Topological classification of continuous selections of five linear functions

Abstract

Continuous selections of linear functions play an important role in Morse theory for piecewise C ²-functions. In this article, the topological properties of continuous selections of linear functions are investigated in detail. These are then utilized to provide a complete classification of all continuous selections of five linear functions. This is done by showing that the first four Betti numbers of a simplicial complex induced by such a function fully determine that function up to topological equivalence. The number of different topological types of continuous selections of linear functions has been known only in the case of four or less selection functions so far. The main result of this article now states that there are exactly 26 different topological types of continuous selections of five linear functions.

Keywords:

• continuous selection of linear functions
• critical point
• topological classification
• Morse theory
• non-smooth optimization

AMS Subject Classifications::

• 26A27
• 90C30

Notes

Notes

1. Two functions f, g : ℝ ⁿ  → ℝ are said to be locally topologically equivalent at (x, y) ∈ ℝ ⁿ  × ℝ ⁿ if there exist open subsets U and V of x and y, respectively, and a homeomorphism φ : U → V, such that φ(x) = y and f ○ φ⁻¹ = g on V.

2. f _max is the pure maximum function defined by .

3. A simple covering is a locally finite covering with the additional property that each set of the covering, and each intersection of two or more sets of the covering, is differentiably contractible. In Euclidean space any convex set is differentiably contractible.

4. Throughout this article, the notation A ≃ B is used for homotopy equivalent spaces A and B and the notation C ≅ D for homeomorphic spaces, C and D.

5. The genus g of a surface is defined to be the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. A surface of genus g is also called a sphere with g handles, because it can be constructed as the connected sum of a sphere and g tori (‘handles’). More details can, for instance, be found in 12.

6. If X is a topological space, we denote by h _i (X) the i-th Betti number of X. It is defined as the rank of the abelian group H _i (X), which is the i-th homology group of X. The homology groups, and hence the Betti numbers, are invariant under homotopy equivalence. For full definitions and further results please refer to 5, 13 or any other textbook on algebraic topology.

7. Let denote the p-th reduced homology group of X and the q-th reduced cohomology group of Y.

8. To determine the homology groups of simplicial complexes, the free program package ‘MOISE’ of R. Andrew Hicks has been used.

9. Throughout this article, let the term handlebody of genus g be defined as a three-dimensional, orientable manifold M with boundary, which contains g pairwise disjoint, properly embedded 2-cells D ₁, … , D _g , such that cutting M along results in a 3-cell. The boundary of a handlebody of genus g is a surface of genus g. In the literature handlebodies of genus g are also called cube with g handles. For a detailed discussion of handlebodies please refer to 6.