Properties of the Brouwer Degree
Robert F. Brown · Birkhäuser Boston eBooks · 2004
This chapter is devoted to the properties of the Brouwer degree that we will need for the Leray-Schauder degree. In all that follows, we assume we have a map $$ f = \bar U \to R^n $$ such that F = f-1(0) is admissible in U, that is, compact and disjoint from ∂U, so the Brouwer degree d(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. The following simple lemma will be helpful in verifying some of those properties.