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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Cubical complex</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>cubical complex</b> (also called <b>cubical set</b> and <b>Cartesian complex</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> composed of <a href="Point_(geometry)" title="Point (geometry)">points</a>, <a href="Line_segment" title="Line segment">line segments</a>, <a href="Square" title="Square">squares</a>, <a href="Cube" title="Cube">cubes</a>, and their <a href="Hypercube" title="Hypercube"><i>n</i>-dimensional counterparts</a>. They are used analogously to <a href="Simplicial_complex" title="Simplicial complex">simplicial complexes</a> and <a href="CW_complex" title="CW complex">CW complexes</a> in the computation of the <a href="Homology_(mathematics)" title="Homology (mathematics)">homology</a> of <a href="Topological_space" title="Topological space">topological spaces</a>. <a href="CAT(k)_space" title="CAT(k) space">Non-positively curved and CAT(0)</a> cube complexes appear with increasing significance in <a href="Geometric_group_theory" title="Geometric group theory">geometric group theory</a>.
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<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>
<div class="mw-heading mw-heading3"><h3 id="With_regular_cubes">With regular cubes</h3></div>
<p>A <b>unit cube</b> (often just called a <b>cube</b>) of dimension <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>
</semantics>
</math></span><img src="./ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}" loading="lazy"></span></b> is the <a href="Metric_space" title="Metric space">metric space</a> obtained as the finite <a href="Product_metric" title="Product metric">(<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l^{2}}</annotation>
</semantics>
</math></span><img src="./d87c123d31160a9e22122d5cfbe786a5c96372de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.748ex; height:2.676ex;" alt="{\displaystyle l^{2}}" loading="lazy"></span>) cartesian product</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}=I^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C_{n}=I^{n}}</annotation>
</semantics>
</math></span><img src="./a5c50598dd6013e03fd4bb129f6932fda101ee67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.414ex; height:2.676ex;" alt="{\displaystyle C_{n}=I^{n}}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> copies of the <a href="Unit_interval" title="Unit interval">unit interval</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation>
</semantics>
</math></span><img src="./87ec65159c44769434523e46928bc1b82681f842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}" loading="lazy"></span>.
</p><p>A <b>face</b> of a unit cube is a subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset {C_{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>⊂<!-- ⊂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F\subset {C_{n}}}</annotation>
</semantics>
</math></span><img src="./47c3e0422c4da8714cd57b6be1fb9b9383b5eb94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.72ex; height:2.509ex;" alt="{\displaystyle F\subset {C_{n}}}" loading="lazy"></span> of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\prod _{i=1}^{n}J_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>=</mo>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F=\prod _{i=1}^{n}J_{i}}</annotation>
</semantics>
</math></span><img src="./e3b358fa81c8ea73b1d0e9284304d63e8f307cb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.286ex; height:6.843ex;" alt="{\displaystyle F=\prod _{i=1}^{n}J_{i}}" loading="lazy"></span>, where for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>i</mi>
<mo>≤<!-- ≤ --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n}</annotation>
</semantics>
</math></span><img src="./abbe58b9b83f8b6ec0da570e2249323a8930ef1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J_{i}}</annotation>
</semantics>
</math></span><img src="./07a98dbd556141d7f0a16eed84fe0ae6467dcd23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.09ex; height:2.509ex;" alt="{\displaystyle J_{i}}" loading="lazy"></span> is either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation>
</semantics>
</math></span><img src="./0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation>
</semantics>
</math></span><img src="./5acdcac635f883f8b4f0a01aa03b16b22f23b124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}" loading="lazy"></span>, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>
</semantics>
</math></span><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"></span>. The dimension of the face <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is the number of indices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}=[0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J_{i}=[0,1]}</annotation>
</semantics>
</math></span><img src="./6098f38c7d820054bb50ad21fa14a18bc14afedf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.841ex; height:2.843ex;" alt="{\displaystyle J_{i}=[0,1]}" loading="lazy"></span>; a face of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-face, is itself naturally a unit elementary cube of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>, and is sometimes called a subcube of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span>. One can also regard <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \emptyset }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi mathvariant="normal">∅<!-- ∅ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \emptyset }</annotation>
</semantics>
</math></span><img src="./63b5cac0509e595a27cef28c6dbb19d0e5a25b2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \emptyset }" loading="lazy"></span> as a face of dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle -1}</annotation>
</semantics>
</math></span><img src="./e03d0251b380d21d87eb1cd07628008e254050c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\textstyle -1}" loading="lazy"></span>.
</p><p>A <b>cubed complex</b> is a metric <a href="Polyhedral_complex" title="Polyhedral complex">polyhedral complex</a> 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 <a href="Isometry" title="Isometry">isometric</a> identifications of faces. One often reserves the term <b>cubical complex</b>, 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.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>A cube complex is said to be <b>finite-dimensional</b> if the dimension of the cubical cells is bounded. It is <b>locally finite</b> if every cube is contained in only finitely many cubes.
