nLab Eilenberg-Mac Lane space

Idea

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

Definition

Characterization

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

Constructions

Via geometric realization of higher groupoids

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

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

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

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

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

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

So

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

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

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

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

Via linearization of spheres

Definition

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

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

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

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

Proposition

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

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

(in particular for $n \geq 1$ 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 $A[S^n]_\ast$ in definition has a canonical continuous action of the orthogonal group $O(n)$, by regarding $S^n \simeq (\mathbb{R}^n)^\ast$ as the one-point compactification of Cartesian space. Hence, in view of prop. , 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 $n$th “ordinary” cohomology of a topological space $X$ with coefficients in $G$ (when $n=1$) or $A$ (generally) is the collection of homotopy classes of maps from $X$ into $K(G,1)$ or $K(A,n)$, respectively:

$$H^1(X,G) = Ho_{Top}(X, K(G,1)) = Ho_{\infty Grpd}(X, \mathbf{B} G)$$

$$H^n(X,A) = Ho_{Top}(X, K(A,n)) = Ho_{\infty Grpd}(X, \mathbf{B}^n A) \,.$$

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

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

$$H^{n-i}(X,A)=\pi_i\mathbf{Top}(X, K(A,n))=\pi_i\mathbf{\infty Grpd}(X, \mathbf{B}^n A),$$

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

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

and

$$\Omega K(A,n)\simeq K(A,n-1)$$

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

Notice that for $G$ a nonabelian group, $H^1(X,G)$ 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 $\mathbf{B}^n A$ 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 $n \in \mathbb{N}$, the ordinary cohomology ring of $K(\mathbb{Z},n)$ with coefficients in the rational numbers is the polynomial algebra on the generator $a \in H^n(K(\mathbb{Z},n),\mathbb{Q}) \simeq \mathbb{Q}$. For all odd $n$ it is the exterior algebra on this generator:

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

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

In p.130 of Eilenberg & MacLane (1954), some low-degree cohomology groups of $K(A,n)$ are computed.

Proposition

There are stable isomorphisms

$$H^n(K(A,n),G) \cong \text{Hom}(A,G)$$

$$H^{m+1}(K(A,m),G)\cong \text{Extabel}(A,G)\subset \text{Ext}(A,G)\cong H^2(K(A,1),G)$$

for $n\geq 1$ and $m\geq 2$.

Proposition

There exists an isomorphism

$$H^4(K(A,2),G)\cong \text{Hom}(\Gamma_4(A),G)$$

where $\Gamma_4(A)$ is the group of quadratic functions from $A$ valued in $G$.

Generalizations

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

Related concepts

• generalized Eilenberg-MacLane space

• Kan-Thurston theorem

• Moore space

• Peterson space

• Eilenberg-MacLane space type

• Eilenberg-MacLane spectrum

References

General

Eilenberg-MacLane spaces originate with:

• Samuel Eilenberg, Saunders Mac Lane, On the Groups $H(\Pi,n)$, I, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 (jstor:1969820)

• Samuel Eilenberg, Saunders Mac Lane, On the Groups $H(\Pi,n)$, II: Methods of Computation, Annals of Mathematics Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 (jstor:1969702)

• Samuel Eilenberg, Saunders Mac Lane, On the Groups $H(\Pi,n)$, III: Operations and Obstructions, Annals of Mathematics Second Series, Vol. 60, No. 3 (Nov., 1954), pp. 513-557 (jstor:1969849)

That Eilenberg-MacLane spaces represent ordinary cohomology is due to:

• Samuel Eilenberg, p. 243 of: Cohomology and Continuous Mappings, Annals of Mathematics Second Series, Vol. 41, No. 1 (Jan., 1940), pp. 231-251 (jstor:1968828)

• Eilenberg-MacLane 54 III, p. 520-521

Early review is in

• Norman Steenrod, Section 19 of: Cohomology operations, and obstructions to extending continuous functions, Advances in Math. 8, 371–416. (1972) (doi:10.1016/0001-8708(72)90004-7, pdf)

Textbook accounts:

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf, doi:10.1007/b97586)

  (EM-spaces are constructed in section 6, the cohomology theory they represent is discussed in section 7.1, and its equivalence to singular cohomology is Corollary 12.1.20)

• Peter May, Chapter 22 of: A concise course in algebraic topology (pdf)

• Anatoly Fomenko, Dmitry Fuchs, §11.7-8 & p 263 in: Homotopical Topology, Graduate Texts in Mathematics 273, Springer (2016) [doi:10.1007/978-3-319-23488-5, pdf]

  (EM-spaces appear in §11.7-8 and that maps to them give ordinary cohomology is proven on p. 263)

Lecture notes:

• Andrew Kobin, Section 7.6 of: Algebraic Topology, 2016 (pdf)

The construction via reduced linearization of spheres is considered in

• Stefan Schwede, Example I.1.14 in Symmetric spectra, 2012 (pdf)

Quick review includes

• Xi Yin, On Eilenberg-MacLane spaces (pdf)

Formalization of Eilenberg-MacLane spaces in homotopy type theory (cf. Eilenberg-MacLane space type):

• Dan Licata, Eric Finster, Eilenberg-MacLane spaces in homotopy type theory, CSL-LICS ‘14 66 (2014) 1-9 [pdf text, Agda HoTT code, web discussion, doi:10.1145/2603088.2603153]

• Floris van Doorn, §42 in: On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory (2018) [arXiv:1808.10690]

• Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types [arXiv:2301.02636]

• David Wärn, Eilenberg-MacLane spaces and stabilisation in homotopy type theory, J. Homotopy Relat. Struct. 18 (2023) 357–368 [arXiv:2301.03685, doi:10.1007/s40062-023-00330-5]

Cohomology

The ordinary cohomology of Eilenberg-MacLane spaces is discussed in

• Alain Clément, Integral Cohomology of Finite Postnikov Towers, 2002 (pdf)

The topological K-theory of EM-spaces is discussed in

• D.W. Anderson, Luke Hodgkin The K-theory of Eilenberg-Maclane complexes, Topology, Volume 7, Issue 3, August 1968, Pages 317-329 (doi:10.1016/0040-9383(68)90009-890009-8))