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In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem (also referred to as Barratt–Priddy–Quillen theorem) expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. It is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction. ==Statement of the theorem== The mapping space is the topological space of all continuous maps from the -dimensional sphere to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint , satisfying , and to have degree 0; this guarantees that the mapping space is connected. The Barratt–Priddy theorem expresses a relation between the homology of these mapping spaces and the homology of the symmetric groups . It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the th homology of this mapping space is ''independent'' of the dimension , as long as . Similarly, Nakaoka (1960) proved that the th group homology of the symmetric group on elements is independent of , as long as . The Barratt–Priddy theorem states that these "stable homology groups" are the same: for there is a natural isomorphism : This isomorphism holds with integral coefficients (in fact with any coefficients, as is made clear in the reformulation below). 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「Barratt–Priddy theorem」の詳細全文を読む スポンサード リンク
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