10 The Work of Stephen Smale in Differential Topology MORRIS W. HIRSCH
Background The theme of this conference is “Unity and Diversity in Mathematics.”
The diversity is evident in the many topics covered. Reviewing Smale's work in ...

If K is a subcomplex of a smooth triangulation of a manifold M and N is a regular neighborhood of K contained in an open neighborhood U of K in M, there is a piecewise regular homeomorphism f of M carrying N onto a smooth submanifold of U, and f reduces to the identity map on K and outside U.

Introduction. LetXbe a submanifold ofY, in either the topological, smooth, or piecewise linear ( =PL) categories. Anormal cell bundleonXinYis a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such thatEis a neighbourhood ofXinYandp: E → Xis a retraction. The triple (Y, X, ξ) is atubular neighbourhood, or briefly, atube. For convenience we may refer to a tube by its cell bundle.

If a smooth foliation of a manifold M has a compact leaf L, conditions on the holonomy of L are given ensuriing that every nearby foliation has a unique compact leaf near L.

This paper reexamines classic attempts at estimating the number of possible genotypes for a species. In the original computations (Hirsch, 1963), the probability that any two human parents will produce two offspring with the same genotype was calculated to be equal to (1/223)2, or over one chance in 70 trillion. The error lies in that this number reflects the number of cells in the matrix of zygotes, not the number of zygotic genotypes. When this is taken into account, the probability of two human parents producing two offspring with the same zygotic genotype is 1 in 160,000—almost a billion times more likely than previously suspected. The complexity of the genetic system is also discussed in the context of the concept of “heritability,” often confused with that of “heredity.” This confusion has led to the wrong view that heritability represents a nature/nurture ratio.