Quotient of a weakly contractible space by a free action
In mathematics, specifically in homotopy theory, a classifying spaceBG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle .[1] As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.
For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg–MacLane space, or a K(G, 1).
^Stasheff, James D. (1971), "H-spaces and classifying spaces: foundations and recent developments", Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), American Mathematical Society, pp. 247–272 Theorem 2, doi:10.1090/pspum/022/0321079, ISBN 978-0-8218-9308-1, MR 0321079
by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifyingspace up to homotopy. For...
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X determines a classifying map from X to RP∞, making L a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used...
of classifyingspaces. The idea that a classifyingspaceclassifies principal bundles can be pushed further. For example, one might try to classify cohomology...
second polynomial is identically zero. The classifyingspace for U(n) is described in the article Classifyingspace for U(n). Special unitary group Projective...
\operatorname {BO} (n)} , Classifyingspace for orthogonal group BO {\displaystyle \operatorname {BO} } , Classifyingspace for infinite orthogonal group...
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finite group acting faithfully is an affine space group. Combining these results shows that classifyingspace groups in n dimensions up to conjugation by...
: H ⟶ G ) {\displaystyle M=(d\colon H\longrightarrow G)\!} has a classifyingspace BM with the property that its homotopy groups are Coker d, in dimension...
arbitrary positive scalar) the problem of classifying topological invariants reduces to the problem of classifying all possible inequivalent choices of Γ...
_{n}\mathbf {RP} ^{n}.} This space is classifyingspace of O(1), the first orthogonal group. The double cover of this space is the infinite sphere S ∞ {\displaystyle...
{\displaystyle \operatorname {BU} (n)} , Classifyingspace for unitary group BU {\displaystyle \operatorname {BU} } , Classifyingspace for infinite unitary group Backup...
structure group a given topological group G, is a specific bundle over a classifyingspace BG, such that every bundle with the given structure group G over M...
G is a discrete group, the classifying topos for G-torsors over a topos is the topos BG of G-sets. The classifyingspace of topological groups in homotopy...
is a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifyingspace for U(1). P U ( H...
left is a classifyingspace for G {\displaystyle G} . It is an Eilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} , i.e., a space whose fundamental...
from M to the classifyingspace such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifyingspace, and the Chern...
space machine produces a group completion of X together with infinite loop space structure. For example, one can take X to be the classifyingspace of...
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