nLab
Eilenberg-Mac Lane space

Context

Topology

Homotopy theory

Contents

Idea

An Eilenberg–Mac Lane space is a connected topological space with nontrivial homotopy groups only in a single degree.

Definition

Characterization

For nn \in \mathbb{N} and GG a group, and an abelian group if n2n \geq 2, then an Eilenberg-MacLane space K(G,n)K(G,n) is a topological space with the property that all its homotopy groups are trivial, except that in degree nn, which is GG.

Constructions

Via geometric realization of higher groupoids

For GG a group, the Eilenberg–Mac Lane space K(G,1)K(G,1) is the image under the homotopy hypothesis Quillen equivalence ||:GrpdTop|-| : \infty Grpd \to Top of the one-object groupoid BG\mathbf{B}G whose hom-set is GG:

K(G,1)=|BG|. K(G,1) = | \mathbf{B} G | \,.

The construction of EM-spaces K(A,n)K(A,n) for an abelian group AA may be given by the Dold-Kan correspondence between chain complexes and simplicial abelian groups: let A[n]A[-n] be the chain complex which is AA in dimension nn and zero elsewhere; the geometric realisation of the corresponding simplicial abelian group is then a K(A,n)K(A,n).

We can include the case n=1n=1 when GG may be nonabelian, by regarding C(G,n)C(G,n) as a crossed complex. Its classifying space B(C(G,n))B(C(G,n)) is then a K(G,n)K(G,n). (This also includes the case n=0n=0 when GG is just a set!) This method also allows for the construction of K(M,n;G,1)K(M,n;G,1) where GG is a group, or groupoid, and MM is a GG-module. This gives a space with π 1=G\pi_1 =G, π n=M\pi_n=M all other homotopy trivial, and with the given operation of π 1\pi_1 on π n\pi_n.

For AA an abelian group, the Eilenberg–Mac Lane space K(A,n)K(A,n) is the image of the ∞-groupoid B nA\mathbf{B}^n A that is the strict ω-groupoid given by the crossed complex [B nA][\mathbf{B}^n A] that is trivial everywhere except in degree nn, where it is AA:

[B nA] =([B nA] n+1[B nA] n[B nA] n1[B nA] 1[B nA] 0) =(*A***). \begin{aligned} [\mathbf{B}^n A] &= ( \cdots \to [\mathbf{B}^n A]_{n+1} \to [\mathbf{B}^n A]_{n} \to [\mathbf{B}^n A]_{n-1} \cdots \to [\mathbf{B}^n A]_{1} \stackrel{\to}{\to} [\mathbf{B}^n A]_{0}) \\ &= ( \cdots \to {*} \to A \to {*} \cdots \to {*} \stackrel{\to}{\to} {*}) \end{aligned} \,.

So

K(A,n)=|B nA|. K(A,n) = |\mathbf{B}^n A| \,.

Therefore Eilenberg–Mac Lane spaces constitute a spectrum: the Eilenberg–Mac Lane spectrum.

In general, if AA is an abelian topological group, then there exist a model for the classifying space A\mathcal{B}A which is an abelian topological group. Iterating this construction, one has a notion of nA\mathcal{B}^n A and a model for it which is an abelian topological group. If moreover AA is discrete, then A=|BA|=K(A,1)\mathcal{B}A=|\mathbf{B}A|=K(A,1), and one inductively sees that nA=|B nA|=K(A,n)\mathcal{B}^n A=|\mathbf{B}^n A|=K(A,n). Therefore one has a model for K(A,n)K(A,n) which is an abelian topological group.

See for instance (May, chapter 16, section 5)

Via linearization of spheres

Definition

For AA an abelian group and nn \in \mathbb{N}, the reduced AA-linearization A[S n] *A[S^n]_\ast of the n-sphere S nS^n is the topological space, whose underlying set is the quotient

kA k×(S n) kA[S n] * \underset{k \in \mathbb{N}}{\sqcup} A^k \times (S^n)^k \longrightarrow A[S^n]_\ast

of the tensor product with AA of the free abelian group on the underlying set of S nS^n, by the relation that identifies every formal linear combination of the (any fixed) basepoint of S nS^n with 0. The topology is the induced quotient topology (of the disjoint union of product topological spaces, where AA is equipped with the discrete topology).

