial complex, namely a oo-simplex Aw (see [Sal]). An 12-manifold is a separable manifold modeled on the Hilbert space 12. Let If be the linear span of the natural orthonormal basis of 12. A separable manifold modeled on the space lf is called an 4f-manifold. In [Sa2, Sa3], it was shown that K is a combinatorial oo-manifold if and only if 1Kim is an 4f-manifold. The main result of this paper is the following MAIN THEOREM. A simplicial complex K is a combinatorial co-manifold if and only if IKI is an 12-manifold. 1. Preliminaries. Let X be a metric space with a metric d. A closed subset A of X is called a Z-set in X if for each E > 0 and each map f: I' -+ X, n E N, Received by the editors May 21, 1986. 1980 Mathematics Subject Clasification (1985 Revision). Primary 57N20, 57Q15, 54E52.

The 𝑙₁-completion of a metric combinatorial ∞-manifold