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 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 \ 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.
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