(Aguilar-Gitler-Prieto 02, def. 6.4.20)

Proposition

For AA a countable abelian group, then the reduced AA-linearization A[S n] *A[S^n]_\ast (def. 1) is an Eilenberg-MacLane space, in that its homotopy groups are

π q(A[S n] *){A ifq=n * otherwise \pi_q(A[S^n]_\ast) \simeq \left\{ \array{ A & if \; q = n \\ \ast & otherwise } \right.

(in particular for n1n \geq 1 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)

Remark

The topological space A[S n] *A[S^n]_\ast in definition 1 has a canonical continuous action of the orthogonal group O(n)O(n), by regarding S n( n) *S^n \simeq (\mathbb{R}^n)^\ast as the one-point compactification of Cartesian space. Hence, in view of prop. 1, these models for EM-spaces lend themselves to the defintiion of Eilenberg-MacLane spectra as orthogonal spectra. See there.

Cohomology

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 nnth “ordinary” cohomology of a topological space XX with coefficients in GG (when n=1n=1) or AA (generally) is the collection of homotopy classes of maps from XX into K(G,1)K(G,1) or K(A,n)K(A,n), respectively:

H 1(X,G)=Ho Top(X,K(G,1))=Ho Grpd(X,BG) H^1(X,G) = Ho_{Top}(X, K(G,1)) = Ho_{\infty Grpd}(X, \mathbf{B} G)
H n(X,A)=Ho Top(X,K(A,n))=Ho Grpd(X,B nA). H^n(X,A) = Ho_{Top}(X, K(A,n)) = Ho_{\infty Grpd}(X, \mathbf{B}^n A) \,.

Here on the right Ho TopHo_{Top} and Ho GrpdHo_{\infty Grpd} denotes the homotopy category of the (∞,1)-categories of topological spaces and of ∞-groupoids, respectively.

Not only the set π 0Top(X,K(A,n))=Ho Top(X,K(A,n))\pi_0\mathbf{Top}(X, K(A,n))=Ho_{Top}(X, K(A,n)) is related to the cohomology of XX with coefficients in AA, but also the higher homotopy groups π iTop(X,K(A,n))\pi_i\mathbf{Top}(X, K(A,n)) are, and in the most obvious way: if XX is a connected CW-complex, then

H ni(X,A)=π iTop(X,K(A,n))=π iGrpd(X,B nA), H^{n-i}(X,A)=\pi_i\mathbf{Top}(X, K(A,n))=\pi_i\mathbf{\infty Grpd}(X, \mathbf{B}^n A),

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

ΩH(X,Y)H(X,ΩY) \Omega\mathbf{H}(X,Y)\simeq \mathbf{H}(X,\Omega Y)

and

ΩK(A,n)K(A,n1) \Omega K(A,n)\simeq K(A,n-1)

one has in every (,1)(\infty,1)-topos, see loop space object. For GG a nonabelian group, Gottlieb proves the following nonabelian analogue of the above result: let XX be a finite dimensional connected CW-complex; for a fixed map f:XK(G,1)f:X\to K(G,1), let C fC_f be the centralizer in G=π 1K(G,1)G=\pi_1 K(G,1) of f *(π 1(X))f_*(\pi_1(X)). Then the connected component of ff in Top(X,K(G,1))\mathbf{Top}(X,K(G,1)) is a K(C f,1)K(C_f,1).

Notice that for GG a nonabelian group, H 1(X,G)H^1(X,G) is a simple (and the most familiar) example of nonabelian cohomology. Nonabelian cohomology in higher degrees is obtained by replacing here the coefficient \infty-groupoids of the simple form B nA\mathbf{B}^n A with more general \infty-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).

Proposition

For all even nn \in \mathbb{N}, the ordinary cohomology ring of K(,n)K(\mathbb{Z},n) with coefficients in the rational numbers is the polynomial algebra on the generator aH n(K(,n),)a \in H^n(K(\mathbb{Z},n),\mathbb{Q}) \simeq \mathbb{Q}. For all odd nn it is the exterior algebra on this generator:

H (K(,n),){[a] neven [a]/(a 2) nodd. H^\bullet(K(\mathbb{Z},n),\mathbb{Q}) \simeq \left\{ \array{ \mathbb{Q}[a] & n \; even \\ \mathbb{Q}[a]/(a^2) & n \; odd } \right. \,.

This is reviewed for instance in (Yin, section 4).

Generalizations

The notion of Eilenberg–Mac Lane object makes sense in every (,1)(\infty,1)-topos, not just in L wheL_{whe}Top. See at Eilenberg-MacLane object.

References

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)

Revised on July 1, 2016 12:58:52 by Urs Schreiber (131.220.184.222)