Fibration of classifying spaces in the cobordism theory of singular maps (Q600706)

Given \(M\) and \(P\) smooth manifolds, a smooth map \(f : M \longrightarrow P\) is a \(\tau\)-map if at every \(p \in P\) the germ \(f\mid_{f^{-1}(p)}\) is equivalent to an element of \(\tau\), where \(\tau\) is a set of isolated stable multisingularities. A \(\tau\)-mapping \(F:W \longrightarrow P \times [0,1]\) is a \(\tau\)-cobordism between \(f_0:M_0 \longrightarrow P \) and \( f_1:M_1 \longrightarrow P \) if \(\partial W = M_0 \cup M_1, F^{-1}(P \times \{j\}) = M_j\) and \(F\mid_{M_{j}}= (f_j,j)\) for \(j=0,1\). This gives naturally an equivalence relation and its set of equivalence classes is denoted by \(Cob_{\tau} (P)\). In this paper, a map \(f:X \longrightarrow Y\) between topological spaces is called a \(\tau\)-map if it is the direct limit of \(\tau\)-maps between manifolds with respect to some direct system of embeddings, which is a useful way to have that the universal \(\tau\)-map \(f_\tau\) will also be a \(\tau\)-map. The author presents a construction of this mapping which allows to reduce the calculation of cobordism groups of singular maps \(Cob_{\tau}(\mathbb{R}^{n+k})\) to that of the homotopy groups \(\pi_{n+k}(X_\tau)\) of the corresponding classifying space, although these groups are very hard to compute. After \textit{A. Szücs} [Geom. Topol. 12, No. 4, 2379--2452 (2008; Zbl 1210.57028)] it is possible to calculate the homotopy groups of \(X_\tau\) in the particular case where \(\tau\) consists of all multisingularities composed from a fixed set of monosingularities. The so-called ``key fibration'' connects these classifying spaces for different sets of allowed singularities. In this paper, the author presents an alternative proof of the existence of such a fibration, making it possible to extend the range of applications to some negative codimension maps.