Nonlinear Equations - UFRJ

[SEC. 3.2: BROUWER DEGREE 39
B
a
Figure 3.1: The four possible cases.
b
Let X = ¯H −1 (ỹ). Then X is a one-dimensional manifold. Its
boundary belongs to ∂M. But by construction, it cannot intersect
[a, b]×∂B. Therefore, if we set Ĥ(t, x) = (t, H(t, x)), we can interpret
deg(g, y) − deg(f, y) =
∑
(b,x)∈∂X
sign det DĤ(b, x)
−
∑
(a,x)∈∂X
sign det DĤ(a, x).
By Proposition 3.4, each of the connected components X i is diffeomorphic
to either the circle S 1 , or a connected subset of the real
line. We claim that each ∂X i has a zero contribution to the sum
above.
There are four possibilities (fig. 3.1) for each connected component
X i : both boundary points in {a} × B, in {b} × B, one in each, or the
component is isomorphic to S 1 (no boundary).
In the first case, let s ↦→ (t(s), x(s)), s 0 ≤ s ≤ s 1 be a (regular)
parameterization of X i .
Because ŷ is a regular value of H, ker DH(x, t) is always one-