2. CHAINS AND TRACES 9

agrees with the algebraic count in the definition of w(γ(1)), at the elements of

u−1(γ(−1))

is opposite to the algebraic count in the definition of w(γ(−1)), and at

the elements of

u−1(γ(0))

∩ R is opposite to the algebraic count in the definition of

ν(γ(0)). Hence

w(γ(1)) = w(γ(−1)) + ν(γ(0)).

In other words the value of ν at a point in α \ β is equal to the value of w slightly

to the left of α minus the value of w slightly to the right of α. Likewise, the value

of ν at a point in β \ α is equal to the value of w slightly to the right of β minus

the value of w slightly to the left of β. This proves Lemma 2.3.

Theorem 2.4. (i) Two elements of D belong to the same connected component

of D if and only if they have the same (α, β)-trace.

(ii) Assume Σ is diffeomorphic to the two-sphere. Let x, y ∈ α ∩ β and let w :

Σ\(α∪β) → Z be a locally constant function. Then Λ = (x, y, w) is an (α, β)-trace

if and only if ∂w is an (x, y)-trace.

(iii) Assume Σ is not diffeomorphic to the two-sphere and let x, y ∈ α ∩ β. If ν is

an (x, y)-trace, then there is a unique two-chain w such that Λ := (x, y, w) is an

(α, β)-trace and ∂w = ν.

Proof. We prove (i). “Only if” follows from the standard arguments in degree

theory as in Milnor [19]. To prove “if”, fix two intersection points

x, y ∈ α ∩ β

and, for X =Σ,α,β, denote by P(x, y; X) the space of all smooth curves γ :[0, 1]→X

satisfying γ(0) = x and γ(1) = y. Every u ∈ D(x, y) determines smooth paths

γu,α ∈ P(x, y; α) and γu,β ∈ P(x, y; β) via

(4) γu,α(s) := u(− cos(πs), 0), γu,β(s) = u(− cos(πs), sin(πs)).

These paths are homotopic in Σ with fixed endpoints. An explicit homotopy is the

map

Fu := u ◦ ϕ : [0,

1]2

→ Σ

where ϕ : [0,

1]2

→ D is the map

ϕ(s, t) := (− cos(πs),t sin(πs)).

By Lemma 2.3, the homotopy class of γu,α in P(x, y; α) is uniquely determined by

να := ∂wu|α\β : α \ β → Z

and that of γu,β in P(x, y; β) is uniquely determined by

νβ := −∂wu|β\α : β \ α → Z.

Hence they are both uniquely determined by the (α, β)-trace of u. If Σ is not

diffeomorphic to the 2-sphere the assertion follows from the fact that each compo-

nent of P(x, y; Σ) is contractible (because the universal cover of Σ is diffeomorphic

to the complex plane). Now assume Σ is diffeomorphic to the 2-sphere. Then

π1(P(x, y;Σ)) = Z acts on π0(D) because the correspondence u → Fu identifies

π0(D) with a space of homotopy classes of paths in P(x, y; Σ) connecting P(x, y; α)

to P(x, y; β). The induced action on the space of two-chains w : Σ \ (α ∪ β) is given

by adding a global constant. Hence the map u → w induces an injective map

π0(D(x, y)) → {2-chains}.