Homotopy Methods and One-Dimensional Manifolds

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.