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Classifying space

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In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically in <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a>, a classifying space BG of a <a href="Topological_group" title="Topological group">topological group</a> G is the quotient of a <a href="Weakly_contractible" title="Weakly contractible">weakly contractible</a> space EG (i.e., a topological space all of whose <a href="Homotopy_group" title="Homotopy group">homotopy groups</a> are trivial) by a proper <a href="Free_action" class="mw-redirect" title="Free action">free action</a> of G. It has the property that any G <a href="Principal_bundle" title="Principal bundle">principal bundle</a> over a <a href="Paracompact" class="mw-redirect" title="Paracompact">paracompact</a> manifold is isomorphic to a <a href="Pullback_bundle" title="Pullback bundle">pullback</a> of the principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EG\to BG}">

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<annotation encoding="application/x-tex">{\displaystyle EG\to BG}</annotation>

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</math><img src="./5fd59bc3c096beb48769e8af1272ecb42ce80425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.807ex; height:2.176ex;" alt="{\displaystyle EG\to BG}" loading="lazy">.<a href="#cite_note-1">[1]</a> As explained later, this means that classifying spaces <a href="Representable_functor" title="Representable functor">represent</a> a set-valued <a href="Functor" title="Functor">functor</a> on the <a href="Homotopy_category" title="Homotopy category">homotopy category</a> of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of <a href="Topological_space" title="Topological space">topological spaces</a>, such as <a href="Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a>. This notion is generalized by the notion of <a href="Classifying_topos" title="Classifying topos">classifying topos</a>. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.

For a <a href="Discrete_group" title="Discrete group">discrete group</a> G, BG is a <a href="Connected_space" title="Connected space">path-connected</a> <a href="Topological_space" title="Topological space">topological space</a> X such that the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of X is isomorphic to G and the higher <a href="Homotopy_groups" class="mw-redirect" title="Homotopy groups">homotopy groups</a> of X are <a href="Trivial_group" title="Trivial group">trivial</a>; that is, BG is an <a href="Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a>, specifically a K(G, 1).

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

An example of a classifying space for the <a href="Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a> G is the <a href="Circle" title="Circle">circle</a> as X. When G is a <a href="Discrete_group" title="Discrete group">discrete group</a>, another way to specify the condition on X is that the <a href="Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a> Y of X is <a href="Contractible" class="mw-redirect" title="Contractible">contractible</a>. In that case the projection map

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon Y\longrightarrow X\ }</annotation>

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</math><img src="./c3c4d53ae2a7160f9feaafcbff73c9122bd21764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.796ex; height:2.176ex;" alt="{\displaystyle \pi \colon Y\longrightarrow X\ }" loading="lazy"></dd></dl>

becomes a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> with structure group G, in fact a <a href="Principal_bundle" title="Principal bundle">principal bundle</a> for G. The interest in the classifying space concept really arises from the fact that in this case Y has a <a href="Universal_property" title="Universal property">universal property</a> with respect to principal G-bundles, in the <a href="Homotopy_category" title="Homotopy category">homotopy category</a>. This is actually more basic than the condition that the higher homotopy groups vanish: the fundamental idea is, given G, to find such a contractible space Y on which G acts <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">freely</a>. (The <a href="Weak_equivalence_(homotopy_theory)" title="Weak equivalence (homotopy theory)">weak equivalence</a> idea of homotopy theory relates the two versions.) In the case of the circle example, what is being said is that we remark that an infinite cyclic group C acts freely on the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> R, which is contractible. Taking X as the <a href="Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> circle, we can regard the projection π from R = Y to X as a <a href="Helix" title="Helix">helix</a> in geometrical terms, undergoing projection from three dimensions to the plane. What is being claimed is that π has a universal property amongst principal C-bundles; that any principal C-bundle in a definite way 'comes from' π.

<div class="mw-heading mw-heading2"><h2 id="Formalism">Formalism</h2></div>

A more formal statement takes into account that G may be a <a href="Topological_group" title="Topological group">topological group</a> (not simply a discrete group), and that <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group actions</a> of G are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the <a href="Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a> construction. In homotopy theory the definition of a topological space BG, the classifying space for principal G-bundles, is given, together with the space EG which is the total space of the <a href="Universal_bundle" title="Universal bundle">universal bundle</a> over BG. That is, what is provided is in fact a <a href="Continuous_mapping" class="mw-redirect" title="Continuous mapping">continuous mapping</a>

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon EG\longrightarrow BG.}</annotation>

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</math><img src="./71b0842b751f0e1cf3dff786345f9ce02824035f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.302ex; height:2.176ex;" alt="{\displaystyle \pi \colon EG\longrightarrow BG.}" loading="lazy"></dd></dl>

