Cubical complex

In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces. Non-positively curved and CAT(0) cube complexes appear with increasing significance in geometric group theory.

All graphs are (homeomorphic to) 1-dimensional cubical complexes.

Definitions

With regular cubes

A unit cube (often just called a cube) of dimension $n\geq 0$ is the metric space obtained as the finite ($l^{2}$) cartesian product $C_{n}=I^{n}$ of $n$ copies of the unit interval $I=[0,1]$.

A face of a unit cube is a subset $F\subset {C_{n}}$ of the form $F=\prod _{i=1}^{n}J_{i}$, where for all $1\leq i\leq n$, $J_{i}$ is either $\{0\}$, $\{1\}$, or $[0,1]$. The dimension of the face $F$ is the number of indices $i$ such that $J_{i}=[0,1]$; a face of dimension $k$, or $k$-face, is itself naturally a unit elementary cube of dimension $k$, and is sometimes called a subcube of $F$. One can also regard ${\textstyle \emptyset }$ as a face of dimension ${\textstyle -1}$.

A cubed complex is a metric polyhedral complex all of whose cells are unit cubes, i.e. it is the quotient of a disjoint union of copies of unit cubes under an equivalence relation generated by a set of isometric identifications of faces. One often reserves the term cubical complex, or cube complex, for such cubed complexes where no two faces of a same cube are identified, i.e. where the boundary of each cube is embedded, and the intersection of two cubes is a face in each cube.

A cube complex is said to be finite-dimensional if the dimension of the cubical cells is bounded. It is locally finite if every cube is contained in only finitely many cubes.

With irregular cubes

An elementary interval is a subset $I\subsetneq \mathbf {R}$ of the form

$I=[l,l+1]\quad {\text{or}}\quad I=[l,l]$

for some $l\in \mathbf {Z}$. An elementary cube $Q$ is the finite product of elementary intervals, i.e.

$Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}$

where $I_{1},I_{2},\ldots ,I_{d}$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube $[0,1]^{n}$ embedded in Euclidean space $\mathbf {R} ^{d}$ (for some $n,d\in \mathbf {N} \cup \{0\}$ with $n\leq d$). A set $X\subseteq \mathbf {R} ^{d}$ is a cubical complex (or cubical set) if it can be written as a union of elementary cubes (or possibly, is homeomorphic to such a set).

Related terminology

Elementary intervals of length 0 (containing a single point) are called degenerate, while those of length 1 are nondegenerate. The dimension of a cube is the number of nondegenerate intervals in $Q$, denoted $\dim Q$. The dimension of a cubical complex $X$ is the largest dimension of any cube in $X$.

If $Q$ and $P$ are elementary cubes and $Q\subseteq P$, then $Q$ is a face of $P$. If $Q$ is a face of $P$ and $Q\neq P$, then $Q$ is a proper face of $P$. If $Q$ is a face of $P$ and $\dim Q=\dim P-1$, then $Q$ is a facet or primary face of $P$.

In algebraic topology

Main article: Cubical homology

In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.

In geometric group theory

Main article: Cubulation (group theory)

Groups acting geometrically by isometries on CAT(0) cube complexes provide a wide class of examples of CAT(0) groups.

The Sageev construction can be understood as a higher-dimensional generalization of Bass-Serre theory, where the trees are replaced by CAT(0) cube complexes. Work by Daniel Wise has provided foundational examples of cubulated groups. Agol's theorem that cubulated hyperbolic groups are virtually special has settled the hyperbolic virtually Haken conjecture, which was the only case left of this conjecture after Thurston's geometrization conjecture was proved by Perelman.

CAT(0) cube complexes

See also: Median graph

See also

• Simplicial complex
• Simplicial homology
• Abstract cell complex