## Abstract

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.

## Degree

MS

## College and Department

Physical and Mathematical Sciences; Mathematics

## Rights

http://lib.byu.edu/about/copyright/

## BYU ScholarsArchive Citation

Dickson, Allen J., "Maximal Surfaces in Complexes" (2005). *Theses and Dissertations*. 545.

https://scholarsarchive.byu.edu/etd/545

## Date Submitted

2005-06-30

## Document Type

Thesis

## Handle

http://hdl.lib.byu.edu/1877/etd871

## Keywords

cubical complex, surface, 2-manifold, genus, graph, n-cube

## Language

English