On the 𝑛-parameter concordance and isotopy theorem

Akiba, Tadatoshi

doi:10.1090/S0002-9939-1976-0433463-2

On the $n$-parameter concordance and isotopy theorem
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Proc. Amer. Math. Soc. 61 (1976), 122-130 Request permission

Abstract:

J. F. P. Hudson [4] proved that “concordance” implies “isotopy". In this paper we show that the $n$-parameter version of Hudson’s result cannot be true without some restrictions. Assuming Millett’s result [6], we can determine a specific dimension where the $n$-parameter version fails. We rely on Kan fibrations and geometric techniques such as the Alexander trick.

References

Tadatoshi Akiba, Homotopy groups of $E(S^{n},S^{n+r})$, Michigan Math. J. 21 (1974), 371–376. MR 368018

Filling in cross sections of

bundles

John F. P. Hudson, Extending piecewise-linear isotopies, Proc. London Math. Soc. (3) 16 (1966), 651–668. MR 202147, DOI 10.1112/plms/s3-16.1.651
J. F. P. Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970), 425–448. MR 259920, DOI 10.2307/1970632
N. H. Kuiper and R. K. Lashof, Microbundles and bundles. II. Semisimplical theory, Invent. Math. 1 (1966), 243–259. MR 216507, DOI 10.1007/BF01452244
Kenneth C. Millett, Piecewise linear concordances and isotopies, Memoirs of the American Mathematical Society, No. 153, American Mathematical Society, Providence, R.I., 1974. MR 0365585
C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9