Finite-cover transfer with inverted degree and sign anti-invariants

On integral singular chains define τ(σ) as the sum of all d lifts of σ. The contractible simplex admits a lift for each point of the fiber over one vertex, and uniqueness of lifts shows these are all the lifts. Restriction to a face bijects these lift sets, so ∂τ=τ∂. This is precisely the chain construction proved in the published rational-transfer supplier; it precedes and is independent of its choice of coefficients. Precomposing R-valued cochains with τ gives T with Tp*=d id and pT=Σ_{g∈G}g. Thus p* is injective. Every pullback is invariant, and for invariant y, p*(d⁻¹Ty)=y. The ring identification follows from multiplicativity of pullback; transfer itself need not be multiplicative. This proves the upgrade over R, including R=Q and R=F_p with p∤d.

For a double cover with deck involution σ and 2 invertible, the sign local system downstairs corresponds to the anti-invariant cochain subcomplex upstairs. To verify this without invoking an unproved transfer with local coefficients, choose a lift of each singular simplex: the local-coefficient cochain assigns a coefficient in its orientation line; changing that lift changes its signed coordinate. Hence it corresponds exactly to an ordinary cochain c satisfying σc=−c. Transport along faces gives the ordinary cochain differential in these coordinates. The idempotents (1±σ)/2 split the entire cochain complex into its two eigenspaces; therefore cohomology of the anti-invariant subcomplex is the anti-invariant part of cohomology. This gives H*(X;O_R) ≅ H*(Y;R)^−.

This applies also when the spaces or cochain groups are infinite: the idempotents, not a finite-dimensional averaging argument, give the splitting.