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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Homotopic PL $ n$-balls are isotopic

Author: Robert M. Dieffenbach
Journal: Proc. Amer. Math. Soc. 37 (1973), 271-280
MSC: Primary 57C35
MathSciNet review: 0310898
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Abstract: The following theorem extends a result of Martin and Rolfsen [Proc. Amer. Math. Soc. 19 (1968), 1290-1292].

Theorem. Let $ {B^n}$ be a PL n-ball, $ {Q^q}$ a $ (2n - q + 1)$-connected PL q-manifold, $ q \geqq n + 2$. Suppose Q is either compact or open and that, for $ i = 0,1,{H_i}:{B^n} \to Q - \dot Q$ is a locally unknotted PL embedding. If there exists a homotopy $ H:{B^n} \times I \to Q$ between $ {H_0}$ and $ {H_1}$ such that $ {H_t}$ is fixed on $ {\dot B^n}$, then there exists a PL ambient isotopy $ {h_t}:Q \to Q$, fixed on $ {H_0}({\dot B^n}) \cup \dot Q$, such that $ {h_1}{H_0} = {H_1}$.

Locally unknotted is taken here to mean that there exists a triangulation (L, K) of $ (Q,{H_i}({B^n}))$ with $ \dot K$ full in K and $ ({\text{lk}}(\nu ,L)$, $ {\text{lk}}(\nu ,K))$ an unknotted sphere pair for all vertices $ \nu \in K - \dot K$.

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Keywords: Embeddings, homotopic embeddings, ambient isotopy, locally unknotted, $ \Delta $-sets, homomorphism spaces
Article copyright: © Copyright 1973 American Mathematical Society

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