Properties of the Brouwer Degree

Abstract

This chapter is devoted to the properties of the Brouwer degree that we will need in order to extend it to the Leray–Schauder degree. In all that follows, we assume that U is an open subset of R n and that we have a map \(f: \overline{U} \rightarrow \mathbf{R}^{n}\) such that \(F = f^{-1}(\mathbf{0})\) is admissible in U, that is, compact and disjoint from ∂ U, so the Brouwer degree deg(f, U) is well defined. The properties of the degree are given names for easy identification; the terminology I’m using for this purpose is pretty much standard. Some of the properties will carry over to the infinite-dimensional case and others are needed in order to make the transition to that more general setting.