Elimination of certain Thom-Boardman singularities of order two

Cobordisms of maps with singularities of given class

“…We have chosen V to be parallelizable. This is justified by the following lemma and codimΣ 3 (n + 1, n + 1) = 9. By the definition of τ ( V , T (f )) in the case ∂V = S n , T (f ) yields the section of Ω 1 (τ ( V , T (f )), ε n+2 V )| CS n , which we denote by s T (f ) .…”

Nonexistence of homotopy equivalences which are C∞ stable or of finite codimension

“…It is known that c(Σ 2i , τ N − f * (τ P )) is equal modulo 2-torsion to the determinant of the i × i matrix whose (s, t) component is the Pontrjagin class p i+s−t (τ N − f * (τ P )) ( [21] and [2,Proposition 5.4]). The following proposition follows from Propositions 1.1 and 1.2.…”

“…Proposition 1.1. (Ando [2]) Let n be a natural number. If there exists a natural number i such that i 3 − i 2 2n 2i 2 , then we have that Cl(Σ i (n, n)) ⊂ Uns(n, n).…”

“…On the other hand, there have appeared since then nonexistence results for C ∞ stable maps and maps of finite codimension outside the Mather's nice range. In particular, those nonexistence conditions have been studied in [7,Chapter VI,§6], [18,2] for C ∞ stable maps, in [9] for maps of bounded codimension and in [19] for maps of finite codimension. In this paper we will study conditions for the nonexistence of those homotopy equivalences between closed oriented n-manifolds.…”

Maps with only Morin singularities and the Hopf invariant one problem

“…The most easily computed part of an obstruction to the existence of such sections is the Thom polynomial, which is the homology class represented by the closure of the set Σ(f ) of the singular points of f of type Σ [22,36]. In fact, in some cases, it is shown that the vanishing of the Thom polynomial implies the existence of maps homotopic to f without the prescribed singularities [2,3,26]. However, the topological location of Σ(f ) in M can be non-trivial even if the Thom polynomial of Σ vanishes.…”