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 π ∗ ( M G ) {\pi _ \ast }(MG) vanishes, where G 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 G -bordism group. In this paper some of the unstable S O SO -bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold X ⊂ C P n X \subset \mathbf {C}{P^n} at ∞ \infty is computed in terms of the characteristic numbers of X X . Examples of nonsmoothable complex cone singularities are given using these computations.