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 \ β) is given
by adding a global constant. Hence the map u w induces an injective map
π0(D(x, y)) {2-chains}.
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