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 \ β) \ α) with compact closure, and the set of two-cells to be the set of connected components of Σ \ 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 \ β) Z (whose support has compact closure in Σ) and a one-chain is a locally constant map \ 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 \ 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 : \ 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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