Close isotopies on piecewise-linear manifolds

Abstract

We say that two maps of a set into a metric space are within 8 everywhere if, for each point in the set, its images under the maps are within 8 of each other.Let P= [-1, 1].We prove the following theorems.Theorem 9.For q-n^3 and e>0, there is a 8>0 such that iff: In -> I" is a proper piecewise-linear embedding with f\bdy (In) the identity and f within 8 of the identity everywhere, then there is a piecewise-linear ambient isotopy of I" that fixes bdy (/'), that moves points less than e, and that takes f to the identity.Theorem 11.Let M be a compact piecewise-linear n-manifold, let M be a compact piecewise-linear n-submanifiold of M, let Q be a piecewise-linear q-manifold, and let n^q -3.Suppose g: M->int(Q) is a piecewise-linear embedding.Let N be a neighborhood of g(M-M) in Q.Then for e>0, there is a 8>0 such that if h: M->-int(0 is a piecewiselinear embedding with h\M=g\M, and with h within o of g everywhere, then there is a piecewise-linear ambient isotopy of Q that is fixed outside N, that moves points ¡ess than e, and that takes h to g. Theorem 14.Let M be a compact piecewise-linear n-manifold, let M be a compact piecewise-linear n-submanifold, and let Q be a piecewise-linear q-manifold.Suppose nfiq -3 and q^5.Suppose g: M-^-int(ß) is a locally-flat embedding such that g\M is piecewise-linear.Let N be a neighborhood of g(M-M) in Q.Then for e > 0, there is an ambient isotopy of Q that is fixed outside N, that moves points less than e, and that takes g to a piecewise-linear embedding of M in Q.The problem underlying Theorems 9 and 11 is the following.Suppose/: In -> I" is a proper p.l. embedding that is the identity on bdy In.In general, /(/") n ({0} x P"1) is just an «-complex.If the vertices of /(/") are moved just enough to get them off{0} x/4"1 but not enough to alter the fact that/is an embedding, then