Eilenberg-MacLane space

Intuition [Beginner]

An Eilenberg-MacLane space $K(A,n)$ is a topological space whose homotopy is concentrated in one single degree: ${\pi }_n(K(A,n))=A$ and all other homotopy groups vanish.

These spaces are the minimal spaces with given homotopy. They serve as the universal home for ordinary cohomology: cohomology classes $H^n(X;A)$ are exactly homotopy classes of maps $X\to K(A,n)$.

The simplest examples: $K(\mathbb{Z},1)=S^1$ (the circle, with ${\pi }_1=\mathbb{Z}$), $K(\mathbb{Z}/2,1)={\mathbb{RP}}^{\infty }$ (infinite real projective space, with ${\pi }_1=\mathbb{Z}/2$), and $K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$ (infinite complex projective space, with ${\pi }_2=\mathbb{Z}$). For higher $n$, the spaces become more complicated and don't have such clean geometric descriptions.

Visual [Beginner]

A space with all homotopy concentrated in a single degree — pictured as a "pure" homotopy object whose only nonvanishing $\pi$-group is at one designated level.

These spaces are the building blocks of homotopy theory: every space can be "decomposed" via Postnikov towers into Eilenberg-MacLane pieces, one per homotopy degree.

Worked example [Beginner]

The space $K(\mathbb{Z},1)=S^1$. By covering-space theory, the universal cover of $S^1$ is the contractible real line $\mathbb{R}$, with deck transformation group $\mathbb{Z}$. So ${\pi }_1(S^1)=\mathbb{Z}$ and ${\pi }_k(S^1)=0$ for $k\ge 2$ (the higher homotopy groups vanish because the universal cover is contractible). This makes $S^1$ exactly the $K(\mathbb{Z},1)$ space.

The space $K(\mathbb{Z}/2,1)={\mathbb{RP}}^{\infty }$. Its universal cover is $S^{\infty }$ (the infinite-dimensional sphere), which is contractible. The deck transformation group is $\mathbb{Z}/2$ (the antipodal action), so ${\pi }_1({\mathbb{RP}}^{\infty })=\mathbb{Z}/2$ with all higher homotopy zero.

The space $K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$. By the Hopf fibration $S^1\to S^{\infty }\to {\mathbb{CP}}^{\infty }$ (after taking infinite-dimensional limits), the long exact sequence gives ${\pi }_2({\mathbb{CP}}^{\infty })=\mathbb{Z}$ and all other homotopy zero.

The cohomology of $K(\mathbb{Z},n)$ is computed by Cartan-Serre and underlies the Steenrod algebra of cohomology operations.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $A$ be an abelian group and $n\ge 1$ an integer. An Eilenberg-MacLane space of type $(A,n)$, denoted $K(A,n)$, is a topological space (more precisely, a homotopy type — usually realised as a CW complex) with:

$${\pi }_n(K(A,n),\ast )=A,{\pi }_k(K(A,n),\ast )=0\text{ for }k\ne n.$$

For $n=0$ the convention is $K(A,0)=A$ as a discrete topological space (so ${\pi }_0=A$ and higher homotopy vanishes).

For $n\ge 1$, $K(A,n)$ is unique up to weak homotopy equivalence (this is not a unique space, but a unique homotopy type). For $n\ge 2$, $A$ must be abelian (because ${\pi }_n$ for $n\ge 2$ is automatically abelian by Eckmann-Hilton).

Loop-space characterisation. $K(A,n)\simeq \Omega K(A,n+1)$, where $\Omega Y$ is the loop space. Equivalently, $K(A,n)$ is the $n$-th space in the Eilenberg-MacLane spectrum $HA=\{K(A,0),K(A,1),K(A,2),\dots \}$ 03.12.04.

Representability of cohomology. For any pointed CW complex $X$, ordinary cohomology with coefficients in $A$ is represented by mapping into $K(A,n)$:

$$H^n(X;A)\cong [X,K(A,n){]}_{\ast }.$$

This is the Brown representability theorem 03.12.04 applied to ordinary singular cohomology, with representing spectrum the Eilenberg-MacLane spectrum.

Examples and computations.

• $K(\mathbb{Z},1)=S^1$.
• $K(\mathbb{Z}/n,1)=$ "lens space at infinity" — the infinite lens space $L^{\infty }(n)=S^{\infty }/(\mathbb{Z}/n)$.
• $K(\mathbb{Z}/2,1)={\mathbb{RP}}^{\infty }$.
• $K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$ (or $BU(1)=B{S}^1$).
• $K(\mathbb{Z},n)$ for $n\ge 3$: no clean geometric description; can be constructed by attaching cells to kill higher homotopy beyond degree $n$.

Cohomology of Eilenberg-MacLane spaces. $H^{\ast }(K(\mathbb{Z}/2,n);\mathbb{Z}/2)$ is the polynomial algebra on Steenrod operations applied to the fundamental class, by Serre's computation. This is the prototype for the Steenrod algebra ${\mathcal{A}}_2$ of cohomology operations.