</p>
<div class="mw-heading mw-heading3"><h3 id="With_irregular_cubes">With irregular cubes</h3></div>
<p>An <b>elementary interval</b> is a subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subsetneq \mathbf {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>⊊<!-- ⊊ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I\subsetneq \mathbf {R} }</annotation>
</semantics>
</math></span><img src="./7dbdf28a05e8f51ea97925e9320ee84a8376a180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.273ex; height:2.676ex;" alt="{\displaystyle I\subsetneq \mathbf {R} }" loading="lazy"></span> of the form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[l,l+1]\quad {\text{or}}\quad I=[l,l]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>l</mi>
<mo>,</mo>
<mi>l</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mspace width="1em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>or</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>I</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>l</mi>
<mo>,</mo>
<mi>l</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I=[l,l+1]\quad {\text{or}}\quad I=[l,l]}</annotation>
</semantics>
</math></span><img src="./76bef99dd86082d70f269a3011b568a6f8cb9709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.691ex; height:2.843ex;" alt="{\displaystyle I=[l,l+1]\quad {\text{or}}\quad I=[l,l]}" loading="lazy"></span></dd></dl>
<p>for some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\in \mathbf {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l\in \mathbf {Z} }</annotation>
</semantics>
</math></span><img src="./0d353c5bcc6f3be58e351d29b528585ade06ed07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.168ex; height:2.176ex;" alt="{\displaystyle l\in \mathbf {Z} }" loading="lazy"></span>. An <b>elementary cube</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> is the finite product of elementary intervals, i.e.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>×<!-- × --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>×<!-- × --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>×<!-- × --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>⊊<!-- ⊊ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}}</annotation>
</semantics>
</math></span><img src="./16e3563293c6d499a63f03d101f5b0640663cfa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.645ex; height:3.176ex;" alt="{\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1},I_{2},\ldots ,I_{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{1},I_{2},\ldots ,I_{d}}</annotation>
</semantics>
</math></span><img src="./abf28f398d373c161f1e8ae0935cedb1644c4d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.482ex; height:2.509ex;" alt="{\displaystyle I_{1},I_{2},\ldots ,I_{d}}" loading="lazy"></span> are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [0,1]^{n}}</annotation>
</semantics>
</math></span><img src="./40160923273b7109968df994dca832b91d957bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle [0,1]^{n}}" loading="lazy"></span> <a href="Embedding" title="Embedding">embedded</a> in <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{d}}</annotation>
</semantics>
</math></span><img src="./9dea41b71d57247b87530b8f06e2e5e01a49c2a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{d}}" loading="lazy"></span> (for some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,d\in \mathbf {N} \cup \{0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>,</mo>
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">N</mi>
</mrow>
<mo>∪<!-- ∪ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n,d\in \mathbf {N} \cup \{0\}}</annotation>
</semantics>
</math></span><img src="./ca8d0eca84034944160b97fd7b372a04ce9136ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.647ex; height:2.843ex;" alt="{\displaystyle n,d\in \mathbf {N} \cup \{0\}}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≤<!-- ≤ --></mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\leq d}</annotation>
</semantics>
</math></span><img src="./d4d2b70142fb297b36d6f85bbc63edc4b4414eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.709ex; height:2.343ex;" alt="{\displaystyle n\leq d}" loading="lazy"></span>).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq \mathbf {R} ^{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>⊆<!-- ⊆ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\subseteq \mathbf {R} ^{d}}</annotation>
</semantics>
</math></span><img src="./c296406609e9d596c8d0485c84208ed43c3a9877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.174ex; height:2.843ex;" alt="{\displaystyle X\subseteq \mathbf {R} ^{d}}" loading="lazy"></span> is a <b>cubical complex</b> (or <b>cubical set</b>) if it can be written as a union of elementary cubes (or possibly, is <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> to such a set).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading4"><h4 id="Related_terminology">Related terminology</h4></div>
<p>Elementary intervals of length 0 (containing a single point) are called <b>degenerate</b>, while those of length 1 are <b>nondegenerate</b>. The <b>dimension</b> of a cube is the number of nondegenerate intervals in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span>, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>dim</mi>
<mo><!-- --></mo>
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \dim Q}</annotation>
</semantics>
</math></span><img src="./050c71192db0da75ac50e82679701397c973086e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.101ex; height:2.509ex;" alt="{\displaystyle \dim Q}" loading="lazy"></span>. The dimension of a cubical complex <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is the largest dimension of any cube in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> are elementary cubes and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\subseteq P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>⊆<!-- ⊆ --></mo>
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q\subseteq P}</annotation>
</semantics>
</math></span><img src="./a5a8d9796569e0a3c1ada1dc1d5c91436e12bda5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle Q\subseteq P}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> is a <b>face</b> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> is a face of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
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</p>
<div class="mw-heading mw-heading2"><h2 id="In_algebraic_topology">In algebraic topology</h2></div>
<p>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 <a href="Singular_homology" title="Singular homology">singular homology</a>, but is <a href="Computability" title="Computability">computable</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="In_geometric_group_theory">In geometric group theory</h2></div>
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<p>Groups acting geometrically by isometries on CAT(0) cube complexes provide a wide class of examples of <a href="CAT(0)_group" title="CAT(0) group">CAT(0) groups</a>.
