In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
, a classifying space ''BG'' of a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
''G'' is the quotient of a
weakly contractible space ''EG'' (i.e. a topological space all of whose
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s are trivial) by a proper
free action of ''G''. It has the property that any ''G''
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
over a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
manifold is isomorphic to a
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of the principal bundle ''EG'' → ''BG''. As explained later, this means that classifying spaces
represent a set-valued
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
on the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s, 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
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
''G'', ''BG'' is, roughly speaking, a
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' such that the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of ''X'' is isomorphic to ''G'' and the higher
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
of ''X'' are
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
, that is, ''BG'' is an
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
, or a ''K(G,1)''.
Motivation
An example of a classifying space for the
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
''G'' is the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
as ''X''. When ''G'' is a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
, another way to specify the condition on ''X'' is that the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
''Y'' of ''X'' is
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
. In that case the projection map
:
becomes a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
with structure group ''G'', in fact a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
for ''G''. The interest in the classifying space concept really arises from the fact that in this case ''Y'' has a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
with respect to principal ''G''-bundles, in the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
. 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 ''
freely''. (The
weak equivalence 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
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
''R'', which is contractible. Taking ''X'' as the
quotient space circle, we can regard the projection π from ''R'' = ''Y'' to ''X'' as a
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helic ...
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.
Formalism
A more formal statement takes into account that ''G'' may be a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
(not simply a ''discrete group''), and that
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
s 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
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
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
universal bundle over ''BG''. That is, what is provided is in fact a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
:
Assume that the homotopy category of
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es 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
:
over a space ''Z'', there is a classifying map φ from ''Z'' to ''BG'', such that γ is the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of π along φ. In less abstract terms, the construction of γ 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
bar construction), that gave concrete descriptions of ''BG'' as a
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
for an arbitrary discrete group. Such constructions make evident the connection with
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
.
Specifically, ''EG'' be the
weak simplicial complex whose ''n-'' simplices are the ordered (''n''+1)-tuples