Properties of the Leray–Schauder Degree

Abstract

You may have noticed that once the Brouwer degree was defined and its properties established, we used it in the chapter that followed only in a sort of formal way. In defining the Leray-Schauder degree we needed to know that there was a well-defined integer, called the Brouwer degree, represented by the symbol d(I∈ - f∈, U∈), but we did not have to specify how that integer was defined. Furthermore, and this is the point I want to emphasize, in the proof that the Leray-Schauder degree is well-defined, which is really a theorem about the Brouwer degree, all we needed to know about that degree was two of its properties: homotopy and reduction of dimension. In this chapter, I will list and demonstrate properties of the Leray-Schauder degree; properties which, basically, are consequences of the corresponding properties of the Brouwer degree. Again all we will need to know about the Brouwer degree is its existence and properties. As in the previous chapter, no homology groups will ever appear. Then, once we have established these properties of the Leray-Schauder degree, that’s all we’ll need to know about that degree for the rest of the book. In other words, after this chapter we can forget how the degree was defined.