310 CHAPTER 13. DEGREE THEORY

13.2 Properties of the DegreeNow that the degree for a continuous function has been defined, it is time to considerproperties of the degree. In particular, it is desirable to prove a theorem about homotopyinvariance which depends only on continuity considerations.

Theorem 13.2.1 If h is in C(Ω× [a,b] ,Rp

), and 0 /∈ h(∂Ω× [a,b]) for each t,

then t→ d (h(·, t) ,Ω,0) is constant for t ∈ [a,b].

Proof: Let 0 < δ = min |h(∂Ω× [a,b])| . By Corollary 13.1.1, there exists hm (·, t) =∑

mk=0 pk (t)h(·, tk) for pk (t) a polynomial in t of degree m such that p0 (a) = 1 but pk (a) =

0 if k ̸= 0 and pm (b) = 1 but pk (b) = 0 if k ̸= m and

maxt∈[a,b]

∥hm (·, t)−h(·, t)∥∞,Ω < δ , t0 = a, tm = b (13.5)

Now replace h(·, tk) with gmk (·) ∈ C∞

(Ω,Rp

)and 0 is a regular value of gm

k and letgm (·, t)≡ ∑

mk=0 pk (t)gm

k (·) where the functions gmk are close enough to h(·, tk) that

maxt∈[a,b]

∥gm (·, t)−h(·, t)∥∞,Ω < δ . (13.6)

gm ∈ C∞(Ω× [a,b] ;Rp

)because all partial derivatives with respect to either t or x are

continuous. Thus gm0 (·) = gm (·,a) , gm

m (·) = gm (·,b) . Also, from the definition of thedegree and Lemma 13.1.13, for small enough ε ,

d (h(·,a) ,Ω,0) = d (gm0 (·) ,Ω,0) =

∫Ω

φ ε (gm (x,a))detD1gm (x,a)dx

=∫

Ω

φ ε (gm (x,b))detD1gm (x,b)dx = d (gmm (·) ,Ω,0) = d (h(·,b) ,Ω,0)

Since a,b are arbitrary, this proves the theorem. ■Now the following theorem is a summary of the main result on properties of the degree.

Theorem 13.2.2 Definition 13.1.6 is well defined and the degree satisfies the fol-lowing properties.

1. (homotopy invariance) If h∈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)

),

thend (f ,Ω1,y)+d (f ,Ω2,y) = d (f ,Ω,y)

3. d (I,Ω,y) = 1 if y ∈Ω.