10 VIN DE SILVA, JOEL W. ROBBIN AND DIETMAR A. SALAMON

This proves (i).

We prove (ii) and (iii). Let w be a two-chain, suppose that ν := ∂w is an (x, y)-

trace, and denote Λ := (x, y, w). Let γα : [0, 1] → α and γβ : [0, 1] → β be as in

Definition 2.1. Then there is a u ∈ D(x, y) such that the map s → u (− cos(πs), 0)

is homotopic to γα and s → u (− cos(πs), sin(πs)) is homotopic to γβ. By definition

the (α, β)-trace of u is Λ = (x, y, w ) for some two-chain w . By Lemma 2.3, we

have

∂w = ν = ∂w

and hence w − w =: d is constant. If Σ is not diffeomorphic to the two-sphere and

Λ is the (α, β)-trace of some element u ∈ D, then u is homotopic to u (as P(x, y;Σ)

is simply connected) and hence d = 0 and Λ = Λ . If Σ is diffeomorphic to the 2-

sphere choose a smooth map v :

S2

→ Σ of degree d and replace u by the connected

sum u := u #v. Then Λ is the (α, β)-trace of u. This proves Theorem 2.4.

Remark 2.5. Let Λ = (x, y, w) be an (α, β)-trace and define

να := ∂w|α\β, νβ := −∂w|β\α.

(i) The two-chain w is uniquely determined by the condition ∂w = να − νβ and its

value at one point. To see this, think of the embedded circles α and β as traintracks.

Crossing α at a point z ∈ α \ β increases w by να(z) if the train comes from the

left, and decreases it by να(z) if the train comes from the right. Crossing β at a

point z ∈ β \ α decreases w by νβ(z) if the train comes from the left and increases

it by νβ(z) if the train comes from the right. Moreover, να extends continuously

to α \ {x, y} and νβ extends continuously to β \ {x, y}. At each intersection point

z ∈ (α ∩ β) \ {x, y} with intersection index +1 (respectively −1) the function w

takes the values

k, k + να(z), k + να(z) − νβ(z), k − νβ(z)

as we march counterclockwise (respectively clockwise) along a small circle surround-

ing the intersection point.

(ii) If Σ is not diffeomorphic to the 2-sphere then, by Theorem 2.4 (iii), the (α, β)-

trace Λ is uniquely determined by its boundary ∂Λ = (x, y, να − νβ).

(iii) Assume Σ is not diffeomorphic to the 2-sphere and choose a universal covering

π : C → Σ. Choose a point x ∈

π−1(x)

and lifts α and β of α and β such that

x ∈ α ∩ β. Then Λ lifts to an (α, β)-trace

Λ = (x, y, w).

More precisely, the one chain ν := να − νβ = ∂w is an (x, y)-trace, by Lemma 2.3.

The paths γα : [0, 1] → α and γβ : [0, 1] → β in Definition 2.1 lift to unique paths

γα : [0, 1] → α and γβ : [0, 1] → β connecting x to y. For z ∈ C\(A∪B) the number

w(z) is the winding number of the loop γα − γ

β

about z (by Rouch´ e’s theorem).

The two-chain w is then given by

w(z) =

z∈π−1(z)

w(z), z ∈ Σ \ (α ∪ β).

To see this, lift an element u ∈ D(x, y) with (α, β)-trace Λ to the universal cover

to obtain an element u ∈ D(x, y) with Λu = Λ and consider the degree.