K(Z,2)

In algebraic topology, homotopy theory, and the theory of classifying spaces, the Eilenberg-MacLane space "K"("Z", 2) (alternatively, $K(mathbb\{Z\},2)$) is the topological space the homotopy groups of which satisfy "π"_"i" = 0 for "i" = 1 and "i" > 2, while π₂ = "Z". Its cohomology ring is "Z" ["x"] , namely the free polynomial ring on a single 2-dimensional generator "x" ∈ H². The generator can be represented in de Rham cohomology by the Fubini-Study 2-form.

Application

An application of K(Z,2) is described at Abstract nonsense.

Manifold model

The space "K"("Z", 2) is one of the rare examples of classifying spaces admitting a manifold model, namely $mathbb\{CP\}^\{infty\}$, the infinite-dimensional complex projective space.

ee also

*Gromov's inequality for complex projective space