The Pontryagin-Thom construction establishes an isomorphism between the cobordism groups of embedded submanifolds in a Euclidean space and the homotopy groups of the Thom space. The authors extend this construction to (stable) maps submitted to arbitrary local and global restrictions. The existence of a Thom-type construction for maps of a given stable type, in particular, enables to describe the cobordism groups of maps with singularities in terms of the homotopy groups of some universal spaces.
In the case of immersions the Pontryagin-Thom construction was already done by Wells by using the well-known Smale-Hirsch-Gromov theory, which reduces the investigation of immersions to algebraic topology. In the general case of singular maps of any given type some results were obtained by Eliashberg, Koschorke, Arnold and Vasiliev, however the problem of proper extension of the Pontryagin-Thom construction to singular maps remained unsolved.
Let \(\tau\) be some set of (multi-) germs of stable codimension \(k\) maps. A smooth map \(f:N\to P\) is called a \(\tau\)-map if for every \(y\in f(N)\) the type of the germ of \(f\) at \(f^{-1}(y)\) is from \(\tau\). The \(\tau\)-maps \(f_1:N_1^m\to P^{m+k}\) and \(f_2:N_2^m\to P^{m+k}\) are called \(\tau\)-cobordant if there is a manifold \(W\) with a boundary the disjoint union of \(N_1\) and \(N_2\), and a \(\tau\)-map \(f:W\to P\times[0,1]\) such that \(f| _{N_1}=f_1\), \(f| _{N_2}=f_2\). The main result of the paper is that for any closed manifold \(P^{m+k}\) there is a bijection between the set \(\text{Cob}_m(P^{m+k};\tau)\) of \(\tau\)-cobordism classes of \(\tau\)-maps into \(P^{m+k}\) and the set \([P^{m+k},X_\tau]\) of homotopy classes of maps into the universal classifying space \(X_\tau\) for \(\tau\)-maps. The universal classifying space is constructed together with universal \(\tau\)-map \(f_\tau:X_\tau\to Y_\tau\) from which any other \(\tau\)-map can be pulled back. This universal \(\tau\)-map can be described very concretely as soon as the maximal compact symmetry groups of the germs occurring in \(\tau\) are understood. The authors give an algorithm for finding these maximal compact subgroups together with some differential topological applications.