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 , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. If then G must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called . 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 . Examples The unit circle is a . The infinite-dimensional complex projective space is a model of . The infinite-dimensional real projective space is a . The wedge sum of k unit circles is a , where is the free group on k generators. The complement to any connected knot or graph in a 3-dimensional sphere is of type ; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos. Any compact, connected, non-positively curved manifold M is a , where is the fundamental group of M. This is a consequence of the Cartan–Hadamard theorem. An infinite lens space given by the quotient of by the free action for is a . This can be shown using covering space theory and the fact that the infinite dimensional sphere is contractible. Note this includes as a . The configuration space of points in the plane is a , where is the pure braid group on strands. Correspondingly, the th unordered configuration space of is a , where denotes the -strand braid group. The infinite symmetric product of a n-sphere is a . More generally is a for all Moore spaces . Some further elementary examples can be constructed from these by using the fact that the product is . For instance