Paths and cylinders
An Eilenberg–Mac Lane space is a connected topological space with nontrivial homotopy groups only in a single degree.
For and a group, and an abelian group if , then an Eilenberg-MacLane space is a topological space with the property that all its homotopy groups are trivial, except that in degree , which is .
Via geometric realization of higher groupoids
For a group, the Eilenberg–Mac Lane space is the image under the homotopy hypothesis Quillen equivalence of the one-object groupoid whose hom-set is :
The construction of EM-spaces for an abelian group may be given by the Dold-Kan correspondence between chain complexes and simplicial abelian groups: let be the chain complex which is in dimension and zero elsewhere; the geometric realisation of the corresponding simplicial abelian group is then a .
We can include the case when may be nonabelian, by regarding as a crossed complex. Its classifying space is then a . (This also includes the case when is just a set!) This method also allows for the construction of where is a group, or groupoid, and is a -module. This gives a space with , all other homotopy trivial, and with the given operation of on .
For an abelian group, the Eilenberg–Mac Lane space is the image of the ∞-groupoid that is the strict ω-groupoid given by the crossed complex that is trivial everywhere except in degree , where it is :
Therefore Eilenberg–Mac Lane spaces constitute a spectrum: the Eilenberg–Mac Lane spectrum.
In general, if is an abelian topological group, then there exist a model for the classifying space which is an abelian topological group. Iterating this construction, one has a notion of and a model for it which is an abelian topological group. If moreover is discrete, then , and one inductively sees that . Therefore one has a model for which is an abelian topological group.
See for instance (May, chapter 16, section 5)
Via linearization of spheres
(Aguilar-Gitler-Prieto 02, def. 6.4.20)
For a countable abelian group, then the reduced -linearization (def. 1) is an Eilenberg-MacLane space, in that its homotopy groups are
(in particular for then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).
(Aguilar-Gitler-Prieto 02, corollary 6.4.23)
With coefficients being EM-spaces
One common use of Eilenberg–Mac Lane spaces is as coefficient objects for “ordinary” cohomology (see e.g. May, chapter 22).
The th “ordinary” cohomology of a topological space with coefficients in (when ) or (generally) is the collection of homotopy classes of maps from into or , respectively:
Here on the right and denotes the homotopy category of the (∞,1)-categories of topological spaces and of ∞-groupoids, respectively.
Not only the set is related to the cohomology of with coefficients in , but also the higher homotopy groups are, and in the most obvious way: if is a connected CW-complex, then
for any choice of base point on the right hand sides. This fact, which appears to have first been remarked by Thom and Federer, is an immediate consequence of the natural homotopy equivalences
one has in every -topos, see loop space object. For a nonabelian group, Gottlieb proves the following nonabelian analogue of the above result: let be a finite dimensional connected CW-complex; for a fixed map , let be the centralizer in of . Then the connected component of in is a .
Notice that for a nonabelian group, is a simple (and the most familiar) example of nonabelian cohomology. Nonabelian cohomology in higher degrees is obtained by replacing here the coefficient -groupoids of the simple form with more general -groupoids.
Of EM spaces
On the other hand there is the cohomology of Eilenberg-MacLane spaces itself. This is in general rich. Classical results by Serre and Henri Cartan are reviewed in (Clement 02, section 2).
For all even , the ordinary cohomology ring of with coefficients in the rational numbers is the polynomial algebra on the generator . For all odd it is the exterior algebra on this generator:
This is reviewed for instance in (Yin, section 4).
The notion of Eilenberg–Mac Lane object makes sense in every -topos, not just in Top. See at Eilenberg-MacLane object.
Textbook references include
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Peter May, A concise course in algebraic topology (pdf)
The construction via reduced linearization of spheres is considered in
Quick review includes
- Xi Yin, On Eilenberg-MacLane spaces (pdf)
Formalization of Eilenberg-MacLane spaces in homotopy type theory is discussed in
The cohomology of Eilenberg-MacLane spaces is discussed in
- Alain Clément, Integral Cohomology of Finite Postnikov Towers, 2002 (pdf)