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Piecewise linear immersions

Published online by Cambridge University Press:  24 October 2008

Anthony Smith
Anthony Smith
Affiliation:
Trinity College, Cambridge
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0. Introduction. An immersion f of one space X in another Y is a continuous map which is locally an embedding; that is, for any x ∈ X there is a neighbourhood N(x) of x such that f|N(x) is an embedding. Thus the image of an immersion may intersect itself, but it can contain no worse singularities. For example, the well-known model of the Klein Bottle is the image of an immersion into ordinary 3-space; the locus of points of self intersection is a circle, where two distinct circles in the Klein Bottle have their images. This paper obtains a sufficient condition for a map of PL manifolds to be deformable into a PL immersion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 45 - 55
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Haefliger, A.Plongements différentiables de variétés dans variétés. Comment Math. Helv. 36 (1962), 47–82.CrossRefGoogle Scholar
(2)Hirsch, M. W. and Zeeman, E. C. Engulfing. Bull. Amer. Math. Soc. 72 (1966), 113–115.CrossRefGoogle Scholar
(3)Hudson, J. F. P.Piecewise linear embeddings. Ann. Math. 85 (1967), 1–31.CrossRefGoogle Scholar
(4)Hudson, J. F. P.PL embeddings of bounded manifolds.Google Scholar
(5)Hudson, J. F. P. and Lickorish, W. B. R. Extending piecewise linear concordances.Google Scholar
(6)Irwin, M. C.Embedding polyhedral manifolds. Ann. Math. 82 (1965), 1–14.CrossRefGoogle Scholar
(7)Irwin, M. C.Polyhedral immersions. Proc. Cambridge Philos. Soc. 62 (1966), 45–50.CrossRefGoogle Scholar
(8)Lickorish, W. B. R. and Siebenmann, L. C. Regular neighbourhoods and the stable range.Google Scholar
(9)Rourke, C. P. The Hauptvermutung according to Sullivan. Part II. Mimeo. notes.Google Scholar
(10)Sanderson, B. J. and Schwarzenberger, R. L. E.Non-immersion theorems for differentiable manifolds. Proc. Cambridge Philos. Soc. 59 (1963), 319–322.CrossRefGoogle Scholar
(11)Stallings, J. The embedding of homotopy types into manifolds. To appear.Google Scholar