The L²-signature of a closed manifold equals its ordinary signature

Apply [F2] to the actual smooth manifold $X$ and $\varphi$. More explicitly, its classifying-map and labelled-bordism arguments produce an acyclic-envelope classifying map and a labelled smooth five-bordism from $X$ with that map to the same $X$ with constant map. Its closed-manifold argument identifies the original $\mathrm{\Gamma }$ intersection signature with the enlarged-target signature by regular-coset block induction and preserves that signature across this actual bordism. This is precisely the geometric intersection signature [F1], rather than a separate analytically defined quantity.

With constant coefficients the finite intersection matrices are the ordinary complex matrices tensored with the regular module; every scalar positive or negative eigenspace has its ordinary multiplicity multiplied by the trace of the identity, namely one. Thus the constant-map signature is $\mathrm{sign}(X)$. The identifications in step 1.1 give ${\mathrm{sign}}_{\mathrm{\Gamma }}^{(2)}(X,\varphi )=\mathrm{sign}(X)$, for nononto maps and arbitrary target cardinality as included in [F2]. If $X$ is disconnected, compactness gives finitely many components and both signatures add; the empty case is zero. This proves the literal smooth Statement without an unproved analytic index or a TOP realization assumption.