Bordism independence of the L²-signature defect

Suppose $W_0,W_1$ are smooth. Glue $W_0$ to $-W_1$ along their boundary collars to obtain a closed smooth four-manifold $X$. The extending homomorphisms agree on the based boundary representation; van Kampen therefore gives the coefficient map on $X$. If paths were used to identify boundary basepoints, send the connecting generators to their stipulated path elements, as in [F3]. By [F2], $\mathrm{sign}(X)=\mathrm{sign}(W_0)-\mathrm{sign}(W_1),{\mathrm{sign}}_{\mathrm{\Gamma }}^{(2)}(X)={\mathrm{sign}}_{\mathrm{\Gamma }}^{(2)}(W_0,{\psi }_0)-{\mathrm{sign}}_{\mathrm{\Gamma }}^{(2)}(W_1,{\psi }_1)\mathrm{.}$ Apply the closed smooth equality in [F3] and rearrange these two identities. The two defects are equal, proving the entire smooth-bound conclusion for the same target of the Statement.

For the full TOP Statement glue the actual collared bounds to obtain the closed oriented TOP four-manifold $X=W_0{\cup }_M(-W_1)$, with the extending coefficient map supplied by van Kampen exactly as in step1.1. The local theorem [F2] supplies both signature-additivity identities on this actual nonsmooth manifold. The external TOP equality [F4] gives ${\mathrm{sign}}_{\mathrm{\Gamma }}^{(2)}(X)=\mathrm{sign}(X)$. Subtract the ordinary identity from the group-signature identity to obtain equality of the two defects. Thus all supplied bounds in the Statement give the same value. Only the closed TOP equality is external; gluing and the resulting algebra are local, and no smooth-only equality is applied to a nonsmooth bound.