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Collapsing K × I

Published online by Cambridge University Press:  24 October 2008

Dallas E. Webster
Dallas E. Webster
Affiliation:
Institute for Advanced Study, Princeton
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Extract

Zeeman(5) once bravely made the following conjecture:

Conjecture. Suppose K is a contractible 2-complex and I is the unit interval [0, 1]. Then K × I collapses.

This conjecture is interesting since it trivially implies the Poincaré conjecture. For if C is a homotopy 3-cell, then it has a contractible 2-dimensional spine, so that C × I would be a collapsible 4-manifold. But then C × I would be a 4-cell, with C = C × 0 in its 3-sphere boundary, and the PL Schoenfliess theorem shows that C is a 3-cell.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 39 - 42
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Lickorish, W. B. R. On collapsing X 2 × I. Topology of manifolds, pp. 157–160 (ed. by Cantrell, J. C. and Edwards, C. H.) (Markham: New York, 1970).Google Scholar
(2)Robinson, J. A. On the collapsibility of K × I (to appear).Google Scholar
(3)Wright, P.Collapsing K × I to vertical segments. Proc. Cambridge Philos. Soc. 69 (1971), 71–74.CrossRefGoogle Scholar
(4)Wright, P.On the collapsibility of K × I m. Quart. J. Math. Oxford (2) 22 (1971), 491–493.CrossRefGoogle Scholar
(5)Zeeman, E. C.On the dunce hat. Topology 2 (1964), 341–358.CrossRefGoogle Scholar