23.7. UNIQUENESS OF THE DEGREE 783

Theorem 23.7.4 You have a function d which has integer values d (g,Ω,y) ∈ Z whenevery /∈ g(∂Ω) for g ∈ C

(Ω;Rp

). Assume it satisfies the following properties which the one

above satisfies.

1. (homotopy invariance) Ifh ∈C

(Ω× [0,1] ,Rp)

and y(t) /∈ h(∂Ω, t) for all t ∈ [0,1] where y is continuous, then

t→ d (h(·, t) ,Ω,y(t))

is constant for t ∈ [0,1] .

2. If Ω⊇Ω1∪Ω2 where Ω1∩Ω2 = /0, for Ωi an open set, then if

y /∈ f(Ω\ (Ω1∪Ω2)

)then

d (f,Ω1,y)+d (f,Ω2,y) = d (f,Ω,y)

3. d (id,Ω,y) = 1 if y ∈Ω.

4. d (f,Ω, ·) is continuous and constant on every connected component of Rp \ f(∂Ω).

5. If y /∈ f(∂Ω), and if d (f,Ω,y) ̸= 0, then there exists x ∈Ω such that f(x) = y.

6. Product formula, Assumption 23.7.2.

Then d is the degree which was defined above. Thus, in a sense, the degree is unique ifwe want it to do these things.

Proof: First note that h → d (h,Ω,y) is continuous on C(Ω,Rp

). Say ∥g−h∥

∞<

dist(h(∂Ω) ,y) . Then if x ∈ ∂Ω, t ∈ [0,1] ,

∥g(x)+ t (h−g)(x)−y∥ = ∥g(x)−h(x)+ t (h−g)(x)+h(x)−y∥≥ dist(h(∂Ω) ,y)− (1− t)∥g−h∥

∞> 0

By homotopy invariance, d (g,Ω,y) = d (h,Ω,y) . By the approximation lemma, if we canidentify the degree for g ∈ C∞

(Ω,Rp

)with y a regular value, y /∈ g(∂Ω) then we know

what the degree is. Say g−1 (y) = {x1, · · · ,xn} . Using the inverse function theorem thereare balls Bir containing xi such that none of these balls of radius r intersect and g is one toone on each. Then from the theorem on the fundamental properties assumed above,

d (g,Ω,y) =n

∑i=1

d (g,Bir,y) (23.7.15)

and by assumption, this is

n

∑i=1

d (g(·+xi) ,B(0,r) ,y) =n

∑i=1

d (g(·+xi)−y,B(0,r) ,0) =n

∑i=1

d (h,B(0,r) ,0)