# Maximal Surfaces in Complexes Maximal Surfaces in Complexes

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.

Advisor:

School:Brigham Young University

School Location:USA - Utah

Source Type:Master's Thesis

Keywords:cubical complex surface 2 manifold genus graph n cube

ISBN:

Date of Publication:06/20/2005