The orientability of small covers and coloring simple polytopes
Nakayama, Hisashi ; Nishimura, Yasuzo
Osaka J. Math., Tome 42 (2005) no. 1, p. 243-256 / Harvested from Project Euclid
Small Cover is an $n$-dimensional manifold endowed with a $\mathbf{Z}_2^n$ action whose orbit space is a simple convex polytope $P$. It is known that a small cover over $P$ is characterized by a coloring of $P$ which satisfies a certain condition. In this paper we shall investigate the topology of small covers by the coloring theory in combinatorics. We shall first give an orientability condition for a small cover. In case $n=3$, an orientable small cover corresponds to a four colored polytope. The four color theorem implies the existence of orientable small cover over every simple convex $3$-polytope. Moreover we shall show the existence of non-orientable small cover over every simple convex $3$-polytope, except the $3$-simplex.
Publié le : 2005-03-14
Classification: 
@article{1153494325,
     author = {Nakayama, Hisashi and Nishimura, Yasuzo},
     title = {The orientability of small covers and coloring simple polytopes},
     journal = {Osaka J. Math.},
     volume = {42},
     number = {1},
     year = {2005},
     pages = { 243-256},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1153494325}
}
Nakayama, Hisashi; Nishimura, Yasuzo. The orientability of small covers and coloring simple polytopes. Osaka J. Math., Tome 42 (2005) no. 1, pp.  243-256. http://gdmltest.u-ga.fr/item/1153494325/