Editing Eilenberg–MacLane space (section)

==Examples==
* The [[unit circle]] <math>S^1</math> is a <math>K(\Z,1)</math>.
* The infinite-dimensional [[complex projective space]] <math>\mathbb{CP}^{\infty}</math> is a model of <math>K(\Z,2)</math>.
* The infinite-dimensional [[real projective space]] <math>\mathbb{RP}^{\infty}</math> is a <math>K(\Z/2,1)</math>.
* The [[wedge sum]] of ''k'' [[unit circle]]s <math>\textstyle\bigvee_{i=1}^k S^1</math> is a <math>K(F_k,1)</math>, where <math>F_k</math> is the [[free group]] on ''k'' generators.
* The complement to any connected [[Knot theory|knot]] or graph in a 3-dimensional sphere <math>S^3</math> is of type <math>K(G,1)</math>; this is called the "[[Aspherical space|asphericity]] of knots", and is a 1957 theorem of [[Christos Papakyriakopoulos]].<ref>{{cite journal |last1=Papakyriakopoulos |first1=C. D. |title=On Dehn's lemma and the asphericity of knots |journal=Proceedings of the National Academy of Sciences |date=15 January 1957 |volume=43 |issue=1 |pages=169–172 |doi=10.1073/pnas.43.1.169 |pmid=16589993 |pmc=528404 |bibcode=1957PNAS...43..169P |doi-access=free }}</ref>
* Any [[compact space|compact]], connected, [[non-positive curvature|non-positively curved]] [[manifold]] ''M'' is a <math>K(\Gamma,1)</math>, where <math>\Gamma=\pi_1(M)</math> is the [[fundamental group]] of ''M''. This is a consequence of the [[Cartan–Hadamard theorem]].
* An infinite [[lens space]] <math> L(\infty, q)</math> given by the quotient of <math>S^\infty</math> by the free action <math> (z \mapsto e^{2\pi i m/q}z) </math> for <math> m \in \Z/q </math> is a <math>K(\mathbb{Z}/q,1)</math>. This can be shown using [[Covering space#Deck transformation|covering space theory]] and the fact that the infinite dimensional sphere is [[Contractible space|contractible]].<ref>{{Cite web|title=general topology - Unit sphere in $\mathbb{R}^\infty$ is contractible?|url=https://math.stackexchange.com/q/282268 |access-date=2020-09-01|website=Mathematics Stack Exchange}}</ref> Note this includes <math>\mathbb{RP}^{\infty}</math> as a <math>K(\Z/2,1)</math>.
* The [[Configuration space (mathematics)|configuration space]] of <math>n</math> points in the plane is a <math>K(P_n,1)</math>, where <math>P_n</math> is the [[Braid_group#Relation_with_symmetric_group_and_the_pure_braid_group|pure braid group]] on <math>n</math> strands.
* Correspondingly, the [[Configuration space (mathematics)| {{var|n}}th unordered configuration space]] of <math> \mathbb{R}^2 </math> is a <math>K(B_n,1)</math>, where <math>B_n</math> denotes the [[Braid group |{{var|n}}-strand braid group]]. <ref>Lucas Williams [https://arxiv.org/pdf/1911.11186.pdf "Configuration spaces for the working undergraduate"], ''arXiv'', November 5, 2019. Retrieved 2021-06-14</ref>
* The [[Symmetric product (topology) |infinite symmetric product]] <math> SP(S^n)</math> of a [[Sphere|''n''-sphere]] is a <math>K(\mathbb{Z},n)</math>. More generally <math> SP(M(G,n)) </math> is a <math> K(G,n) </math> for all [[Moore space (algebraic topology) | Moore spaces]] <math> M(G,n) </math>.

Some further elementary examples can be constructed from these by using the fact that the product <math>K(G,n) \times K(H,n)</math> is <math>K(G\times H,n)</math>. For instance the [[Torus#n-dimensional_torus| {{var|n}}-dimensional Torus]] <math>\mathbb{T}^n</math> is a <math> K(\Z^n, 1)</math>.