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Polyhedral immersions

Published online by Cambridge University Press:  24 October 2008

M. C. Irwin
M. C. Irwin
Affiliation:
University of Liverpool
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Extract

1. We work in the category of compact polyhedral spaces and polyhedral maps ((5)). Given spaces M and Q and a map f: M → Q, we define Ur(f) to be the set of points x ∈ M such that f−1f(x) contains at least r points. Sr(f) is the closure of Ur(f) in M. f is an embedding if it is a homeomorphism into, and an immersion if it is locally an embedding. We shall call f a simple immersion if S3(f) = ø and the connected components of S2(f) are individually embedded by f. Obviously a simple immersion is an immersion. If M and Q are manifolds (as they will be for the rest of the paper)f: M → Q is proper if it takes ∂M, the boundary of M, into ∂Q. In (2) the following result was proved:

Theorem 1. Let Mmbe(2m−q)-connected and Qq be(2m−q + 1)-connected, m ≤ q−3. Then any proper map f: M → Q which embeds ∂M is homotopic rel ∂M to an embedding.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 1 , January 1966 , pp. 45 - 50
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

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