Prime-power characters admit a multiple bounding bordism

On [F1]'s branched-cover summand send a class to its character value, identifying its cyclic image with $C_d$; send $\tilde{\mu }$ to zero in that factor. In the other factor use the integral free projection. This is the stated homomorphism $\eta$ on actual integral homology and hence on the fundamental group. It is represented by a continuous map to the specified $K(G,1)$: on a finite CW presentation send each one-cell to a loop with the assigned group image, fill each two-cell because its relation has trivial image, and extend over higher cells because the target's higher homotopy groups vanish. The extensions agree on cell boundaries; their finite pasting is continuous. AC permits these finite choices, and changing them alters the map only within its homotopy class by the same cell-by-cell argument.

Its geometric class belongs to [F2]'s finite group. Some positive integer $r$ kills that element (indeed the explicit calculation permits $r=d$). By [F3], zero means that the disjoint union of those $r$ actual marked copies bounds a compact smooth oriented four-manifold with a map to $K(G,1)$ restricting to their maps. This is exactly the required bordism datum, with the positive orientations and both prescribed character coordinates unchanged.