14 2. PRELIMINARIES AND OUTLINE OF PART 1

While, necessarily, the index of f with respect to g is an integer, the fractional

index of f on g|[a,b] need not be. We shall have occasion to use fractional index in

the proof of Theorem 3.2.2.

Proposition 2.1.2. Let g : S1 → C be a map with g(S1) = S, and suppose

f : S → C has no fixed points on S. Let a = b ∈ S1 with [a, b] denoting the

counterclockwise subarc on S1 from a to b (so [a, b] and (b, a) are complementary

arcs and S1 = [a, b] ∪ [b, a]). Then ind(f, g) = ind(f, g|[a,b]) + ind(f, g|[b,a]).

2.2. Variation

In this section we introduce the notion of variation of a map on an arc and

relate it to winding number.

Definition 2.2.1 (Junctions). The standard junction JO is the union of the

three rays JO i = {z ∈ C | z = reiπ/2, r ∈ [0, ∞)}, JO

+

= {z ∈ C | z = r, r ∈ [0, ∞)},

JO

−

= {z ∈ C | z =

reiπ,

r ∈ [0, ∞)}, having the origin O in common. A junction

(at v) Jv is the image of JO under any orientation-preserving homeomorphism

h : C → C where v = h(O). We will often suppress h and refer to h(JO)

i

as Jv,

i

and

similarly for the remaining rays in Jv. Moreover, we require that for each bounded

neighborhood W of v, d(Jv

+

\ W, Jv

i

\ W ) 0.

Definition 2.2.2 (Variation on an arc). Let S ⊂ C be a simple closed curve,

f : S → C a map and A = [a, b] a subarc of S such that f(a),f(b) ∈ T (S) and

f(A) ∩ A = ∅. We define the variation of f on A with respect to S, denoted

var(f, A, S), by the following algorithm:

(1) Let v ∈ A and let Jv be a junction with Jv ∩ S = {v}.

(2) Counting crossings: Consider the set M = f −1(Jv) ∩ [a, b]. Each time

a point of f −1(Jv +) ∩ [a, b] is immediately followed in M, in the counter-

clockwise order on [a, b] ⊂ S, by a point of f

−1(Jv), i

count +1 and each

time a point of f

−1(Jv) i

∩ [a, b] is immediately followed in M by a point

of f

−1(Jv +),

count −1. Count no other crossings.

(3) The sum of the crossings found above is the variation var(f, A, S).

Note that f

−1(Jv +)

∩ [a, b] and f

−1(Jv) i

∩ [a, b] are disjoint closed sets in [a, b].

Hence, in (2) in the above definition, we count only a finite number of crossings

and var(f, A, S) is an integer. Of course, if f(A) does not meet both Jv

+

and

Jv,i

then var(f, A, S) = 0.

If α : S → C is any map such that α|A = f|A and α(S \ (a, b)) ∩ Jv = ∅, then

var(f, A, S) = win(α, S, v). In particular, this condition is satisfied if α(S \(a, b)) ⊂

T (S) \ {v}. The invariance of winding number under suitable homotopies implies

that the variation var(f, A, S) also remains invariant under such homotopies. That

is, even though the specific crossings in (2) in the algorithm may change, the sum

remains invariant. We will state the required results about variation below without

proof. Proofs can be obtained directly by using the fact that var(f, A, S) is integer-

valued and continuous under suitable homotopies.

Proposition 2.2.3 (Junction Straightening). Let S ⊂ C be a simple closed

curve, f : S → C a map and A = [a, b] a subarc of S such that f(a),f(b) ∈ T (S)

and f(A)∩A = ∅. Any two junctions Jv and Ju with u, v ∈ A and Jw ∩S = {w} for

w ∈ {u, v} give the same value for var(f, A, S). Hence var(f, A, S) is independent

of the particular junction used in Definition 2.2.2.