Topological five-dimensional bordisms preserve ordinary and L2 signatures

Use [F1] for $Y,X$ and $(Y,X)$ with $R=\mathbb{C}G$ and $A=\mathcal{N}G$. Write $V={\mathcal{H}}^2(X)$ for completed reduced cohomology. Transport the middle intersection form of $X$ to $V$ by its bounded cap isomorphism, and denote it by $Q$. It is a bounded Hermitian form with boundedly invertible operator, since $X$ is closed. Let $r:{\mathcal{H}}^2(Y)\to V$ be restriction and $\delta :V\to {\mathcal{H}}^3(Y,X)$ the completed connecting map of the pair. These are bounded maps of finite Hilbert $A$ modules. Set $U=\overline{\mathrm{im}r}$, an actual closed submodule; no assertion that $\mathrm{im}r$ is closed is made.

Cap duality on $Y$ and Riesz duality give a bounded nondegenerate scalar sesquilinear pairing ${\mathcal{H}}^2(Y)\times {\mathcal{H}}^3(Y,X)\to \mathbb{C}$: take evaluation of the first class on the cap of the second with $[Y,X]$, followed by the canonical trace, initially on algebraic harmonic parts. Riesz and the bounded cap isomorphism extend this pairing to the Hilbert completions and identify either factor boundedly with the Hilbert dual of the other. No $A$-valued pairing is asserted on arbitrary ${\mathrm{\ell }}^2$ vectors. The boundary identity identifies $Q(ra,b)$ with this pairing of $a$ and $\delta b$, up to the fixed conjugation convention, for all $a\in {\mathcal{H}}^2(Y)$ and $b\in V$. Concretely, for an actual degree-two boundary cocycle $\beta$ extend it to a cochain $\tilde{\beta }$ on $Y$. The relative cocycle $d\tilde{\beta }$ represents $\delta \beta$, and the cohomology-first cap formula gives $\mathrm{\partial }(\tilde{\beta }\cap [Y,X])=\beta \cap [X]-(d\tilde{\beta })\cap [Y,X]$. Thus $i_{\ast }{\mathrm{P}\mathrm{D}}_X\beta ={\mathrm{P}\mathrm{D}}_{Y,X}\delta \beta$ in absolute homology. Pairing with $a$ proves the asserted identity on algebraic harmonic parts. Such parts are dense, all displayed maps and pairings are bounded, and [F1] transfers this actual identity to the completed modules. This uses cochains and the finite orientation chain, with no geometric transverse intersections in dimension four or five.

By step 2.1 and nondegeneracy of the perfect pairing, $Q(ra,b)=0$ for every $a$ if and only if $\delta b=0$. Continuity replaces $\mathrm{im}r$ by its closure without changing this annihilator, giving $U^{{\perp }_Q}=\ker \delta$. Weak exactness of the actual pair sequence in [F1] gives $\ker \delta =\overline{\mathrm{im}r}=U$. Therefore $U$ is a closed self-annihilating, hence coisotropic, submodule, and $Q{\mathrm{\mid }}_U$ is identically zero. These are equalities of Hilbert-closed submodules; no algebraic exactness has been promoted to closed-range exactness.

Apply [F2] to the invertible closed-boundary form $Q$ and $U$. Its signature is the signature of the zero restriction, namely zero. The bounded cap change of coordinates preserves inertia, so this is precisely ${\mathrm{sign}}_G^{(2)}(X)$. Apply the same argument to the trivial group: its complex Hermitian intersection form is the complexification of the real symmetric form, with equal positive and negative multiplicities, and hence gives the ordinary signature zero. All arguments allow finite disconnected boundaries and empty boundary; in the latter case the middle module is zero.

A disjoint union is an orthogonal direct sum of the finite-projective complexes and forms; orientation reversal negates cap and the form. Thus $X=X_1\bigsqcup (-X_0)$ in step 4.1 yields the two asserted bordism-invariance equalities. For an embedded coefficient subgroup $G\hookrightarrow K$, construct the same finite models over $\mathbb{C}G$ and extend the matrices to $\mathbb{C}K$. Every left-$G$ matrix decomposes on ${\mathrm{\ell }}^2(K)={⨁}_{G\gamma }{\mathrm{\ell }}^2(G\gamma )$ into identical copies of its ${\mathrm{\ell }}^2(G)$ operator. Kernel, harmonic and positive/negative projections have the same identity-coordinate diagonal trace. By [F1] this preserves the signatures on both ends and on $X$, for arbitrary cardinalities of $K$. No equality between ordinary and $L^2$ signatures on arbitrary closed four-manifolds and no TOP bordism realization theorem was assumed.