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Homotopic PL $ n$-balls are isotopic


Author: Robert M. Dieffenbach
Journal: Proc. Amer. Math. Soc. 37 (1973), 271-280
MSC: Primary 57C35
DOI: https://doi.org/10.1090/S0002-9939-1973-0310898-7
MathSciNet review: 0310898
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Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [1] M. M. Cohen, A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189-229. MR 40 #2052. MR 0248802 (40:2052)
  • [2] J. F. P. Hudson, Piecewise-linear topology, Benjamin, New York, 1969. MR 40 #2094. MR 0248844 (40:2094)
  • [3] -, Extending piecewise-linear isotopies, Proc. London Math. Soc. (3) 16 (1966), 651-668. MR 34 #2020. MR 0202147 (34:2020)
  • [4] J. F. P. Hudson and E. C. Zeeman, On regular neighbourhoods, Proc. London Math. Soc. (3) 14 (1964), 719-745. MR 29 #4063. MR 0166790 (29:4063)
  • [5] L. S. Husch, Homotopy groups of PL-embedding spaces, Pacific J. Math. 33 (1970). 149-155. MR 42 #1126. MR 0266219 (42:1126)
  • [6] L. S. Husch and T. B. Rushing, Restrictions of isotopies and concordances, Michigan Math. J. 16 (1969), 303-307. MR 41 #7689. MR 0263084 (41:7689)
  • [7] C. Lacher, Locally flat strings and half-strings, Proc. Amer. Math. Soc. 18 (1967), 299-304. MR 35 #3670. MR 0212805 (35:3670)
  • [8] J. Martin and D. Rolfsen, Homotopic arcs are isotopic, Proc. Amer. Math. Soc. 19 (1968), 1290-1292. MR 38 #719. MR 0232394 (38:719)
  • [9] C. P. Rourke and B. J. Sanderson, $ \Delta $-sets I: Homotopy theory, Quart. J. Math. 22 (1971), 321-328. MR 0300281 (45:9327)
  • [10] E. C. Zeeman, Linking spheres, Abh. Math. Sem. Univ. Hamburg 24 (1960), 149-153. MR 22 #8513. MR 0117739 (22:8513)
  • [11] S.-T. Hu, Homotopy theory, Pure and Appl. Math., vol. 8, Academic Press, New York, 1959. MR 21 #5186. MR 0106454 (21:5186)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310898-7
Keywords: Embeddings, homotopic embeddings, ambient isotopy, locally unknotted, $ \Delta $-sets, homomorphism spaces
Article copyright: © Copyright 1973 American Mathematical Society

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