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
= 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) :=
: ∂D ∩ R → α, z), for z ∈ α \ β,
: ∂D ∩
→ β, z), for z ∈ β \ α.
Here we orient the one-manifolds D∩R and
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.