Let A' be a ^-dimensional compactum and /: X -» M" a map into a piecewise linear n-manifold. n > k + 3. The main result of this paper asserts that if X is locally (2k ^-connected and / is (2k n + l)-connected, then / is homotopic to a CE equivalence. In particular, every ^--dimensional, /-connected, locally /--connected compactum is CE equivalent to a compact subset of R2*~r as long as r < k 3....

This paper pays a visit to famous contractible open 3-manifold $W^3$ proposed by R. H. Bing in 1950's. By the finiteness theorem \cite{Hak68}, Haken proved that can embed no compact 3-manifold. However, until now, question about whether more general space such as compact, locally connected and 1-connected metric 3-space was not known. Using techniques developed Sternfeld's 1977 PhD thesis \cite...