CHAPTER 2

Chains and Traces

Define a cell complex structure on Σ by taking the set of zero-cells to be the set

α ∩ β, the set of one-cells to be the set of connected components of (α \ β) ∪ (β \ α)

with compact closure, and the set of two-cells to be the set of connected components

of Σ \ (α ∪ β) with compact closure. (There is an abuse of language here as the

“two-cells” need not be homeomorphs of the open unit disc if the genus of Σ is

positive and the “one-cells” need not be arcs if α ∩ β = ∅.) Define a boundary

operator ∂ as follows. For each two-cell F let

∂F = ±E,

where the sum is over the one-cells E which abut F and the plus sign is chosen

iff the orientation of E (determined from the given orientations of α and β) agrees

with the boundary orientation of F as a connected open subset of the oriented

manifold Σ. For each one-cell E let

∂E = y − x

where x and y are the endpoints of the arc E and the orientation of E goes from

x to y. (The one-cell E is either a subarc of α or a subarc of β and both α and

β are oriented one-manifolds.) For k = 0, 1, 2 a k-chain is defined to be a formal

linear combination (with integer coeﬃcients) of k-cells, i.e. a two-chain is a locally

constant map Σ \ (α ∪ β) → Z (whose support has compact closure in Σ) and

a one-chain is a locally constant map (α \ β) ∪ (β \ α) → Z (whose support has

compact closure in α ∪ β). It follows directly from the definitions that

∂2F

= 0 for

each two-cell F .

Each u ∈ D determines a two-chain w via

(1) w(z) := deg(u, z), z ∈ Σ \ (α ∪ β).

and a one-chain ν via

(2) ν(z) :=

deg(u

∂D∩R

: ∂D ∩ R → α, z), for z ∈ α \ β,

− deg(u

∂D∩S1

: ∂D ∩

S1

→ β, z), for z ∈ β \ α.

Here we orient the one-manifolds D∩R and

D∩S1

from −1 to +1. For any one-chain

ν : (α \ β) ∪ (β \ α) → Z denote

να := ν|α\β : α \ β → Z, νβ := ν|β\α : β \ α → Z.

Conversely, given locally constant functions να : α \ β → Z (whose support has

compact closure in α) and νβ : β \ α → Z (whose support has compact closure in

β), denote by ν = να − νβ the one-chain that agrees with να on α \ β and agrees

with −νβ on β \ α.

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