Assume that the homotopy category of <a href="CW_complex" title="CW complex">CW complexes</a> is the underlying category, from now on. The classifying property required of BG in fact relates to π. We must be able to say that given any principal G-bundle

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<annotation encoding="application/x-tex">{\displaystyle \gamma \colon Y\longrightarrow Z\ }</annotation>

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</math><img src="./7b26064c4de82885b679290dee449330e91fc24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.426ex; height:2.676ex;" alt="{\displaystyle \gamma \colon Y\longrightarrow Z\ }" loading="lazy"></dd></dl>

over a space Z, there is a classifying map φ from Z to BG, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is the <a href="Pullback_of_a_bundle" class="mw-redirect" title="Pullback of a bundle">pullback</a> of π along φ. In less abstract terms, the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.

For this to be a useful concept, there evidently must be some reason to believe such spaces BG exist. The early work on classifying spaces introduced constructions (for example, the <a href="Bar_construction" class="mw-redirect" title="Bar construction">bar construction</a>), that gave concrete descriptions of BG as a <a href="Simplicial_complex" title="Simplicial complex">simplicial complex</a> for an arbitrary discrete group. Such constructions make evident the connection with <a href="Group_cohomology" title="Group cohomology">group cohomology</a>.

Specifically, let EG be the <a href="Delta_set" title="Delta set">weak simplicial complex</a> whose n- simplices are the ordered (n+1)-tuples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [g_{0},\ldots ,g_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle [g_{0},\ldots ,g_{n}]}</annotation>

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</math><img src="./c18f1541a37bdfd5e7125bb41fb09017696e307e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.963ex; height:2.843ex;" alt="{\displaystyle [g_{0},\ldots ,g_{n}]}" loading="lazy"> of elements of G. Such an n-simplex attaches to the (n−1) simplices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [g_{0},\ldots ,{\hat {g}}_{i},\ldots ,g_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle [g_{0},\ldots ,{\hat {g}}_{i},\ldots ,g_{n}]}</annotation>

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</math><img src="./053e69ad316489ad225ba505d556e952be0112dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.172ex; height:2.843ex;" alt="{\displaystyle [g_{0},\ldots ,{\hat {g}}_{i},\ldots ,g_{n}]}" loading="lazy"> in the same way a standard simplex attaches to its faces, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}_{i}}</annotation>

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</math><img src="./93c138fd9c628b35a7b5477b7798926df931674f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.031ex; height:2.676ex;" alt="{\displaystyle {\hat {g}}_{i}}" loading="lazy"> means this vertex is deleted. The complex EG is contractible. The group G acts on EG by left multiplication,

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<annotation encoding="application/x-tex">{\displaystyle g\cdot [g_{0},\ldots ,g_{n}]=[gg_{0},\ldots ,gg_{n}],}</annotation>

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</math><img src="./0bdb2c4cbf761aa2cecc315b7b2e7a5178449990.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.698ex; height:2.843ex;" alt="{\displaystyle g\cdot [g_{0},\ldots ,g_{n}]=[gg_{0},\ldots ,gg_{n}],}" loading="lazy"></dd></dl>

and only the identity e takes any simplex to itself. Thus the action of G on EG is a covering space action and the quotient map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EG\to EG/G}">

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<annotation encoding="application/x-tex">{\displaystyle EG\to EG/G}</annotation>

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</math><img src="./5386873dec98b84998af9b2100dc863f563996fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.808ex; height:2.843ex;" alt="{\displaystyle EG\to EG/G}" loading="lazy"> is the universal cover of the orbit space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG=EG/G}">

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<annotation encoding="application/x-tex">{\displaystyle BG=EG/G}</annotation>

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</math><img src="./086c289fd7730560134fc5a816567855683d916a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.281ex; height:2.843ex;" alt="{\displaystyle BG=EG/G}" loading="lazy">, and BG is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(G,1)}">

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<annotation encoding="application/x-tex">{\displaystyle K(G,1)}</annotation>

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</math><img src="./f30ac500e56f9a311b1e02891755822a53a99af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.898ex; height:2.843ex;" alt="{\displaystyle K(G,1)}" loading="lazy">.<a href="#cite_note-2">[2]</a>

In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether a certain functor is <a href="Representable_functor" title="Representable functor">representable</a>: the <a href="Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a> from the homotopy category to the <a href="Category_of_sets" title="Category of sets">category of sets</a>, defined by

