Contracted k-tessellations of closed surfaces

✍ Scribed by Paola Bandieri

• Book ID: 103056977
• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 344 KB
• Volume: 122
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)90283-y

✦ Synopsis

In this paper we study tilings of closed surfaces by means of polygons with k edges, which are minimal with respect to the number of vertices.

Throughout

this paper, spaces and maps will be piecewise-linear (PL), in the sense of or [l 11.

From now on, we shall denote by Tg (resp. U,) the orientable (resp. non-orientable) surface of genus .4 (resp. h).

For basic notations and results relatively to polytopes we refer to [S]; we shall need often to consider 2-cell embeddings (see ) of polygons into closed surfaces, we shall also use the term 'k-gon' to indicate a region of the embedding.

Definition 1. Let F be a surface. A ball complex ([ 111) & will be said to be a pseudo k-tessellation of F iff each 2-ball of rk considered with all its faces, is isomorphic with a closed k-gon (k 3 3). Remark 1. Note that a pseudo k-tessellation is a finite polyhedron in the sense of [6] and a map in the sense of [13]. Definition 2. A pseudo k-tessellation & of a surface F will be called a contracted k-tessellation iff Y, has exactly k vertices (O-balls).