Degree theory

Degree theory is a part of mathematics which I have had little to do with up to now. The one result related to degree theory which has come up repeatedly in connection with my research interests in the past is the Brouwer fixed point theorem. I had an early contact with it through Hirsch’s book ‘Differential Topology’, which I mentioned in a previous post. One formulation of the theorem is that any continuous mapping from the closed unit ball centred at the origin in n-dimensional Euclidean space to itself has a fixed point. The statement is obviously topologically invariant and so we can replace the closed ball by any topological space homeomorphic to it. Since any closed bounded convex subset of Euclidean space is homeomorphic to a closed ball we get an apparently more general formulation concerning convex sets. For n=1 the theorem is easily proved using the intermediate value theorem. I find that it is already not intuitive for n=2. There are various different proofs.

A step towards one type of proof is as follows. Let B be the open unit ball in R^n. If the mapping is called f consider for any point x of the closed unit ball \bar B the straight line joining x and f(x). Extend it in the direction beyond x until it meets the boundary \partial B and call the intersection point \phi(x). Then \phi is continuous, it maps \bar B onto \partial B and its restriction to \partial B is the identity. A map of this type is called a retraction. Thus to prove the Brouwer fixed point theorem it is enough the prove the ‘no retraction theorem’, i.e. that there is no retraction of the unit ball onto the unit sphere. My aim here is not to present a proof of the theorem which is as simple is possible but instead to use one proof of it (the one given in Smoller’s book) to try and throw some more light on degree theory.

The degree is an integer d(f,B,y_0) which is associated to a continuous mapping f:\bar B\to R^n and a point y_0\in R^n\setminus f(\partial B). The proof of the no retraction theorem consists of putting together the following three statements. The first is that (i) since y_0 does not belong to the image of \partial B we have d(f,B,y_0)=d(I,B,y_0), where I is the identity mapping. The second is that (ii) d(I,B,y_0)=1. The third is that (iii) the statement (ii) implies that d(f,B,y_0)=1 and y_0\in f(B), a contradiction. It remains to consider how (i)-(iii) are proved. Central properties of the degree is that it varies continuously under certain types of deformations and that it is an integer. These two things together show that it is left unchanged by these deformations. The proof of (i) is as follows. The mappings f and I agree on \partial D and y_0 does not belong to the image of \partial D under either of them. It is possible to join these two mappings by a homotopy given by tf+(1-t)I. The degree at a given point is left unchanged by a homotopy whose image does not meet that point (iv). This property holds in the present case. Thus (i) follows from (iv). In the proof of (iv) in Smoller’s book it is assumed that the restriction of the homotopy to each fixed value of t is C^1. Thus there is a gap in the argument at this point. In the book it is filled by Theorem 12.7 which says that various properties of the degree proved for C^1 functions also hold for continuous functions. The differentiability is used in the proof of (iv) via the fact that the degree can be expressed as the integral of the pull-back of a suitable differential form under the mapping. Property (ii) follows from the fact that at a regular point y_0 the degree is equal to an expression computed from the determinant of the linearization of the mapping at the inverse images of y_0. In the case of the identity mapping every point is regular and the computation is trivial. When y_0\notin f(B) it follows that y_0 is regular and again the computation is trivial and gives the result zero. This completes the proof of (iii).

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