<dl><dd>h(Z) = set of isomorphism classes of principal G-bundles on Z.</dd></dl>

The abstract conditions being known for this (<a href="Brown's_representability_theorem" title="Brown's representability theorem">Brown's representability theorem</a>) ensure that the result, as an <a href="Existence_theorem" title="Existence theorem">existence theorem</a>, is affirmative and not too difficult.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ol><li>The <a href="Circle" title="Circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>

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</math><img src="./60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"> is a classifying space for the <a href="Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation>

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</math><img src="./89f4f38f32c2068bca9dc701d13b03dd4a5d52ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}" loading="lazy"> The total space is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mathbb {Z} =\mathbb {R} .}">

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<annotation encoding="application/x-tex">{\displaystyle E\mathbb {Z} =\mathbb {R} .}</annotation>

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</math><img src="./9743a48a5b92294a991f145b4476e3d0d57d3055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.749ex; height:2.176ex;" alt="{\displaystyle E\mathbb {Z} =\mathbb {R} .}" loading="lazy"></li>

<li>The <a href="Torus" title="Torus">n-torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{n}}</annotation>

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</math><img src="./533e91762d91f473171e75226e4c0fe059325e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {T} ^{n}}" loading="lazy"> is a classifying space for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation>

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</math><img src="./a9b5de7ced4588982b574fe19894aec6a3ca4c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}" loading="lazy">, the <a href="Free_abelian_group" title="Free abelian group">free abelian group</a> of rank n. The total space is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mathbb {Z} ^{n}=\mathbb {R} ^{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle E\mathbb {Z} ^{n}=\mathbb {R} ^{n}.}</annotation>

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</math><img src="./3ee48d08d8250a16e59f1d3a50f23bb806b9d0f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.186ex; height:2.343ex;" alt="{\displaystyle E\mathbb {Z} ^{n}=\mathbb {R} ^{n}.}" loading="lazy"></li>

<li>The wedge of n circles is a classifying space for the <a href="Free_group" title="Free group">free group</a> of rank n.</li>

<li>A <a href="Closed_manifold" title="Closed manifold">closed</a> (that is, <a href="Compact_space" title="Compact space">compact</a> and without boundary) connected <a href="Surface_(topology)" title="Surface (topology)">surface</a> S of <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> at least 1 is a classifying space for its <a href="Fundamental_group" title="Fundamental group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(S).}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{1}(S).}</annotation>

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</math><img src="./d6991bfb93f7214aff5797d6c2abdd5af0b7a8d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.335ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(S).}" loading="lazy"></li>

<li>A <a href="Closed_manifold" title="Closed manifold">closed</a> (that is, <a href="Compact_space" title="Compact space">compact</a> and without boundary) connected <a href="Hyperbolic_manifold" title="Hyperbolic manifold">hyperbolic manifold</a> M is a classifying space for its <a href="Fundamental_group" title="Fundamental group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{1}(M)}</annotation>

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</math><img src="./630292b85b0cf5ba68896c5711b07677efdb04cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(M)}" loading="lazy">.</li>

<li>A finite locally connected <a href="CAT(0)_space" class="mw-redirect" title="CAT(0) space">CAT(0)</a> <a href="Cubical_complex" title="Cubical complex">cubical complex</a> is a classifying space of its <a href="Fundamental_group" title="Fundamental group">fundamental group</a>.</li>

<li>The <a href="Real_projective_space#Infinite_real_projective_space" title="Real projective space">infinite-dimensional projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {RP} ^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {RP} ^{\infty }}</annotation>

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</math><img src="./7875ef4f8d5cf05563256a4848493221cd8cf5a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.974ex; height:2.343ex;" alt="{\displaystyle \mathbb {RP} ^{\infty }}" loading="lazy"> (the direct limit of finite-dimensional projective spaces) is a classifying space for the cyclic group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z} .}</annotation>

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</math><img src="./8106bbd8ca8a697b473708e8caedaf187544822d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.775ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{2}=\mathbb {Z} /2\mathbb {Z} .}" loading="lazy"> The total space is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\mathbb {Z} _{2}=S^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle E\mathbb {Z} _{2}=S^{\infty }}</annotation>

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</math><img src="./3322e57702ca3a372bc6fd9df5e3225aeb87c9e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.876ex; height:2.676ex;" alt="{\displaystyle E\mathbb {Z} _{2}=S^{\infty }}" loading="lazy"> (the direct limit of spheres <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle S^{n}.}</annotation>

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</math><img src="./92958fea8ef643336942ade776d4910aa8c8cd9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.387ex; height:2.343ex;" alt="{\displaystyle S^{n}.}" loading="lazy"> Alternatively, one may use Hilbert space with the origin removed; it is contractible).</li>

