Pontryagin–Thom construction for approximation of mappings by embeddings

Abstract

Let n⩾3 and d< n−3 2 be positive integers, f :S n→S n be a C 0-mapping, and J: S n⊂ R 2n−d 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 O ̌ (f) to discrete and isotopic realizability (realizability as an embedding) of the mapping J∘ f. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y)∈S n×S n∣f(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.