Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T02:19:59.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Polyhedral immersions

Published online by Cambridge University Press:  24 October 2008

M. C. Irwin
M. C. Irwin
Affiliation:
University of Liverpool
Get access Rights & Permissions [Opens in a new window]

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

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)Alexander, J. W.The combinatorial theory of complexes. Ann. of Math. 30 (1930), 292–320.CrossRefGoogle Scholar
(2)Irwin, M. C.Embeddings of polyhedral manifolds. Ann. of Math. 82 (1965), 1–14.CrossRefGoogle Scholar
(3)Penrose, R., Whitehead, J. H. C. and Zeeman, E. C.Imbeddings of manifolds in euclidean space. Ann. of Math. 73 (1961), 613–623.CrossRefGoogle Scholar
(4)Whitehead, J. H. C.Simplicial spaces, nuclei and m-groups. Proc. London Math. Soc. 45 (1939), 243–327.CrossRefGoogle Scholar
(5)Zeeman, E. C. Polyhedral n-manifolds. Topology of 3-manifolds and related topics, pp. 57–70 (Prentice-Hall, 1962).Google Scholar
(6)Zeeman, E. C. Isotopies and knots in manifolds. Topology of 3-manifold and related topics, pp. 187–193 (Prentice-Hall, 1962).Google Scholar