5.4 Eilenberg-MacLane spaces

Eilenberg-MacLane spaces are key players in algebraic topology. They're special spaces with just one non-zero homotopy group, helping us link homotopy theory and group theory. These spaces are crucial for defining cohomology groups and classifying spaces up to homotopy equivalence.

These spaces have practical applications in computing cohomology groups for various spaces. Examples include infinite-dimensional real and complex projective spaces, which are Eilenberg-MacLane spaces for specific groups. They're also used in constructing Postnikov towers and in obstruction theory.

Eilenberg-MacLane Spaces

Definition and Role in Algebraic Topology

• An Eilenberg-MacLane space, denoted $K(G, n)$, is a topological space with a single non-trivial homotopy group, specifically the $n$th homotopy group isomorphic to the group $G$
• Eilenberg-MacLane spaces provide a way to study the relationship between homotopy theory and group theory in algebraic topology
• The Eilenberg-MacLane theorem guarantees the existence of Eilenberg-MacLane spaces for any abelian group $G$ and any positive integer $n$
• Eilenberg-MacLane spaces define the cohomology groups of a space $X$, denoted $H^n(X; G)$, as the set of homotopy classes of maps from $X$ to $K(G, n)$
• Studying Eilenberg-MacLane spaces allows for the classification of spaces up to homotopy equivalence based on their homotopy groups

Applications and Examples

• Eilenberg-MacLane spaces are used to compute the cohomology groups of various spaces, such as spheres, projective spaces, and lens spaces
• The infinite-dimensional real projective space $\mathbb{RP}^\infty$ is an example of an Eilenberg-MacLane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$
• The infinite-dimensional complex projective space $\mathbb{CP}^\infty$ is an example of an Eilenberg-MacLane space $K(\mathbb{Z}, 2)$
• The classifying space of a discrete group $G$, denoted $BG$, is an Eilenberg-MacLane space $K(G, 1)$

Construction of Eilenberg-MacLane Spaces

Trivial and Cyclic Groups

• For the trivial group $G = \{0\}$, the Eilenberg-MacLane space $K(\{0\}, n)$ is contractible for all $n \geq 0$
• For the integers $G = \mathbb{Z}$ and $n = 1$, the Eilenberg-MacLane space $K(\mathbb{Z}, 1)$ is homotopy equivalent to the circle $S^1$
• For the integers $G = \mathbb{Z}$ and $n > 1$, the Eilenberg-MacLane space $K(\mathbb{Z}, n)$ can be constructed as an infinite-dimensional CW complex with a single $n$-cell in each dimension $\geq n$
• For a finite cyclic group $G = \mathbb{Z}/m\mathbb{Z}$ and $n = 1$, the Eilenberg-MacLane space $K(\mathbb{Z}/m\mathbb{Z}, 1)$ is homotopy equivalent to the lens space $L(m, 1)$

Direct Sums and Products

• For a direct sum of abelian groups $G = H \oplus K$, the Eilenberg-MacLane space $K(G, n)$ is homotopy equivalent to the product $K(H, n) \times K(K, n)$
• The product of Eilenberg-MacLane spaces $K(G, n) \times K(H, m)$ is an Eilenberg-MacLane space $K(G \oplus H, n)$ if $n = m$ and is not an Eilenberg-MacLane space otherwise
• The loop space of an Eilenberg-MacLane space $\Omega K(G, n+1)$ is homotopy equivalent to the Eilenberg-MacLane space $K(G, n)$

Eilenberg-MacLane Spaces and Cohomology

Cohomology Groups and Homotopy Classes of Maps

• The $n$th cohomology group of a space $X$ with coefficients in an abelian group $G$, denoted $H^n(X; G)$, is isomorphic to the set of homotopy classes of maps from $X$ to the Eilenberg-MacLane space $K(G, n)$
• The isomorphism between $H^n(X; G)$ and $[X, K(G, n)]$ is natural, meaning it is compatible with the induced homomorphisms of cohomology groups and the induced maps of homotopy classes
• The cohomology groups of an Eilenberg-MacLane space $K(G, n)$ are given by $H^i(K(G, n); H) = G$ for $i = n$ and $H^i(K(G, n); H) = 0$ for $i \neq n$, where $H$ is any abelian group

Cup Product and Cohomology Ring Structure

• The cup product in cohomology corresponds to the composition of maps between Eilenberg-MacLane spaces, providing a geometric interpretation of the cohomology ring structure
• The cohomology ring $H^*(K(G, n); \mathbb{Z})$ of an Eilenberg-MacLane space $K(G, n)$ is isomorphic to the exterior algebra $\Lambda_{\mathbb{Z}}(G)$ if $n$ is odd and to the divided power algebra $\Gamma_{\mathbb{Z}}(G)$ if $n$ is even
• The cohomology ring $H^*(K(G, n); \mathbb{F}_p)$ of an Eilenberg-MacLane space $K(G, n)$ with coefficients in a field $\mathbb{F}_p$ is isomorphic to the polynomial algebra $\mathbb{F}_p[x]$ if $n$ is even and $p \nmid |G|$, and to the truncated polynomial algebra $\mathbb{F}_p[x]/(x^p)$ if $n$ is even and $p \mid |G|$

Classifying Homotopy Types with Eilenberg-MacLane Spaces

Homotopy Equivalence and Whitehead Theorem

• Two spaces $X$ and $Y$ are of the same homotopy type if and only if there exist maps $f: X \to Y$ and $g: Y \to X$ such that the compositions $f \circ g$ and $g \circ f$ are homotopic to the respective identity maps
• The Whitehead theorem states that the homotopy type of a connected CW complex $X$ is determined by its homotopy groups $\pi_n(X)$ for all $n \geq 1$
• A connected CW complex $X$ is homotopy equivalent to a product of Eilenberg-MacLane spaces $\prod_n K(\pi_n(X), n)$ if and only if $X$ is a $K(\pi, 1)$ space, i.e., $\pi_i(X) = 0$ for $i > 1$

Postnikov Towers and Obstruction Theory

• The homotopy type of a simply connected space $X$ is determined by its cohomology ring $H^*(X; \mathbb{Z})$ and its Postnikov tower, which consists of a sequence of fibrations involving Eilenberg-MacLane spaces
• The Postnikov tower of a space $X$ is a sequence of spaces $X_n$ and maps $p_n: X \to X_n$ such that $\pi_i(X_n) = \pi_i(X)$ for $i \leq n$ and $\pi_i(X_n) = 0$ for $i > n$, and the homotopy fiber of $p_n$ is the Eilenberg-MacLane space $K(\pi_{n+1}(X), n+1)$
• The obstruction to the existence of a map between two spaces $X$ and $Y$ can be described using the cohomology groups of $X$ with coefficients in the homotopy groups of $Y$, which are related to the Eilenberg-MacLane spaces $K(\pi_n(Y), n)$
• The obstruction to extending a map $f: X^{(n)} \to Y$ from the $n$-skeleton of a CW complex $X$ to the $(n+1)$-skeleton lies in the cohomology group $H^{n+1}(X; \pi_n(Y))$, which can be interpreted as the set of homotopy classes of maps from $X$ to the Eilenberg-MacLane space $K(\pi_n(Y), n+1)$