Cubic complex

In mathematics , a cubic complex is a polyhedral complex built from points , segments , squares , cubes, and hypercubes of any dimensions.

definition

An elementary interval is a set of the form or for an integer . ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle I=\left[l,l+1\right]}$ ${\displaystyle I=\left[l,l\right]}$ ${\displaystyle l\in \mathbb {Z} }$

An elementary cube is a product of elementary intervals. ${\displaystyle Q\subset \mathbb {R} ^{d}}$ ${\displaystyle Q=I_{1}\times \ldots \times I_{d}}$

A cubic complex is a union of elementary cubes, the intersection of which is either empty or a common face. ${\displaystyle X}$

Metric

On a cubic complex is a metric defined by each elementary cube with the Euclidean metric of the is provided and the distance between two points in various elementary cubes is then defined as the shortest length of a composite of straight lines in elementary cubes path. ${\displaystyle \mathbb {R} ^{d}}$

literature

• Daniel Wise: From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry , CBMS Regional Conference Series in Mathematics 2012; 141 pp; softcover, ISBN 0-8218-8800-5