Definitions

# K(Z,2)

In algebraic topology, homotopy theory, and the theory of classifying spaces, the Eilenberg-MacLane space K(Z, 2) (alternatively, $K\left(mathbb\left\{Z\right\},2\right)$) is the topological space the homotopy groups of which satisfy πi = 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 $mathbb\left\{CP\right\}^\left\{infty\right\}$, the infinite-dimensional complex projective space.