<li>The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mathbb {Z} _{n}=S^{\infty }/\mathbb {Z} _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B\mathbb {Z} _{n}=S^{\infty }/\mathbb {Z} _{n}}</annotation>

</semantics>

</math><img src="./d9987a3dd2860d11c49af467a79fca6f2814bc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.96ex; height:2.843ex;" alt="{\displaystyle B\mathbb {Z} _{n}=S^{\infty }/\mathbb {Z} _{n}}" loading="lazy"> is the classifying space for the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{n}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n}.}</annotation>

</semantics>

</math><img src="./87f39fa381c1cf2fb4d4a156e7320955d794c21a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.416ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{n}.}" loading="lazy"> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{\infty }}</annotation>

</semantics>

</math><img src="./bc6ba232b7e000afed285fec1f303447da8bf617.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.397ex; height:2.343ex;" alt="{\displaystyle S^{\infty }}" loading="lazy"> is understood to be a certain subset of the infinite dimensional Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{\infty }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{\infty }}</annotation>

</semantics>

</math><img src="./228f02204428653a7c0fd06092cbf0619e70dab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{\infty }}" loading="lazy"> with the origin removed; the cyclic group is considered to act on it by multiplication with roots of unity.</li>

<li>The unordered <a href="Configuration_space_(mathematics)" title="Configuration space (mathematics)">configuration space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})}">

<msub>

<mi>UConf</mi>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})}</annotation>

</semantics>

</math><img src="./b9ced90af3c12b3b13a0192a4a000e9363eed6ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.348ex; height:3.176ex;" alt="{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})}" loading="lazy"> is the classifying space of the <a href="Braid_group" title="Braid group">Artin braid group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy">,<a href="#cite_note-3">[3]</a> and the ordered configuration space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}</annotation>

</semantics>

</math><img src="./ee311322cc043f28c5c200291d73b0da2ab0c62e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.605ex; height:3.176ex;" alt="{\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}" loading="lazy"> is the classifying space for the pure Artin braid group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{n}.}</annotation>

</semantics>

</math><img src="./11a9a5ad583cfce82782ab280c38516688dff442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.358ex; height:2.509ex;" alt="{\displaystyle P_{n}.}" loading="lazy"></li>

<li>The (unordered) <a href="Configuration_space_(mathematics)" title="Configuration space (mathematics)">configuration space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{\infty })}">

<msub>

<mi>UConf</mi>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{\infty })}</annotation>

</semantics>

</math><img src="./fe364cb9dce38b1f7f0e161037ad12e4ba3d02b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.169ex; height:2.843ex;" alt="{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{\infty })}" loading="lazy"> is a classifying space for the symmetric group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{n}.}</annotation>

</semantics>

</math><img src="./a5a7e377e63c1f493aa1f3470f8af79e77c0c503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.29ex; height:2.509ex;" alt="{\displaystyle S_{n}.}" loading="lazy"><a href="#cite_note-4">[4]</a></li>

<li>The infinite dimensional complex <a href="Projective_space" title="Projective space">projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {CP} ^{\infty }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {CP} ^{\infty }}</annotation>

</semantics>

</math><img src="./18765e2dc0706599b9e7733ffe0d528ff7fccc76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.974ex; height:2.343ex;" alt="{\displaystyle \mathbb {CP} ^{\infty }}" loading="lazy"> is the classifying space BS1 for the circle S1 thought of as a compact topological group.</li>

<li>The <a href="Grassmannian" title="Grassmannian">Grassmannian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Gr(n,\mathbb {R} ^{\infty })}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Gr(n,\mathbb {R} ^{\infty })}</annotation>

</semantics>

</math><img src="./b0c8a2211fda4fbfd29d64241b59655d63aace73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.667ex; height:2.843ex;" alt="{\displaystyle Gr(n,\mathbb {R} ^{\infty })}" loading="lazy"> of n-planes in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\infty }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\infty }}</annotation>

</semantics>

</math><img src="./da5e5160fa2811da2c516b0fa543236c5cf707fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\infty }}" loading="lazy"> is the classifying space of the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(n). The total space is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EO(n)=V(n,\mathbb {R} ^{\infty })}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle EO(n)=V(n,\mathbb {R} ^{\infty })}</annotation>

</semantics>

</math><img src="./5cfb2821ce63f5313ab270cfb4049f9ef269516e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.43ex; height:2.843ex;" alt="{\displaystyle EO(n)=V(n,\mathbb {R} ^{\infty })}" loading="lazy">, the <a href="Stiefel_manifold" title="Stiefel manifold">Stiefel manifold</a> of n-dimensional orthonormal frames in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\infty }.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\infty }.}</annotation>

