Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T06:01:47.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Imbedding and isotopy of spheres in manifolds

Published online by Cambridge University Press:  24 October 2008

J. Levine
J. Levine
Affiliation:
Department of Mathematics, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).

(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.

(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 433 - 437
Copyright
Copyright © Cambridge Philosophical Society 1964

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1) Haefliger, A. Plongements différentiables de variétés dans variétés. Comment. Math. Helv. 36 (1961), 47–82.CrossRefGoogle Scholar
(2) Haefliger, A. Knotted (4k − 1)-spheres in 6k−space. Ann. of Math. 75 (1962), 452–466.CrossRefGoogle Scholar
(3) Haefliger, A. Plongements de variétés dans le domaine stable. Séminaire Bourbaki (1962/1963), no. 245.Google Scholar
(4) Irwin, M. C. Combinatorial embeddings of manifolds. Bull. American Math. Soc. 68 (1962), 25–27.CrossRefGoogle Scholar
(5) Shapiro, A. Obstructions to the imbedding of a complex in Euclidean space, I. The first obstruction. Ann. Math. 66 (1957), 256–269.CrossRefGoogle Scholar
(6) Smale, S. Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. 74 (1961), 391–406.CrossRefGoogle Scholar
(7) Smale, S. On the structure of manifolds. American J. Math. 84 (1962), 387–399.CrossRefGoogle Scholar
(8) Whitney, H. The self-intersections of a smooth n−manifold in 2n−space. Ann. of Math. 45 (1944), 220–246.CrossRefGoogle Scholar
(9) Zeeman, E. C. Polyhedral n−manifolds: II. Embeddings, in Topology of 3-manifolds and related topics, ed. Fort, M. K. Jr (Prentice-Hall, 1962), pp. 64–70.Google Scholar
(10) Zeeman, E. C. Isotopies and knots in manifolds, in Topology of 3-manifolds and related topics, ed. Fort, M. K. Jr (Prentice-Hall, 1962), pp. 187–94.Google Scholar