In mathematics, specifically algebraic topology, an EilenbergâMacLane space [note 1] is a topological space with a single nontrivial homotopy group.
Let G be a group and n a positive integer. A connected topological space X is called an EilenbergâMacLane space of type , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that , EilenbergâMacLane 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 name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
As such, an EilenbergâMacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.
A generalised EilenbergâMaclane space is a space which has the homotopy type of a product of EilenbergâMaclane spaces .
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 .
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.
Another simplicial construction, in terms of classifying spaces and universal bundles, is given in J. Peter May's book. [5]
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 â 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.
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.
A constructive proof of this theorem can be found here, [6] another making use of the relation between omega-spectra and generalized reduced cohomology theories can be found here [7] and the main idea is sketched later as well.
The loop space of an EilenbergâMacLane space is again an EilenbergâMacLane 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 EilenbergâMacLane spaces with various n form an omega-spectrum, called an "EilenbergâMacLane 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.
In a more general context, Brown representability says that every reduced cohomology theory on based CW-complexes comes from an omega-spectrum.
For a fixed abelian group there are maps on the stable homotopy groups
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.
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
Every connected CW-complex possesses a Postnikov tower, that is an inverse system of spaces:
such that for every :
Dually there exists a Whitehead tower, which is a sequence of CW-complexes:
such that for every :
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]
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]
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.
The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence
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.
{{
cite book}}
: CS1 maint: location (
link)
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.
In mathematics, specifically algebraic topology, an EilenbergâMacLane space [note 1] is a topological space with a single nontrivial homotopy group.
Let G be a group and n a positive integer. A connected topological space X is called an EilenbergâMacLane space of type , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that , EilenbergâMacLane 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 name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
As such, an EilenbergâMacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.
A generalised EilenbergâMaclane space is a space which has the homotopy type of a product of EilenbergâMaclane spaces .
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 .
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.
Another simplicial construction, in terms of classifying spaces and universal bundles, is given in J. Peter May's book. [5]
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 â 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.
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.
A constructive proof of this theorem can be found here, [6] another making use of the relation between omega-spectra and generalized reduced cohomology theories can be found here [7] and the main idea is sketched later as well.
The loop space of an EilenbergâMacLane space is again an EilenbergâMacLane 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 EilenbergâMacLane spaces with various n form an omega-spectrum, called an "EilenbergâMacLane 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.
In a more general context, Brown representability says that every reduced cohomology theory on based CW-complexes comes from an omega-spectrum.
For a fixed abelian group there are maps on the stable homotopy groups
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.
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
Every connected CW-complex possesses a Postnikov tower, that is an inverse system of spaces:
such that for every :
Dually there exists a Whitehead tower, which is a sequence of CW-complexes:
such that for every :
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]
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]
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.
The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence
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.
{{
cite book}}
: CS1 maint: location (
link)
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.