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
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 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. (See e.g. ) In this context it is therefore conventional to write the name without a space.] is a
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 ...
with a single nontrivial
homotopy group.
Let ''G'' be a group and ''n'' a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. A
connected topological space ''X'' is called an Eilenberg–MacLane space of type
, if it has ''n''-th
homotopy group isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to ''G'' and all other homotopy groups
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 ...
. If
then ''G'' must be
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
. Such a space exists, is a
CW-complex, and is unique up to a
weak homotopy equivalence, therefore any such space is often just called
.
The name is derived from
Samuel Eilenberg and
Saunders Mac Lane, who introduced such spaces in the late 1940s.
As such, an Eilenberg–MacLane space is a special kind 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 ...
that in
homotopy theory can be regarded as a building block for CW-complexes via
fibrations in a
Postnikov system
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
. These spaces are important in many contexts in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, including computations of
homotopy groups of spheres, definition of
cohomology operations, and for having a strong connection to
singular cohomology.
A generalised Eilenberg–Maclane space is a space which has the homotopy type of a
product of Eilenberg–Maclane spaces
.
Examples
* The
unit circle is a
.
* The infinite-dimensional
complex projective space is a model of
.
* The infinite-dimensional
real projective space is a
.
* The
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the q ...
of ''k''
unit circles
is a
, where
is the
free group on ''k'' generators.
* The complement to any connected knot or graph in a 3-dimensional sphere
is of type
; this is called the "
asphericity of knots", and is a 1957 theorem of
Christos Papakyriakopoulos
Christos Dimitriou Papakyriakopoulos (), commonly known as Papa (Greek: Χρήστος Δημητρίου Παπακυριακόπουλος ; June 29, 1914 – June 29, 1976), was a Greek mathematician specializing in geometric topology.
Early li ...
.
* Any
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, connected,
non-positively curved manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'' is a
, where
is the
fundamental group of ''M''. This is a consequence of the
Cartan–Hadamard theorem.
* An infinite
lens space given by the quotient of
by the free action
for
is a
. This can be shown using
covering space theory and the fact that the infinite dimensional sphere 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 ...
. Note this includes
as a
.
* The
configuration space of
points in the plane is a
, where
is the
pure braid group on
strands.
* Correspondingly the
th unordered configuration space of
is a
, where
denotes the
-strand braid group.
* The
infinite symmetric product of a
n-sphere is a
. More generally
is a
for all
Moore spaces .
Some further elementary examples can be constructed from these by using the fact that the product
is
. For instance the
-dimensional Torus is a
.
Remark on constructing Eilenberg–MacLane spaces
For
and
an arbitrary
group the construction of
is identical to that of the
classifying space of the group
. Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional.
There are multiple techniques for constructing higher Eilenberg-Maclane spaces. One of which is to construct a
Moore space for an abelian group
: Take the
wedge of ''n''-
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s, one for each generator of the group ''A'' and realise the relations between these generators by attaching ''(n+1)''-cells via corresponding maps in
of said wedge sum. Note that the lower homotopy groups
are already trivial by construction. Now iteratively kill all higher homotopy groups
by successively attaching cells of dimension greater than
, and define
as
direct limit under inclusion of this iteration.
Another useful technique is to use the geometric realization of
simplicial abelian groups. This gives an explicit presentation of simplicial abelian groups which represent Eilenberg-Maclane spaces.
Another simplicial construction, in terms of
classifying spaces and
universal bundles, is given in
J. Peter May's book.
Since taking the loop space lowers the homotopy groups by one slot, we have a canonical homotopy equivalence
, hence there is a fibration sequence
:
.
Note that this is not a cofibration sequence ― the space
is not the homotopy cofiber of
.
This fibration sequence can be used to study the cohomology of
from
using the
Leray spectral sequence In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
Definition
Let f:X\to Y be a conti ...
. This was exploited by
Jean-Pierre Serre while he studied the homotopy groups of spheres using the
Postnikov system
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
and spectral sequences.
Properties of Eilenberg–MacLane spaces
Bijection between homotopy classes of maps and cohomology
An important property of
's is that for any abelian group ''G'', and any based CW-complex ''X'', the set