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.
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