Topology of Hulls

Abstract

We begin by stating some basic results about Morse theory. A nice description of what we need about Morse theory is given in Part I of Milnor’s text [Mi2]. We will restrict our attention to open subsets M of ℝ^s . Let ϕ be a smooth function on M. A point p ∈ M is a critical point of ϕ if ∂ϕ/∂x_k(p) = 0 for 1 ≤ k ≤ s. A critical point p is nondegenerate if the (real) Hessian matrix (∂²ϕ/∂x_j∂x_k(p)) is nonsingular. Then one defines the index of ϕ at p to be the number of negative eigenvalues of this matrix. Nondegenerate critical points are necessarily isolated. A Morse function ρ on M is a smooth real function such that M^a ≡ {x ∈ M : ρ(x) ≤ a} is compact for all a, all critical points of ρ are nondegenerate, and ρ(p1) ≠ ρ(p2) for critical points p1 ≠ p2. The following lemma produces Morse functions.