Standard moves for standard polyhedra and spines.

*(English)*Zbl 0672.57004
Topology, 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391-414 (1988).

[For the entire collection see Zbl 0638.00026.]

A finite 2-dimensional CW-complex P is called a standard (or special) polyhedron if the link of any vertex of P is homeomorphic to a circle with three radii and the link of any other point of its l-skeleton is homeomorphic to a circle with one diameter. Three transformations of standard polyhedra are defined, called standard moves. Moves I and III change a small neighbourhood of a vertex, move II changes a small neighbourhood of an edge. Let P, Q be two standard polyhedra. The following two results are proved: 1) P can be 3-deformed (in the sense of J. H. C. Whitehead) to Q if and only if P can be obtained from Q by a finite sequence of moves I, II, III and its inverses; 2) If P is a spine of a 3-manifold then Q is a spine of the same manifold if and only if P and Q can be related by moves I, II and its inverses only.

Reviewer’s remark: Both results were obtained also by the reviewer, who used them to prove that the Zeeman conjecture for standard polyhedra is equivalent to the union of PoincarĂ© and Andrews-Curtis conjectures [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No.5, 1104-1116 (1987; Zbl 0642.57003); Sib. Mat. Zh. 28, No.6(166), 66-80 (1987; Zbl 0638.57002)].

A finite 2-dimensional CW-complex P is called a standard (or special) polyhedron if the link of any vertex of P is homeomorphic to a circle with three radii and the link of any other point of its l-skeleton is homeomorphic to a circle with one diameter. Three transformations of standard polyhedra are defined, called standard moves. Moves I and III change a small neighbourhood of a vertex, move II changes a small neighbourhood of an edge. Let P, Q be two standard polyhedra. The following two results are proved: 1) P can be 3-deformed (in the sense of J. H. C. Whitehead) to Q if and only if P can be obtained from Q by a finite sequence of moves I, II, III and its inverses; 2) If P is a spine of a 3-manifold then Q is a spine of the same manifold if and only if P and Q can be related by moves I, II and its inverses only.

Reviewer’s remark: Both results were obtained also by the reviewer, who used them to prove that the Zeeman conjecture for standard polyhedra is equivalent to the union of PoincarĂ© and Andrews-Curtis conjectures [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No.5, 1104-1116 (1987; Zbl 0642.57003); Sib. Mat. Zh. 28, No.6(166), 66-80 (1987; Zbl 0638.57002)].

Reviewer: S.V.Matveev

##### MSC:

57M20 | Two-dimensional complexes (manifolds) (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |