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. 7

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