Maximal Surfaces in Complexes

• Main Author: Dickson, Allen J.
• Format: Others
• Published: BYU ScholarsArchive 2005
• Subjects: cubical complex; surface; 2-manifold; genus; graph; n-cube; Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/545; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1544&context=etd

Cubical complexes are defined in a manner analogous to that for simplicial complexes, the chief difference being that cubical complexes are unions of cubes rather than of simplices. A very natural cubical complex to consider is the complex C(k_1,...,k_n) where k_1,...,k_n are nonnegative integers. This complex has as its underlying space [0,k_1]x...x[0,k_n] subset of R^n with vertices at all points having integer coordinates and higher dimensional cubes formed by the vertices in the natural way. The genus of a cubical complex is defined to be the maximum genus of all surfaces that are subcomplexes of the cubical complex. A formula is given for determining the genus of the cubical complex C(k_1,...,k_n) when at least three of the k_i are odd integers. For the remaining cases a general solution is not known. When k_1=...=k_n=1 the genus of C(k_1,...,k_n) is shown to be (n-4)2^{n-3}+1 which is equivalent to the genus of the graph of the n-cube. Indeed, the genus of the complex and the genus of the graph of the 1-skeleton of the complex, are shown to be equal when at least three of the k_i are odd, but not equal in general.