Unstable bordism groups and isolated singularities

Abstract:

An isolated singularity of an embedded submanifold can be topologically smoothed if and only if a certain obstruction element in ${\pi _ \ast }(MG)$ vanishes, where $G$ is the group of the normal bundle. In fact this obstruction lies in a certain subgroup which is referred to here as the unstable $G$-bordism group. In this paper some of the unstable $SO$-bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold $X \subset \mathbf {C}{P^n}$ at $\infty$ is computed in terms of the characteristic numbers of $X$. Examples of nonsmoothable complex cone singularities are given using these computations.