Nonlinear Equations - UFRJ

38 [CH. 3: TOPOLOGY AND ZERO COUNTING
For y ∈ Y f , we define:
deg(f, y) =
∑
x∈f −1 (y)
sign det Df(x).
Theorem 3.6. Under the conditions of Lemma 3.5, deg(f, y) does
not depend on the choice of y ∈ Y f .
We define the Brouwer degree deg(f) of f as deg(f, y) for y ∈ Y f .
Before proving theorem 3.6, we need a few preliminary definitions.
Let F be the space of mappings satisfying the conditions of
Lemma 3.5, namely the smooth maps f : B → R n extending to a C 1
map ¯f : B → R n .
A smooth homotopy on F is a smooth map f : [0, 1] × B → R n ,
extending to a C 1 map ¯f on [0, 1] × B. We say that f and g ∈ F
are smoothly homotopic if and only if there is a smooth homotopy
H : [a, b] × B → R n with H(a, x) ≡ f(x) and H(b, x) ≡ g(x).
Lemma 3.7. Assume that f and g ∈ F are smoothly homotopic, and
that y ∈ Y f ∩ Y g . Then,
deg(f; y) = deg(g; y).
Proof. Let H : [a, b] × B → R n be the smooth homotopy between f
and g. Let Y be the set of regular values of H, not in H([a, b] × ∂B).
Then Y has full measure in R n .
Consider the manifold M = [a, b] × B. It admits an obvious
orientation as a subset of R n+1 . Its boundary is
∂M = ({a} × B) ∪ ({b} × B) ∪ ([a, b] × ∂B)
Now, H |{a,b}×B is smooth and admits y as a regular value. Therefore,
there is an open neighborhood U ∋ y so that all ỹ ∈ U is a
regular value for H |{a,b}×B .
Because B is compact, we can take U small enough so that the
number of preimages of ỹ in {a}×B (and also on {b}×B) is constant.
Since Y has full measure, there is ỹ ∈ U regular value for H, and also
for H |{a,b}×B .