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 cell-like approximation theorem in dimension 5

“…This theorem follows from 8.3 and the techniques established by Miller in [20] with special care taken to measure controls in the decomposition space.…”

“…The second theorem is due to Hudson [19] and the third is a controlled version of the same result. The fourth is primarily attributed to Connelly [10] and Miller [20] with contributions made by Cobb [8], Akin [1], and Bryant and Seebeck [5] (see [3]). …”

On maps with unstable singularities

“…In view of an analogy between Lemmas 4.1 and 7.6 below, one can regard 3.7 as a geometric version of the Freudental Suspension Theorem. The proof of 3.7 in [Ed1] is somewhat similar to the proof of the Penrose-Whitedead-Zeeman-Irwin Embedding Theorem, meanwhile Miller proves a generalization of 3.7 in [Mill3] using his controlled version (see [Mill1]) of sunny collapsing (see §5).…”

“…It seems that Hudson's original proof of CIIT [Hu] (as well as Lickorish's proof of the case Q = S m [Li, Theorem 6]) does not work to prove 1.13a (compare to remarks in [Mill1,Introduction], [ReS2,§2]). In [Ro] Rourke sketched a new proof of CIIT, and in [KeL, last paragraph] it was 'expected that, when the details of Rourke's proof are published, they will apply' to prove 1.13a.…”

“…Theorem 3.3. (a) [Ch1], [Mill1], [Co] Let X n be a compact polyhedron, Q m a PL manifold, m − n ≥ 3, and f : X ֒→ Q a PL embedding. For each ε > 0 there exists δ > 0 such that any PL embedding f ′ :…”