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Any Spine of the Cube is 2-Collapsible

Published online by Cambridge University Press:  20 November 2018

Robert Edwards  and
David Gillman
Robert Edwards
Affiliation:
University of California, Los Angeles, California
David Gillman
Affiliation:
University of California, Los Angeles, California
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1. Introduction. M. Cohen [1] denned a polyhedron K to be n-collapsible if K × In PL collapses. He proved that any spine of the cube B3 is 3-collapsible. This was a step directed toward the Zeeman Conjecture [4], which asserts that every compact contractible 2-polyhedron is 1-collapsible. In this paper we improve the result of Cohen by one dimension (Theorem 3): Any spine of the cube is 2-collapsible. The central question of 1-collapsibility remains unanswered.

Gillman and Rolfsen [3] have shown that any standard spine of the cube is 1-collapsible. Conjecture: If K is any spine of the cube, then K × I collapses to a standard spine of the cube. This would imply our main theorem. Lacking a proof of this conjecture, we must resort to an argument independent of [3].

THEOREM 1. Let A1, A2, …, Anbe a finite collection of pairwise disjoint contractible PL subsets of the cube. Then the decomposition obtained by shrinking each Ai to a point is 1-collapsible.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 1 , 01 February 1983 , pp. 43 - 48
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Cohen, M., Dimension estimates in collapsing X × Iq, Topology 14 (1975), 253–256“Google Scholar
2. Cohen, M., A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189–229.Google Scholar
3. Gillman, D. and Rolfsen, D., The Zeeman Conjecture for standard spines is equivalent to the Poincaré Conjecture, Topology, accepted for publication.Google Scholar
4. Zeeman, E. C., On the dunce hat, Topology 2 (1964), 341–358.Google Scholar