Isotopic and continuous realizability of maps in the metastable range

Abstract

A continuous map of a compact -polyhedron into an orientable piecewise linear -manifold, , is discretely (isotopically) realizable if it is the uniform limit of a sequence of embeddings , (respectively, of an isotopy , ), and is continuously realizable if any embedding sufficiently close to can be included in an arbitrarily small such isotopy. It was shown by the author that for , , all maps are continuously realizable, but for , there are maps that are discretely realizable, but not isotopically. The first obstruction to the isotopic realizability of a discretely realizable map lies in the kernel of the canonical epimorphism between the Steenrod and Čech -dimensional homologies of the singular set of . It is known that for , , this obstruction is complete and is continuously realizable if and only if the group is trivial. In the present paper it is established that is continuously realizable if and only if is trivial even in the metastable range, that is, for , . The proof uses higher cohomology operations. On the other hand, for each a map is constructed that is discretely realizable and has zero obstruction to the isotopic realizability, but is not isotopically realizable, which fact is detected by the Steenrod square. Thus, in order to determine whether a discretely realizable map in the metastable range is isotopically realizable one cannot avoid using the complete obstruction in the group of Koschorke-Akhmet'ev bordisms.