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The author was launched on the investigations of this paper and others to follow by a seminar study of the series of papers by E. E. Moise entitled Ajfine Structures in 3-manifolds. Some of the results are isotopic extensions or versions of corresponding results in that series and the author is indebted to Moise for some of the techniques of proof. The separable metric space M is an n-manifold (n-manifold with boundary) if every point p of M has a neighborhood homeomorphic to Euclidean w-space, E" (a neighborhood whose closure is a closed topological w-cell). If M is an w-manifold with boundary, the boundary of M, denoted by Bd(M), is defined to be the set of points of M which do not have a neighborhood in M homeomorphic to En. This kind of boundary is to be distinguished from the point set boundary of M in a containing space. The set M-Bd(M) is called the interior of M and is denoted by Int(Af). If M is homeomorphic to a locally finite simplicial complex, M is said to be triangulable. Moise [5] has shown that every 3-manifold is triangulable and Bing [l ] has proved the corresponding result for 3-manifolds with boundary. The same notation will frequently be used for the space and for its triangulation (i.e., the associated complex) wherever no confusion is likely to occur. Let K and M be triangulated spaces and / a mapping of K into M. Then / is said to be piecewise linear if there exist subdivisions K' and M' of K and M, respectively, such that / maps each simplex of K' in a linear (i.e., affine) fashion onto a simplex of M'. If / is in fact a homeomorphism and /' is another such homeomorphism, then a piecewise linear isotopy between / and /' is defined to be a mapping G(x, t) of AX [0, 1] into M such that for all x in K, G(x, 0) =f(x), G(x, 1) =f'(x), and for every t in the closed interval [0, l], G(x, t) is a piecewise linear homeomorphism. If KEM and / is the identity, G is called an isotopic deformation of K. If there is a fixed subdivision K' of K on which G(x, /) is piecewise linear for every t in [0, l], then

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