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In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space〔Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space.〕 is a topological space with a single nontrivial homotopy group. As such, 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 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 ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type ''K''(''G'', ''n''), if it has ''n''-th homotopy group π''n''(''X'') isomorphic to ''G'' and all other homotopy groups trivial. If ''n'' > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just ''K''(''G'', ''n''). ==Examples== * The unit circle S1 is a ''K''(Z,1). * The infinite-dimensional complex projective space P∞(C) is a model of ''K''(Z,2). This is one of the rare examples of a ''K''(''G'',''n'') admitting a manifold model for ''n''>1. Its cohomology ring is Z(), namely the free polynomial ring on a single 2-dimensional generator ''x'' ∈ ''H''2. The generator can be represented in de Rham cohomology by the Fubini–Study 2-form. An application of ''K''(Z,2) is described at Abstract nonsense. * The infinite-dimensional real projective space P∞(R) is a ''K''(Z2, 1). * The wedge sum of ''k'' unit circles is a ''K''(''G'', ''1'') for ''G'' the free group on ''k'' generators. * The complement to any knot in a 3-dimensional sphere S3 is of type ''K''(''G'', 1); this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos. Some further elementary examples can be constructed from these by using the fact that the product ''K''(''G'', ''n'') × ''K''(''H'', ''n'') is ''K''(''G'' × ''H'', ''n''). A ''K''(''G'', ''n'') can be constructed stage-by-stage, as a CW complex, starting with a wedge of ''n''-spheres, one for each generator of the group ''G'', and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Eilenberg–MacLane space」の詳細全文を読む スポンサード リンク
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