Eilenberg–MacLane space

• The unit circle ${\displaystyle S^{1}}$ is a ${\displaystyle K(\mathbb {Z} ,1)}$.
• The infinite-dimensional complex projective space ${\displaystyle \mathbb {CP} ^{\infty }}$ is a model of ${\displaystyle K(\mathbb {Z} ,2)}$. Its cohomology ring is ${\displaystyle \mathbb {Z} [x]}$, namely the free polynomial ring on a single 2-dimensional generator ${\displaystyle x}$ in degree 2. The generator can be represented in de Rham cohomology by the Fubini–Study 2-form. An application of ${\displaystyle K(\mathbb {Z} ,2)}$ is described as abstract nonsense.
• The infinite-dimensional real projective space ${\displaystyle \mathbb {RP} ^{\infty }}$ is a ${\displaystyle K(\mathbb {Z} /2,1)}$.
• The wedge sum of k unit circles ${\displaystyle \textstyle \bigvee _{i=1}^{k}S^{1}}$ is a ${\displaystyle K(F_{k},1)}$, where ${\displaystyle F_{k}}$ is the free group on k generators.
• The complement to any knot in a 3-dimensional sphere ${\displaystyle S^{3}}$ is of type ${\displaystyle K(G,1)}$; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.[1]
• Any compact, connected, non-positively curved manifold M is a ${\displaystyle K(\Gamma ,1)}$, where ${\displaystyle \Gamma =\pi _{1}(M)}$ is the fundamental group of M.
• An infinite lens space given by the quotient ${\displaystyle S^{\infty }/(\mathbb {Z} /q)=L(\infty ,q)}$ is a ${\displaystyle K(\mathbb {Z} /q,1)}$. This can be shown using the long exact sequence on homotopy groups for the fibration ${\displaystyle \mathbb {Z} /q\to S^{\infty }\to L(\infty ,q)}$ since ${\displaystyle \pi _{1}(S^{\infty })=0}$ because the infinite sphere is contractible.[2] Note this includes ${\displaystyle \mathbb {RP} ^{\infty }}$ as a ${\displaystyle K(\mathbb {Z} /2,1)}$.
• The configuration space of ${\displaystyle n}$ points in the plane is a ${\displaystyle K(P_{n},1)}$, where ${\displaystyle P_{n}}$ is the pure braid group on ${\displaystyle n}$ strands.

Some further elementary examples can be constructed from these by using the fact that the product ${\displaystyle K(G,n)\times K(H,n)}$ is ${\displaystyle K(G\times H,n)}$.

A ${\displaystyle K(G,n)}$ can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group G, and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy. The corresponding chain complex is given by the Dold–Kan correspondence.

The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence

${\displaystyle 0\rightarrow K(\mathbb {Z} ,2)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}$

with ${\displaystyle {\text{String}}(n)}$ the string group, and ${\displaystyle {\text{Spin}}(n)}$ the spin group. The relevance of ${\displaystyle K(\mathbb {Z} ,2)}$ lies in the fact that there are the homotopy equivalences

${\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }$

for the classifying space ${\displaystyle B\mathbb {Z} }$, and the fact ${\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)}$. Notice that because the complex spin group is a group extension

${\displaystyle 0\to K(\mathbb {Z} ,1)\to {\text{Spin}}^{\mathbb {C} }(n)\to {\text{Spin}}(n)\to 0}$

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space ${\displaystyle K(\mathbb {Z} ,2)}$ is an example of a higher group. It can be thought of the topological realization of the groupoid ${\displaystyle \mathbf {B} U(1)}$ whose object is a single point and whose morphisms are the group ${\displaystyle U(1)}$. Because of these homotopical properties, the construction generalizes: any given space ${\displaystyle K(\mathbb {Z} ,n)}$ can be used to start a short exact sequence that kills the homotopy group ${\displaystyle \pi _{n+1}}$ in a topological group.

• Brown representability theorem, regarding representation spaces
• Moore space, the homology analogue.
• Homology sphere