</semantics>

</math><img src="./6d0d5264c0c1be2da6b073065c6760226bea298d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.2ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\infty }.}" loading="lazy"></li></ol>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

This still leaves the question of doing effective calculations with BG; for example, the theory of <a href="Characteristic_class" title="Characteristic class">characteristic classes</a> is essentially the same as computing the <a href="Cohomology_group" class="mw-redirect" title="Cohomology group">cohomology groups</a> of BG, at least within the restrictive terms of homotopy theory, for interesting groups G such as <a href="Lie_group" title="Lie group">Lie groups</a> (H. Cartan's theorem). As was shown by the <a href="Bott_periodicity_theorem" title="Bott periodicity theorem">Bott periodicity theorem</a>, the <a href="Homotopy_group" title="Homotopy group">homotopy groups</a> of BG are also of fundamental interest.

An example of a classifying space is that when G is cyclic of order two; then BG is <a href="Real_projective_space" title="Real projective space">real projective space</a> of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>, with G acting via v going to −v, and allowing for <a href="Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a> in choosing BG. This example shows that classifying spaces may be complicated.

In relation with <a href="Differential_geometry" title="Differential geometry">differential geometry</a> (<a href="Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a>) and the theory of <a href="Grassmannian" title="Grassmannian">Grassmannians</a>, a much more hands-on approach to the theory is possible for cases such as the <a href="Unitary_group" title="Unitary group">unitary groups</a> that are of greatest interest. The construction of the <a href="Thom_complex" class="mw-redirect" title="Thom complex">Thom complex</a> MG showed that the spaces BG were also implicated in <a href="Cobordism_theory" class="mw-redirect" title="Cobordism theory">cobordism theory</a>, so that they assumed a central place in geometric considerations coming out of <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>. Since <a href="Group_cohomology" title="Group cohomology">group cohomology</a> can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much <a href="Homological_algebra" title="Homological algebra">homological algebra</a>.

Generalizations include those for classifying <a href="Foliation" title="Foliation">foliations</a>, and the <a href="Classifying_topos" title="Classifying topos">classifying toposes</a> for logical theories of the predicate calculus in <a href="Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a> that take the place of a 'space of models'.

<ul><li><a href="Classifying_space_for_O(n)" title="Classifying space for O(n)">Classifying space for O(n)</a>, BO(n)</li>

<li><a href="Classifying_space_for_U(n)" title="Classifying space for U(n)">Classifying space for U(n)</a>, BU(n)</li>

<li><a href="Classifying_space_for_SO(n)" title="Classifying space for SO(n)">Classifying space for SO(n)</a></li>

<li><a href="Classifying_space_for_SU(n)" title="Classifying space for SU(n)">Classifying space for SU(n)</a></li>

<li><a href="Classifying_stack" class="mw-redirect" title="Classifying stack">Classifying stack</a></li>

<li><a href="Borel's_theorem" title="Borel's theorem">Borel's theorem</a></li>

<li><a href="Equivariant_cohomology" title="Equivariant cohomology">Equivariant cohomology</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFStasheff1971" class="citation cs2"><a href="Jim_Stasheff" title="Jim Stasheff">Stasheff, James D.</a> (1971), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p-wCCAAAQBAJ&pg=PA247">"H-spaces and classifying spaces: foundations and recent developments"</a>, <a rel="nofollow" class="external text" href="http://www.ams.org/books/pspum/022/">Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970)</a>, <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, pp. 247–272 Theorem 2, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpspum%2F022%2F0321079">10.1090/pspum/022/0321079</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-9308-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0321079">0321079</a></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHatcher2002" class="citation book cs1"><a href="Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a> (2002). Algebraic topology. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 89. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-79160-X</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/45420394">45420394</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFArnold1969" class="citation book cs1">Arnold, Vladimir I. (1969). "The cohomology ring of the colored braid group". Vladimir I. Arnold — Collected Works. Springer. pp. 183–6. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-31031-7_18">10.1007/978-3-642-31031-7_18</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-31030-0</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/classifying+space">"classifying space in nLab"</a>. ncatlab.org. Retrieved 2017-08-22.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFMay1999" class="citation book cs1">May, J.P. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g8SG03R1bpgC&pg=PA3">A Concise Course in Algebraic Topology</a>. University of Chicago Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-226-51183-2</bdi>.</cite></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/classifying+space">Classifying space</a> at the <a href="NLab" title="NLab">nLab</a></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Classifying_space">"Classifying space"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-23" href="https://en.wikipedia.org/wiki/?title=Classifying_space&oldid=1297056813">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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