Any Spine of the Cube is 2-Collapsible

Edwards, Robert ; Gillman, David

Canadian journal of mathematics, Tome 35 (1983) no. 1, pp. 43-48

Résumé

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 B 3 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 A 1, A 2, ..., An be a finite collection of pairwise disjoint contractible PL subsets of the cube. Then the decomposition obtained by shrinking each A i to a point is 1-collapsible.

Détail

DOI : 10.4153/CJM-1983-003-x

Bibliographie

[1] 1. Cohen, M., Dimension estimates in collapsing X × Iq, Topology 14 (1975), 253–256“ Google Scholar

[2] 2. Cohen, M., A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189–229. Google Scholar

[3] 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] 4. Zeeman, E. C., On the dunce hat, Topology 2 (1964), 341–358. Google Scholar