The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. . Let be a closed oriented manifold (topological, piecewise-linear, smooth, etc.) and let be its orientation (here is the -dimensional homology group of ). Any continuous mapping defines an element . The Steenrod problem consists of describing those homology classes of , called realizable, which are obtained in this way, i.e. which take the form for a certain from the given class. All elements of the groups , , are realizable by a smooth manifold. Any element of the group , , is realizable by a mapping of a Poincaré complex . Moreover, any cycle can be realized by a pseudo-manifold. Non-orientable manifolds can also be considered, and every homology class modulo (i.e. element of ) can be realized by a non-oriented smooth singular manifold .
Thus, for smooth the Steenrod problem consists of describing the form of the homomorphism , where is the oriented bordism group of the space. The connection between the bordisms and the Thom spaces (cf. Thom space) , discovered by R. Thom , clarified the Steenrod problem by reducing it to the study of the mappings . A non-realizable class has been exhibited, where is the Eilenberg–MacLane space . For any class , some multiple is realizable (by a smooth manifold); moreover, can be chosen odd.
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Steenrod problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Steenrod_problem&oldid=24570