Pontryagin–Thom construction for approximation of mappings by embeddings

Let n3 and be positive integers, f :Sⁿ→Sⁿ be a C⁰-mapping, and denote the standard embedding. As an application of the Pontryagin–Thom construction in the special case of the two-point configuration space, we construct complete algebraic obstructions O(f) and to discrete and isotopic realizability (realizability as an embedding) of the mapping Jf. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y)Sⁿ×Sⁿf(x)=f(y), x≠y}.

A standard method of solving problems in differential topology is to translate them into homotopy theory by means of bordism theory and Pontryagin–Thom construction. By this method we give a generalization of the van-Kampen–Skopenkov obstruction to discrete realizability of f and the van-Kampen–Melikhov obstruction to isotopic realizability of f. The latter are complete only in the case d=0 and are the images of our obstructions under a Hurewicz homomorphism.

We consider several examples of computation of the obstructions.