Approximating Stable Homeomorphisms by Piecewise Linear Ones

Connell, E. H.

doi:10.2307/1970346

Cited by 41 publications

(31 citation statements)

References 5 publications

(9 reference statements)

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“…Partition Cl(Int f(S n~1 )) into annular regions R iy together with a small cell C about the origin, in such a way that each boundary sphere of each Ri is parallel to f(S n~~l ) and the parallel distance between any two consecutive spheres (i.e., the constant L of the definition of parallel embeddings) is 2e. These results, together with those of Connell [5] and Bing [l], can be used to show that the problem of approximating homeomorphisms of S n , n^Sj by p.w.l. ones is equivalent to approximating locally flat embeddings of (« -1)-spheres by p.w.l.…”

Section: F(x)mentioning

confidence: 82%

A uniform generalized Schoenflies theorem

Wright¹

1968

Bull. Amer. Math. Soc.

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Section: F(x)mentioning

confidence: 82%

A uniform generalized Schoenflies theorem

Wright¹

1968

Bull. Amer. Math. Soc.

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“…Using earlier work of Connell [7] we can prove Theorem 1. Let Mn be a compact PL manifold, n ¥= 4; ifn = 5 suppose 8Af is empty.…”

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confidence: 90%

“…When n > 6, Connell shows on p. 331 of [7] that any nontrivial symmetric radial expansion of I" which fixes a neighborhood of 3/" can serve as h0. The restriction n > 6 arises from the radial engulfing lemma used in [7]. But in [3], Bing proves a stronger engulfing lemma which makes ConneU's proof valid for n > 4; see [3, p. 3] and [7, p. 337].…”

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confidence: 99%

On the Space of Piecewise Linear Homeomorphisms of a Manifold

Geoghegan¹,

Haver²

1976

Proceedings of the American Mathematical Society

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“…By induction on the number of elements in the handle decomposition of M, it is sufficient to prove Theorem 11 (q -n,n) for the special case that M=M-HuH (where H is an «-cell, M-H is an «-manifold, and M-Hn H is a solid torus), that TV is a neighborhood of H in ß, and that g\M-H=h\M-H. Connell [3] and Bing [1] have shown that a stable homeomorphism of R" onto itself, q S; 5, can be ambient isotoped to a piecewise-linear one that is an approximation in the sense of the statement of this proposition. Let Proof of Theorem 14.…”

Section: By Lemma 3 P Sunny Collapses In Iikxiq~k) -C2 To/(d) Thus mentioning

confidence: 99%

Close isotopies on piecewise-linear manifolds

Miller¹

1970

Trans. Amer. Math. Soc.

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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.

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Approximating Stable Homeomorphisms by Piecewise Linear Ones

Cited by 41 publications

References 5 publications

A uniform generalized Schoenflies theorem

A uniform generalized Schoenflies theorem

On the Space of Piecewise Linear Homeomorphisms of a Manifold

Close isotopies on piecewise-linear manifolds

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