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Regular neighbourhoods and mapping cylinders

Published online by Cambridge University Press:  24 October 2008

C. Kearton
C. Kearton
Affiliation:
Corpus Christi College, Cambridge
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Extract

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 1 , July 1971 , pp. 15 - 18
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Cohen, M. M.Simplicial structures and transverse cellularity. Ann. of Math. 85 (1967), 218–245.Google Scholar
(2)Zeeman, E. C.Unknotting combinatorial balls. Ann. of Math. 78 (1963), 501–526.CrossRefGoogle Scholar