Let G be a group and n a positive
integer. A
connected topological space X is called an Eilenberg–MacLane space of type , if it has n-th
homotopy groupisomorphic to G and all other homotopy groups
trivial. Assuming that G is
abelian in the case that , Eilenberg–MacLane spaces of type always exist, and are all weak homotopy equivalent. Thus, one may consider as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a " or as "a model of ". Moreover, it is common to assume that this space is a CW-complex (which is always possible via CW approximation).
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.[1]
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.[2] Note this includes as a .
Some further elementary examples can be constructed from these by using the fact that the product is . For instance the
n-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-
spheres, 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.[4] This gives an explicit presentation of simplicial abelian groups which represent Eilenberg-Maclane spaces.
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. This was exploited by
Jean-Pierre Serre while he studied the homotopy groups of spheres using the
Postnikov system 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 of based homotopy classes of based maps from X to is in natural bijection with the n-th
singular cohomology group of the space X. Thus one says that the are
representing spaces for singular cohomology with coefficients in G. Since
there is a distinguished element corresponding to the identity. The above bijection is given by the pullback of that element . This is similar to the
Yoneda lemma of
category theory.
The
loop space of an Eilenberg–MacLane space is again an Eilenberg–MacLane space: . Further there is an adjoint relation between the loop-space and the reduced suspension: , which gives the structure of an abelian group, where the operation is the concatenation of loops. This makes the bijection mentioned above a group isomorphism.
Also this property implies that Eilenberg–MacLane spaces with various n form an
omega-spectrum, called an "Eilenberg–MacLane spectrum". This spectrum defines via a reduced cohomology theory on based CW-complexes and for any reduced cohomology theory on CW-complexes with for there is a natural isomorphism , where denotes reduced singular cohomology. Therefore these two cohomology theories coincide.
induced by the map . Taking the direct limit over these maps, one can verify that this defines a reduced homology theory
on CW complexes. Since vanishes for , agrees with reduced singular homology with coefficients in G on CW-complexes.
Functoriality
It follows from the
universal coefficient theorem for cohomology that the Eilenberg MacLane space is a quasi-functor of the group; that is, for each positive integer if is any homomorphism of abelian groups, then there is a
non-empty set
satisfying
where denotes the homotopy class of a continuous map and
Relation with Postnikov/Whitehead tower
Every connected CW-complex possesses a
Postnikov tower, that is an inverse system of spaces:
such that for every :
there are commuting maps , which induce isomorphism on for ,
for ,
the maps are fibrations with fiber .
Dually there exists a
Whitehead tower, which is a sequence of CW-complexes:
With help of
Serre spectral sequences computations of higher
homotopy groups of spheres can be made. For instance and using a Whitehead tower of can be found here,[8] more generally those of using a Postnikov systems can be found here. [9]
Cohomology operations
For fixed natural numbers m,n and abelian groups G,H exists a bijection between the set of all
cohomology operations and defined by , where is a
fundamental class.
As a result, cohomology operations cannot decrease the degree of the cohomology groups and degree preserving cohomology operations are corresponding
to coefficient homomorphism . This follows from the
Universal coefficient theorem for cohomology and the
(m-1)-connectedness of .
Some interesting examples for cohomology operations are
Steenrod Squares and Powers, when are
finite cyclic groups. When studying those the importance of the cohomology of with coefficients in becomes apparent quickly;[10] some extensive tabeles of those groups can be found here. [11]
Group (co)homology
One can define the
group (co)homology of G with coefficients in the group A as the singular (co)homology of the Eilenberg-MacLane space with coefficients in A.
with the
string group, and the
spin group. The relevance of lies in the fact that there are the homotopy equivalences
for the
classifying space, and the fact . Notice that because the complex spin group is a group extension
,
the String group can be thought of as a "higher" complex spin group extension, in the sense of
higher group theory since the space is an example of a higher group. It can be thought of the topological realization of the
groupoid whose object is a single point and whose morphisms are the group . Because of these homotopical properties, the construction generalizes: any given space can be used to start a short exact sequence that kills the homotopy group in a
topological group.
^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.
MR13312) In this context it is therefore conventional to write the name without a space.
The Cartan seminar contains many fundamental results about Eilenberg-Maclane spaces including their homology and cohomology, and
applications for calculating the homotopy groups of spheres.