Finding a boundary for a Menger manifold

We give a characterization of k-dimensional (k > 1) Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann. In [5] Chapman and Siebenmann considered the problem of putting a boundary on a Hilbert cube manifold (similar problems in the cases of smooth and piecewise linear manifolds were considered in [2, 14]; a parametrical version of the above problem was considered in [13]). It was proved there that if a Q-manifold M satisfies certain minimal necessary homotopy-theoretic conditions (finite type and tameness at oc), then there are two obstructions ro(M) and zTo (M) to M having a boundary. The first one is an element of the group lim {Xr 1 (M A): A C M compact}, where o7r 1 is the projective class group functor. If ac, (M) = 0, then the second obstruction can be defined as an element of the first derived limit of the inverse system lim {7fh7r, (MA): A C M compact}, where 7tfh7r, is the Whitehead group functor. Further, it was proved in [5] that the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms of the group lim {1h 7r (M A): A C M compact}. In the present paper we carry out a similar program for the problem of putting boundaries on 1un~1-manifolds, where n1n+l denotes the (n + l)-dimensional universal Menger compactum (a 4un+I-manifold M admits a boundary if there exists a compact ,1n+l -manifold N such that M = N Z , where Z is a Z-set in N; in this case we shall say that N is a compactification of M corresponding to the boundary Z, and conversely, Z is a boundary of M corresponding to the compactification N. We recall also that a closed subset Z of a space X is said to be a Z-set if for each open cover ( E cov(X) there is a map f: X -* X Z E/-close to the identity map of X ). Having in mind a deep analogy between the theories of ,un+l-manifolds and Q-manifolds [1, 6-9] it is not surprising that the corresponding results are valid in the case of 1Un+lmanifolds as well. However, it should be observed that the situation in the last case is much simpler. For example, the analogies of the above-described Received by the editors September 23, 1992. 1991 Mathematics Subject Classification. Primary 57Q12, 57N99; Secondary 55P55.