Eilenberg-MacLane space

In mathematics, an Eilenberg-MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory. These spaces are important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

Let π be a group and n a positive integer. A connected topological space X is called an EilenbergMac Lane space of type K ( π , n ), if it has n -th homotopy group π n(X) isomorphic to π and all other homotopy groups trivial. If n > 1 then π must be abelian. Then an EilenbergMacLane space exists, as a CW complex, and is unique up to a Whitehead theorem. By abuse of language, any such space is often called just K ( π , n ).

A K ( π , n ) can be constructed stage-by-stage, as a CW complex, starting with a wedge sum of spheres, one for each generator of the group π , and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy.

An important property of K ( π , n ) is that, for abelian π and any topological space X , the set

:[ X , K ( π , n )]

of homotopy classes of based maps from X to K ( π , n ) is in natural bijection with n -th cohomology group

: H n( X ; π )

of the space X . Thus one says that the K ( π , n ) are representing spaces for cohomology with coefficients in π.

Every CW complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration with fibers the EilenbergMac Lane spaces.

There is a method due to Jean-Pierre Serre which allows one, at least theoretically, to compute homotopy groups of spaces using spectral sequence for special fibrations with EilenbergMac Lane spaces for fibers.

See also: Moore space, the homology analogue.