Eilenberg–MacLane space

In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group.

Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type ${\displaystyle K(G,n)}$, if it has n-th homotopy group ${\displaystyle \pi _{n}(X)}$ isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that ${\displaystyle n>1}$, Eilenberg–MacLane spaces of type ${\displaystyle K(G,n)}$ always exist, and are all weak homotopy equivalent. Thus, one may consider ${\displaystyle K(G,n)}$ as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a ${\displaystyle K(G,n)}$" or as "a model of ${\displaystyle K(G,n)}$". Moreover, it is common to assume that this space is a CW-complex (which is always possible via CW approximation).

The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.

A generalised Eilenberg–Maclane space is a space which has the homotopy type of a product of Eilenberg–Maclane spaces ${\displaystyle \prod _{m}K(G_{m},m)}$.