Modifications and Cobounding Manifolds

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications. The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M 1 and M2 are two members of a pencil of hyperplane sections.