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 , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that , Eilenberg–MacLane spaces of type always exist, and are all weak homotopy equivalent. Thus, one may consider as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a " or as "a model of ". 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 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 .