</p><p>The Sageev construction can be understood as a higher-dimensional generalization of <a href="Bass%E2%80%93Serre_theory" title="Bass–Serre theory">Bass-Serre theory</a>, where the trees are replaced by CAT(0) cube complexes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Work by Daniel Wise has provided foundational examples of cubulated groups.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Agol's theorem that cubulated hyperbolic groups are virtually special has settled the hyperbolic <a href="Virtually_Haken_conjecture" title="Virtually Haken conjecture">virtually Haken conjecture</a>, which was the only case left of this conjecture after Thurston's <a href="Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> was proved by Perelman.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="CAT(0)_cube_complexes">CAT(0) cube complexes</h3></div>
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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Median_graph" title="Median graph">Median graph</a></div>
<div class="mw-heading mw-heading4"><h4 id="Gromov's_theorem">Gromov's theorem</h4></div>
<div class="mw-heading mw-heading4"><h4 id="Hyperplanes">Hyperplanes</h4></div>
<div class="mw-heading mw-heading3"><h3 id="CAT(0)_cube_complexes_and_group_actions">CAT(0) cube complexes and group actions</h3></div>
<div class="mw-heading mw-heading4"><h4 id="The_Sageev_construction">The Sageev construction</h4></div>
<div class="mw-heading mw-heading4"><h4 id="RAAGs_and_RACGs">RAAGs and RACGs</h4></div>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
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<ul><li><a href="Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>
<li><a href="Simplicial_homology" title="Simplicial homology">Simplicial homology</a></li>
<li><a href="Abstract_cell_complex" title="Abstract cell complex">Abstract cell complex</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><div id="Topology1035" style="font-size:114%;margin:0 4em"><a href="Topology" title="Topology">Topology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="General_topology" title="General topology">General (point-set)</a></li>
<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>
<li><a href="Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li>
<li><a href="Continuum_(topology)" title="Continuum (topology)">Continuum</a></li>
<li><a href="Differential_topology" title="Differential topology">Differential</a></li>
<li><a href="Geometric_topology" title="Geometric topology">Geometric</a>
<ul><li><a href="Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>
<li><a href="Homology_(mathematics)" title="Homology (mathematics)">Homology</a>
<ul><li><a href="Cohomology" title="Cohomology">cohomology</a></li></ul></li>
<li><a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>
<li><a href="Digital_topology" title="Digital topology">Digital</a></li></ul>
</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="Klein_bottle" title="Klein bottle"></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Open_set" title="Open set">Open set</a> / <a href="Closed_set" title="Closed set">Closed set</a></li>
<li><a href="Interior_(topology)" title="Interior (topology)">Interior</a></li>
<li><a href="Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>
<li><a href="Topological_space" title="Topological space">Space</a>
<ul><li><a href="Compact_space" title="Compact space">compact</a></li>
<li><a href="Connected_space" title="Connected space">connected</a></li>
<li><a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>
<li><a href="Metric_space" title="Metric space">metric</a></li>
<li><a href="Uniform_space" title="Uniform space">uniform</a></li></ul></li>
<li><a href="Homotopy" title="Homotopy">Homotopy</a>
<ul><li><a href="Homotopy_group" title="Homotopy group">homotopy group</a></li>
<li><a href="Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>
<li><a href="Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>
<li><a href="CW_complex" title="CW complex">CW complex</a></li>
<li><a href="Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>
<li><a href="Manifold" title="Manifold">Manifold</a></li>
<li><a href="Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>
<li><a href="Second-countable_space" title="Second-countable space">Second-countable space</a></li>
<li><a href="Cobordism" title="Cobordism">Cobordism</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>
<li><a href="Betti_number" title="Betti number">Betti number</a></li>
<li><a href="Winding_number" title="Winding number">Winding number</a></li>
<li><a href="Chern_class" title="Chern class">Chern number</a></li>
<li><a href="Orientability" title="Orientability">Orientability</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>
<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>
<li><a href="Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>
<li><a href="Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>
<li><a href="Tychonoff's_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>
<li><a href="Urysohn's_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>
<li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li>
<li><span class="noviewer" typeof="mw:File"></span> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw external" title="wikibooks:Topology">Wikibook</a></li>
<li><span class="noviewer" typeof="mw:File"></span> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw external" title="wikiversity:Topology">Wikiversity</a></li>
<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"></span></span> <a href="List_of_topology_topics" title="List of topology topics">Topics</a>
<ul><li><a href="List_of_general_topology_topics" title="List of general topology topics">general</a></li>
<li><a href="List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>
<li><a href="List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>
<li><span class="noviewer" typeof="mw:File"><span title="List-Class article"></span></span> <a href="List_of_important_publications_in_mathematics" class="mw-redirect" title="List of important publications in mathematics">Publications</a></li></ul>
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