Postnikov towers. Every connected CW complex $X$ has a Postnikov tower: a sequence of spaces $X^{(n)}$ for $n\ge 0$ with $X^{(n)}\to X^{(n-1)}$ a principal fibration with fibre $K({\pi }_n(X),n)$, and $X\to X^{(n)}$ inducing iso on ${\pi }_k$ for $k\le n$. The decomposition into Eilenberg-MacLane fibres is the homotopy-theoretic analogue of the chain-complex decomposition into degree pieces.

Key theorem with proof [Intermediate+]

Theorem (existence and uniqueness of Eilenberg-MacLane spaces). For any abelian group $A$ and any $n\ge 1$, there exists a CW complex $K(A,n)$ with ${\pi }_n(K(A,n))=A$ and ${\pi }_k(K(A,n))=0$ for $k\ne n$. Moreover, $K(A,n)$ is unique up to weak homotopy equivalence.

Proof sketch (existence). Build $K(A,n)$ by attaching cells to kill homotopy beyond degree $n$ and to realise $A$ at degree $n$.

Step 1: Realise ${\pi }_n=A$. Choose a free resolution $0\to F_1\to F_0\to A\to 0$ of $A$ as a $\mathbb{Z}$-module. Let $F_0$ have basis indexed by some set $S_0$ and $F_1$ by $S_1$ with attaching map $\partial :F_1\to F_0$. Construct a Moore space $M(A,n)$ by taking a wedge of $n$-spheres indexed by $S_0$ (giving ${\pi }_n=F_0$ on $n$-skeleton) and attaching $(n+1)$-cells indexed by $S_1$ along maps representing $\partial$ (collapsing the ${\pi }_n$ to $A$).

Step 2: Kill higher homotopy. For each $k>n$ and each generator of ${\pi }_k$ at this stage, attach a $(k+1)$-cell along the generating map to make the generator null-homotopic. Repeat for each generator at each degree. The result is a CW complex with ${\pi }_n=A$ and ${\pi }_k=0$ for $k>n$ (higher homotopy is killed by the cell attachments).

Step 3: Verify ${\pi }_k=0$ for $k<n$. The construction starts with cells of dimension $\ge n$, so $K(A,n)$ has only cells of dimension $\ge n$ (after Step 2). Hence ${\pi }_k=0$ for $k<n$ by cellular approximation.

Proof sketch (uniqueness). Suppose $K,K^{\prime }$ are two CW complexes with the homotopy of $K(A,n)$. The identity map of ${\pi }_n$ gives a map $K\to K^{\prime }$ on the $n$-skeletons. Obstructions to extending to higher cells lie in $H^{k+1}(K;{\pi }_k(K^{\prime }))=H^{k+1}(K;0)=0$ for $k>n$. So the map extends, and similarly its inverse. The obtained map is a weak equivalence by construction (induces iso on all ${\pi }_k$). $\square$

This existence-and-uniqueness theorem makes $K(A,n)$ a well-defined homotopy type — unique up to weak homotopy equivalence, even though no specific topological space is privileged.

Bridge. The construction here builds toward 03.06.03 (stiefel-whitney classes), where the same data is upgraded, and the symmetry side is taken up in 03.06.04 (pontryagin and chern classes). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — Mathlib lacks Eilenberg-MacLane spaces in their canonical homotopy form.

[object Promise]

A formalisation would establish the existence theorem (cell-by-cell construction), uniqueness (obstruction theory), the loop-space relation, and the cohomology representability statement.

Advanced results [Master]

Cohomology operations. A cohomology operation of type $(A,n)\to (B,m)$ is a natural transformation $H^n(-;A)\to H^m(-;B)$. By representability, this is the same as a homotopy class of maps $K(A,n)\to K(B,m)$, equivalently an element of $H^m(K(A,n);B)$. The Steenrod algebra ${\mathcal{A}}_p$ at prime $p$ is the algebra of cohomology operations on mod-$p$ cohomology, with multiplication = composition.

Steenrod squares. The basic mod-2 cohomology operations are Steenrod squares ${\mathrm{Sq}}^i:H^n(-;\mathbb{Z}/2)\to H^{n+i}(-;\mathbb{Z}/2)$, satisfying the Cartan formula and Adem relations. The Steenrod algebra ${\mathcal{A}}_2$ is generated by $\{{\mathrm{Sq}}^i{\}}_{i\ge 1}$ subject to these relations.

Eilenberg-MacLane spectrum. The collection $HA=\{K(A,0),K(A,1),K(A,2),\dots \}$ with structure maps $\Sigma K(A,n)\to K(A,n+1)$ forms a spectrum 03.12.04 representing ordinary cohomology with coefficients in $A$. $H\mathbb{Z}$ is one of the most important spectra in algebraic topology.

