CHAPTER 2
Preliminaries and outline of Part 1
In this chapter we give the formal definitions and describe the results of part 1
in more detail. By a map f : X Y we will always mean a continuous function.
Let p : R S1 denote the covering map p(x) = e2πix. Let g : S1 S1 be
a map. By the degree of the map g, denoted by degree(g), we mean the number
ˆ(1) g ˆ(0), g where ˆ g : R R is a lift of the map g to the universal covering space
R of
S1
(i.e., p ˆ g = g p). It is well-known that degree(g) is independent of the
choice of the lift.
2.1. Index
Let g :
S1
C be a map and f :
g(S1)
C a fixed point free map. Define the
map v :
S1

S1
by
v(t) =
f(g(t)) g(t)
|f(g(t)) g(t)|
.
Then the map v :
S1

S1
lifts to a map v : R R. Define the index of f with
respect to g, denoted ind(f, g) by
ind(f, g) = v(1) v(0) = degree(v).
Note that ind(f, g) measures the net number of revolutions of the vector f(g(t))−
g(t) as t travels through the unit circle one revolution in the positive direction.
Remark 2.1.1. The following basic facts hold.
(a) If g :
S1
C is a constant map with
g(S1)
= c and f(c) = c, then
ind(f, g) = 0.
(b) If f is a constant map and f(C) = w with w g(S1), then ind(f, g) =
win(g, S1,w), the winding number of g about w. In particular, if f : S1
T (S1) \ S1 is a constant map, then ind(f, id|S1 ) = 1, where id|S1 is the
identity map on S1.
Note also, that for a simple closed curve S and a point w T (f(S )) we have
win(f, S , w) = 0. Suppose S C is a simple closed curve and A S is a subarc
of S with endpoints a and b. Then we write A = [a, b] if A is the arc obtained by
traveling in the counter-clockwise direction from the point a to the point b along
S. In this case we denote by the linear order on the arc A such that a b. We
will call the order the counterclockwise order on A. Note that [a, b] = [b, a].
More generally, for any arc A = [a, b] S1, with a b in the counterclockwise
order, define the fractional index [Bro90] of f on the sub-path g|[a,b] by
ind(f, g|[a,b]) = v(b) v(a).
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