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:''Not to be confused with Intersectionality theory.'' In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. ==Topological intersection form== For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in . Stated precisely, there is a bilinear form : given by : with : This is a symmetric form for even (so doubly even), in which case the signature of is defined to be the signature of the form, and an alternating form for odd (so singly even). These can be referred to uniformly as ε-symmetric forms, where respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an , though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with coefficients instead. These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism – see intersection form (4-manifold). By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative -dimensional submanifolds , for the Poincaré duals of and . Then is the oriented intersection number of and , which is well-defined because of the dimensions of and . This explains the terminology ''intersection form''. 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「Intersection theory」の詳細全文を読む スポンサード リンク
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