Brown representability and Eilenberg-MacLane. Eilenberg-MacLane spaces are the special case of Brown representability for ordinary cohomology. The spectrum $HA$ has ${\pi }_n(HA)=A$ at $n=0$ and zero elsewhere — making it the "minimal" spectrum representing $H^{\ast }(-;A)$.

Postnikov towers. Every space $X$ has a tower of approximations $X^{(0)}\leftarrow X^{(1)}\leftarrow \cdots$ where $X^{(n)}$ has homotopy of $X$ truncated at degree $n$. The fibres of consecutive towers are Eilenberg-MacLane spaces. Postnikov towers are the homotopy-theoretic analogue of filtration by degree.

Cohomology with twisted coefficients. For ${\pi }_1(X)$-modules, generalised Eilenberg-MacLane constructions $K(A,n)$ with twisted coefficients give equivariant cohomology and underlie modern equivariant homotopy theory.

Applications to characteristic-class theory. Stiefel-Whitney classes $w_i$ correspond to maps $BO(n)\to K(\mathbb{Z}/2,i)$, hence to elements of $H^i(BO(n);\mathbb{Z}/2)$. Chern classes $c_i$ correspond to $BU(n)\to K(\mathbb{Z},2i)$. The classifying-space cohomology rings of $BO(n)$ and $BU(n)$ are exactly the Eilenberg-MacLane representability of these characteristic classes.

Synthesis. This construction generalises the pattern fixed in 02.01.01 (topological space), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Existence. Sketched in §"Key theorem". The cell-attachment construction produces $K(A,n)$ as a CW complex.

Uniqueness. Proved via obstruction theory in §"Key theorem".

Loop-space relation. Proved in Exercise 3.

Cohomology representability (sketch). Brown's theorem applied to $H^n(-;A)$ gives a representing spectrum, which by uniqueness and the homotopy-group profile is the Eilenberg-MacLane spectrum $HA$. The level-$n$ space $K(A,n)$ then represents $H^n$ specifically.

Postnikov tower (sketch). Proved in Exercise 7. The tower exists by repeated application of the cell-attachment killing of homotopy.

Steenrod algebra structure. Cohomology operations form an algebra under composition; the algebra is generated by Steenrod squares (mod 2) or reduced powers (mod $p$) subject to the Adem relations. The algebra structure follows from the multiplicative structure of cohomology operations and the Yoneda-style composition of natural transformations.

Connections [Master]

• Topological space 02.01.01 — the underlying setting.

• Homotopy and homotopy group 03.12.01 — $K(A,n)$ defined by its homotopy profile.

• Spectrum 03.12.04 — Eilenberg-MacLane spaces form the Eilenberg-MacLane spectrum.

• Suspension 03.12.03 — relates consecutive Eilenberg-MacLane spaces.

• Stiefel-Whitney classes 03.06.03, Pontryagin and Chern classes 03.06.04 — characteristic classes via maps to Eilenberg-MacLane spaces.

• Whitehead tower 03.12.07 — by conn:449.whitehead-postnikov-dual, Whitehead tower equivalent to dual Postnikov tower for connectivity (equivalence). Postnikov truncates from above using $K({\pi }_n,n)$ as fibres; the Whitehead tower truncates from below. The two towers bracket the homotopy structure of any simply-connected space.

Historical & philosophical context [Master]

Samuel Eilenberg and Saunders Mac Lane introduced the spaces $K(\Pi ,n)$ in their 1945 paper Relations between homology and homotopy groups of spaces, motivated by the classification of group extensions. Serre's 1951 thesis computed the cohomology of these spaces and established their role in stable homotopy theory.

The Postnikov tower (Postnikov, 1951) made Eilenberg-MacLane spaces the building blocks of homotopy theory: every space decomposes into Eilenberg-MacLane fibres glued by $k$-invariants. This structural decomposition underlies obstruction theory, cohomology operations, and the modern theory of homotopy types via $\infty$-groupoids.

In the modern $\infty$-categorical formalism, $K(A,n)$ is the $n$-th truncation of the abelian group $A$ viewed as a homotopical object — a "fully Postnikov tower" perspective. Eilenberg-MacLane spaces sit at the heart of the connection between stable homotopy theory, derived algebra, and arithmetic geometry.

Bibliography [Master]

• Hatcher, A., Algebraic Topology, Cambridge University Press, 2002. §4.K.
• May, J. P., A Concise Course in Algebraic Topology, University of Chicago Press, 1999. Ch. 22.
• Switzer, R. M., Algebraic Topology — Homotopy and Homology, Springer Classics, 1975. §10.
• Eilenberg, S. & Mac Lane, S., "Relations between homology and homotopy groups of spaces", Annals of Mathematics 46 (1945), 480–509.
• Serre, J.-P., "Cohomologie modulo 2 des complexes d'Eilenberg-MacLane", Comm. Math. Helv. 27 (1953), 198–232.

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Wave 5 unit #3. Eilenberg-MacLane space — the universal space with homotopy concentrated in one degree; representing space for ordinary cohomology and building block for Postnikov towers.