Elimination of \(B_2\) singularities. I (Q6588179)

A \(C^{\infty}\) map \(f\) between manifolds is stable if there exists a neighbourhood in the Whitney \(C^{\infty}\) topology such that any map in the neighbourhood is equivalent under diffeomorphisms in source and target to \(f\). Stable maps from a 3-manifold with boundary to a surface without boundary can have singularities of type fold, cusp, boundary fold, boundary cusp and \(B_2\).\N\NThe problem of elimination of singularities by \(C^{\infty}\) homotopies has been extensively studied in different dimensions of source and target ever since Whitney analyzed the elimination of cross-caps for maps from \(n\)-manifolds to \(\mathbb R^{2n-1}\). In this paper the author proves that if the Euler characteristic of all the boundary components of \(M\) is even, then any stable map is homotopic to a stable map with no \(B_2\) points. Otherwise, it is homotopic to a stable map with one \(B_2\) point.