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In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data. Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric interpretation. Equivariant assembly maps are used to formulate the Farrell–Jones conjectures in K- and L-theory. ==Homotopy-theoretical viewpoint== It is a classical result that for any generalized homology theory on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum such that : where . The functor from spaces to spectra has the following properties: * It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that is homotopy-invariant. * It preserves homotopy co-cartesian squares. This reflects that fact that has Mayer-Vietoris sequences, an equivalent characterization of excision. * It preserves arbitrary coproducts. This reflects the disjoint-union axiom of . A functor from spaces to spectra fulfilling these properties is called excisive. Now suppose that is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation from some excisive functor to such that is a homotopy equivalence. If we denote by the associated homology theory, it follows that the induced natural transformation of graded abelian groups is the universal transformation from a homology theory to , i.e. any other transformation from some homology theory factors uniquely through a transformation of homology theories . Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction. 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「assembly map」の詳細全文を読む スポンサード リンク
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