Chapter 23

Degree TheoryThis chapter is on the Brouwer degree, a very useful concept with numerous and importantapplications. The degree can be used to prove some difficult theorems in topology suchas the Brouwer fixed point theorem, the Jordan separation theorem, and the invariance ofdomain theorem. A couple of these big theorems have been presented earlier, but when youhave degree theory, they get much easier. Degree theory is also used in bifurcation theoryand many other areas in which it is an essential tool. The degree will be developed forRp first. When this is understood, it is not too difficult to extend to versions of the degreewhich hold in Banach space. There is more on degree theory in the book by Deimling [38]and much of the presentation here follows this reference. Another more recent book whichis really good is [43]. This is a whole book on degree theory.

The original reference for the approach given here, based on analysis, is [62] and datesfrom 1959. The degree was developed earlier by Brouwer and others using different meth-ods.

To give you an idea what the degree is about, consider a real valued C1 function definedon an interval I, and let y ∈ f (I) be such that f ′ (x) ̸= 0 for all x ∈ f−1 (y). In this case thedegree is the sum of the signs of f ′ (x) for x ∈ f−1 (y), written as d ( f , I,y).

y

In the above picture, d ( f , I,y) is 0 because there are two places where the sign is 1 andtwo where it is −1.

The amazing thing about this is the number you obtain in this simple manner is a spe-cialization of something which is defined for continuous functions and which has nothingto do with differentiability. An outline of the presentation is as follows. First define thedegree for smooth functions at regular values and then extend to arbitrary values and finallyto continuous functions. The reason this is possible is an integral expression for the degreewhich is insensitive to homotopy. It is very similar to the winding number of complex anal-ysis. The difference between the two is that with the degree, the integral which ties it alltogether is taken over the open set while the winding number is taken over the boundary,although proofs of in the case of the winding number sometimes involve Green’s theoremwhich involves an integral over the open set.

In this chapter Ω will refer to a bounded open set.

Definition 23.0.1 For Ω a bounded open set, denote by C(Ω)

the set of functions which

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