Homotopy Methods and One-Dimensional Manifolds

Abstract

In this chapter, Sard’s theorem (Proposition 4.55 of Part I) plays a central role. Before reading this chapter, one should look again at this theorem as well as Definition 4.52 about regular values. For didactical reasons, we use a parametrized version of Sard’s theorem already in Section 78.2, and present the proof afterwards in Section 78.7. The definition of the fixed-point index and the mapping degree of Section 78.6, however, only requires Sard’s theorem and not the parametrized version. Sard’s theorem is one of the most important theorems in modern mathematics. It gives a precise formulation of the following philosophy: Most situations in nature are generic, i.e., not degenerate.

The idea of considering the measure of the set of critical values of one function or of several functions is due to Marston Morse (1939).

Arthur Sard (1942)

The purpose of this note is to introduce a nonlinear version of Fredholm operators, and to prove that in this context Sard’s theorem (1942) holds if zero measure is replaced by first category. Strictly speaking, our result is a generalization of a theorem of Brown (1935), an earlier special case of Sard’s theorem.

Steve Smale (1965)

We illustrate that most existence theorems using degree theory are in principle relatively constructive.

Shui-Nee Chow, John Mallet-Paret, and James A. Yorke (1978)

The term continuation method derives from a familiar class of numerical methods dating back at least to Lahaye (1935), and also known as embedding methods. It is important to emphasize a distinction between classical embedding methods and the present continuation methods. The classical methods require that the homotopy parameter shall vary monotonically and the effort to follow a homotopy curve is abandoned when a critical point of the homotopy parameter, i.e., a turning point, is encountered. In contrast, the present continuation methods have faith and proceed beyond such critical points.

Eugene Allgower and Kurt Georg (1980)