A 2-manifold of genus 8 without the W v-property

D. W. BARNETTE

A 2-MANIFOLD

OF GENUS

8 WITHOUT

THE

Wv-PROPERTY

ABSTRACT.We construct the first known polyhedral 2-manifold M that does not have the w~property, i.e. there are two verticesx and y in M such that any path from x to y must revisit some face of the manifold. 1. INTRODUCTION One of the most important unsolved problems in the field of convexity is the Hirsch conjecture, that between any two vertices of a convex d-dimensional polytope (hereafter to be called a d-polytope), with n facets is a path with at most n-d edges. The truth of the Hirsch conjecture is implied by the truth of the ~ conjecture, that between any two vertices of a d-polytope is a path whose intersections with each facet is connected. For any cell complex c~ (either geometric or topological) we shall say that has the ~ p r o p e r t y provided that between any two vertices there is a path whose intersection with each face is connected. It is known that d-polytopes for d ~< 3 have the Ww property E4]. Mani and Walkup [6] have constructed a 3-sphere without the W~ property and L a r m a n [5] has constructed a 2-cell complex without the Wv property. The W~ property holds for polyhedral maps on the projective plane [1], the torus [2] and for the Klein bottle and double torus [3]. In this paper we construct polyhedral maps on 2-dimensional manifolds (both orientable and nonorientable) of genus 8 without the V¢~ property.

2. DEFINITIONS Suppose a graph G is embedded in a 2-manifold or 2-pseudomanifold M. The faces of G are defined to be the connected components of M - G . We say that G is a polyhedral map provided the closure of each face is a closed cell, each vertex has valence at least 3 and any two faces that intersect, do so on a single vertex or a single edge. When the faces have this intersection property we say that the faces meet properly. A path in a 2-manifold is a Wv path provided its intersection with each face is connected. If a path has a disconnected intersection with a face we say that the path revisits that face. Geometriae Dedicata 46:211-214, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

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D . w . BARNETTE 3. THE COUNTEREXAMPLES

We begin with the polyhedral map MI on the sphere shown in Figure 1. From this we construct a map M 2 o n a pseudomanifold by identifying vertices with the same label (see figure). .X

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~5/

2

3 Fig. 1.

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6

A 2-MANIFOLD OF GENUS g

213

Fig. 2. We shall show that M 2 is a polyhedral map. First we note that the faces F1 . . . . . F 4 separate the two sets of labeled vertices, thus no face has two vertices with the same label and thus the (closed) faces remain closed cells after identifications. Also note that no pair of faces meeting in M1 will have two distinct vertices with the same label on each, thus no improperly meeting faces are created in this way. The only other way that an improper meeting could occur would be if two faces not meeting in M~ each have the same labeling for two of their vertices. To see that this does not happen, note that for the labeled vertices on faces meeting x, if two of these vertices lie on any face then they lie on a face meeting x. We can also make the same observation for labeled vertices on faces meeting y. Thus if two faces are to meet improperly then a face meeting x and a face meeting y would meet improperly. The reader can easily check that these faces meet properly (after identifications). Before we construct our manifold of genus 8 we shall show that M2 does not have the W~ property for vertices x and y. Suppose P is a Wv path from x to y in M2. Suppose the first edge of P is x z (the argument is the same for the other cases). Since vertex z is identified with vertex w we are on the face F5 when we reach vertex z. Since P is a Wv path that ends at y, the remaining portion of P must be a path along F 5. Such a path will pass through either vertex s or vertex t. If it passes through s then the outside face in Figure 1 is revisited. If P passes through t then face F 6 is revisited. We shall now modify M 2 t o get a manifold M3. We shall describe the modification at the vertices z and w. The modifications at the other vertices are similar. Instead of identifying z and w, we replace these two vertices by two small triangles, identify the triangles, and then remove the triangle from the cell complex (this has the effect of producing a hole in M3). There are several ways to identify the two triangles. We require that they be identified such that in the resulting surface the neighbor of x on the triangle will lie on F 5. We then do the same for the other vertices, making note of the x - y

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D . W . BARNETTE

s y m m e t r y in M 1 (e.g. for the vertices labeled 1, we want the resulting neighbor of y to lie on the face that contains x and v). To see that this creates a polyhedral m a p we note that two faces meet in M 2 if and only if they meet in M a. The only possible change in the way two faces intersect would be that two faces might meet on a vertex in M2 but on an edge in M a. This does not change the property of faces meeting properly, thus M 3 is a polyhedral map. Since eight identifications were made, M3 has genus 8. The p r o o f that M 3 does not have the Wv property is n o w the same as for m 2. We note that this m e t h o d gives us both an orientable and a nonorientable example. A nonorientable example can easily be constructed from the orientable one by the obvious change in the way two triangles are identified.

REFERENCES 1. 2. 3. 4. 5. 6.

Barnette, D,, 'W, paths in the projective plane', Discrete Math. 62 (1986), 127-131. Barnette, D., 'I,V~paths on the torus' (to appear in Discrete Comput. Geom.). Engelhardt, L., 'Some problems on paths in graphs', Ph.D. thesis, Univ. of Washington, 1988. Klee, V., 'Paths on polyhedra I', J. Soc. Indust. Appl. Math. 13 (1965), 946-956. Larman, D., 'Paths on polytopes', Proc. London Math. Soc. 20 (1970), 161-178. Mani, P. and Walkup, D., 'A 3-sphere counterexample to the Wv-pathconjecture', Math. Oper. Res. 5 (4) (1980), 595 598. A u t h o r ' s address:

D. W. Barnette, D e p a r t m e n t of Mathematics, University of California, Davis, California 95616-8633,

U.S.A. (Received, May 7, 1992; revised version, July 10, 1992)