Cheeger–Gromov uniform bound for rho invariants

For smooth $M$, choose the one $W_M$ and degree-two presentation in [F2] before choosing a target or representation. For each $\varphi :{\pi }_{1}M\to \mathrm{\Gamma }$, functoriality of the acyclic envelope extends the representation over this same bound after the injective enlargement $\mathrm{\Gamma }\hookrightarrow A(\mathrm{\Gamma })$. The actual coefficient chain-homotopy identities extend under the resulting ring homomorphism, so they still model its singular complex; no flatness of that homomorphism is needed. The extended degree-two term is the range of the same size-$N$ idempotent, of von Neumann dimension at most $N$. Its reduced homology and the positive and negative spectral subspaces have dimension at most $N$, hence $\mathrm{\mid }{\mathrm{sign}}_{A(\mathrm{\Gamma })}^{(2)}{W}_M\mathrm{\mid }\le N$. Ordinary augmentation gives $\mathrm{\mid }\mathrm{sign}{W}_M\mathrm{\mid }\le N$ too. Independence in [F2] therefore yields $\mathrm{\mid }{\rho }_{\mathrm{s}\mathrm{m}}(M,\varphi )\mathrm{\mid }\le 2N$, uniformly over every target cardinality and every homomorphism, including nononto maps. For finitely many components sum these constants; the empty manifold has constant zero.

For the literal TOP reading choose one compatible smooth structure on $M$ by [F3], once before choosing $\mathrm{\Gamma }$ or $\varphi$. Step1.1 supplies its finite constant $2N$. Any rho defined by a collared TOP bound agrees with the smooth bounding-defect value by [F3] and the bordism-independence lemma; thus the same constant bounds it. Finitely many connected components use the sum of their constants. This proves the full stated signature-defect convention with an explicitly external TOP comparison. It does not claim a new analytic eta-to-defect theorem for an unrelated definition of rho.