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String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Chas and Sullivan in 1999 (see Chas & Sullivan 1999). ==Motivation== While the singular cohomology of a space has always a product structure, this is not true for the singular homology of a space. Nevertheless, it is possible to construct such a structure for an oriented manifold M of dimension d. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes and , take their product and make it transversal to the diagonal . The intersection is then a class in , the intersection product of x and y. One way to make this construction rigorous is to use stratifolds. Another case, where the homology of a space has a product, is the (based) loop space of a space X. Here the space itself has a product : by going first the first loop and then the second. There is no analogous product structure for the free loop space LX of all maps from to X since the two loops need not have a common point. A substitute for the map m is the map : where ''Map(8, M)'' is the subspace of , where the value of the two loops coincides at 0 and is defined again by composing the loops. (Here "8" denotes the topological space "figure 8", i.e. the wedge of two circles.) 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「string topology」の詳細全文を読む スポンサード リンク
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