Topological finite-projective L2 duality and whole-component boundary gluing

Apply [F1] over $R$ to each of the finitely many compact manifolds in a proposed finite diagram, obtaining $C(M;R)\simeq E(M)$ with bounded finite-projective $E$, actual chain maps $a:C\to E$, $b:E\to C$, and actual homotopies for $ba\simeq 1_C$, $ab\simeq 1_E$. Realize each projective term as $e{R}^r$ for a supplied finite idempotent $e$, using [F1]'s explicit presentation. Its conjugate right dual is $e^{\ast }{R}^r$, after the finite dual-coordinate identification. Every map or homotopy between two such finite projective terms is a finite rectangular matrix: compose it with the source idempotent and include its target in the ambient finite free module. This assertion includes dual maps; it does not identify the dual of an infinite singular direct sum with a direct sum.

For an inclusion $u:C(B)\to C(M)$ use the finite model map $u_E=a_{M}u{b}_B$ and its mapping cone, with differential $d(y,x)=(d_{M}y+u_{E}x,-d_{B}x)$. This cone is bounded finite-projective and compares to the actual singular cone. For example a comparison to the singular cone is $(y,x)\mapsto (b_{M}y-H_{M}u{b}_{B}x,b_{B}x)$ when $1-b_M{a}_M=d{H}_M+H_{M}d$: substituting gives a chain map because $u{b}_B-b_M{u}_E=d(H_{M}u{b}_B)+(H_{M}u{b}_B)d$. Its component maps induce homology isomorphisms, so the cone comparison is a homology isomorphism by the exact mapping-cone sequence. For the pair, the singular inclusion is a split injection in every degree, using the actual subset of singular generators lying in $B$. The map from its cone to the quotient $C(M,B)$ is a homology isomorphism, with kernel the contracted cone of the identity of $C(B)$. Under [A1] the comparison cones are chain contractible: a bounded-below acyclic complex of projectives splits first onto its lowest term, leaving a projective kernel, and then recursively onto that kernel. Free singular terms are projective here by choosing preimages of their basis elements using AC. Thus these are actual chain homotopy comparisons, not just homology identifications.

The same construction applies to a finite gluing graph of embedded compact pieces whose distinct seams join different pieces and are disjoint closed whole boundary components. Take collar-enlarged open neighborhoods $U_v$ of the pieces with no triple intersections and their disjoint seam overlaps $U_e$. The small singular chain sequence is $0\to {⨁}_{e}C(U_e)\to {⨁}_{v}C(U_v)\to C^{\mathrm{s}\mathrm{m}}(X)\to 0.$ Its first map is the two inclusions with opposite signs. It is degreewise split by assigning a small simplex to the least piece containing it; a duplicated simplex is exactly in one seam overlap. The supplied collar deformations retract $U_v$ onto $M_v$ and $U_e$ onto $N_e$, with their actual coefficient comparisons; the displayed exact sequence is the open-neighborhood sequence before those homotopy replacements. Small subdivision and its homotopy [F5] compare the last term with all singular chains. Taking finite models and the finite matrix of signed inclusion maps therefore gives a bounded finite-projective cone model for the entire diagram, by step 2.1. Dualizing these actual chain homotopy comparisons gives cochain comparisons too. The first-vertex free bases identify the scalar-extended singular complex literally with the singular complex of the extended $A$ fiber. Tensoring with $A$ preserves the comparisons because their homotopy identities are actual identities, requiring no flatness assumption. On finite projectives the dual coefficient comparison is $\overline{{\mathrm{Hom}}_R(E,R)}{\otimes }_{R}A\cong \overline{{\mathrm{Hom}}_R(E,A)}$, verified directly on the idempotent coordinates; thus no infinite product is interchanged with tensor.

Transfer cap with one finite relative fundamental representative $z$ using these chain comparisons. The absolute cochain cap lands in relative chains, and relative cochain cap lands in absolute chains. They are chain maps with the cap signs of [F5]: in the chain reindexing of a degree-$p$ dual cochain use differential $(-1{)}^{p+1}\delta$, so $\mathrm{\partial }(\bar{\alpha }\cap z)=(-1{)}^{p+1}\overline{\delta \alpha }\cap z$ after the appropriate relative quotient. Conjugate the intrinsic left duals to right modules via [F2]. For example the absolute-to-relative map on finite models is the composite of the dual of $a_M$, cap with $z$, and the relative comparison back to $E(M,\mathrm{\partial }M)$. Its induced homology is an isomorphism by [F2] and steps 1.1, 2.1 and 3.1. Both ends are bounded finite-projective, so its mapping cone is bounded acyclic finite-projective and is contractible: split its lowest term and continue through only finitely many degrees. All resulting maps and homotopies are finite matrices over $R$. This proves a finite-projective cap chain homotopy equivalence, before any Hilbert completion.

Each finite $R$ matrix acts boundedly on finite sums of ${\mathrm{\ell }}^2(G)$, with norm bounded by the finite sums of absolute values of its group coefficients. An idempotent $e$ has closed Hilbert range, since it is the kernel of $1-e$; its orthogonal range projection $p$ lies in a matrix algebra over $A$ by [F3]. The identities $ep=p$, $pe=e$ give $e{A}^r=p{A}^r$. Hence the finite projective complexes and all cap maps and their inverses and homotopies from step 4.1 complete to bounded Hilbert-module chain maps. Bounded homotopies send closures of boundaries into closures of boundaries and induce inverse maps on completed reduced homology. Thus completed cap duality is a bounded isomorphism, without a closed-range assertion for any differential.

In a completed degree term $p{\mathrm{\ell }}^2(G{)}^r$, let $p_Z$ project onto $\ker d_i$ and $p_B$ onto $\overline{\mathrm{im}{d}_{i+1}}$. These are matrix projections in $A$, $p_B\le p_Z\le p$, and the algebraic matrix kernel is exactly $p_Z{A}^r$: check columns for one inclusion and use $d_i{p}_Z=0$ for the other. Harmonic projection gives an algebraic homology surjection onto $P_i=(p_Z-p_B)A^r$ with kernel $p_B{A}^r\mathrm{/}\mathrm{im}{d}_{i+1}$, of dimension zero by [F3]. Inclusion of harmonic algebraic cycles splits that surjection. The map $P_i\to (p_Z-p_B){\mathrm{\ell }}^2(G{)}^r$, sending each operator column to its value on ${\delta }_e$, is injective: an operator annihilating ${\delta }_e$ annihilates all its right translates. Its image is dense because group-ring columns are Hilbert dense. This gives exactly the algebraic/projective/completed dictionary in the Statement, with no false surjectivity onto the completed Hilbert module.

All pair and finite-graph homology and cohomology sequences from step 3.1 are algebraically exact after tensoring with $A$: their finite cone sequences are degreewise split and their actual comparisons are homotopy equivalences. Their maps on harmonic parts are bounded, since inclusions, cone connectors and degreewise projections are finite matrices followed by orthogonal harmonic projections. Write each algebraic term as $P\oplus T$ as in step 6.1. Every map $T\to P^{\mathrm{\prime }}$ is zero, since its image is zero-dimensional whereas every nonzero vector in a finite projection module generates a positive-dimensional submodule by [F3]. For consecutive maps $f,g$, let $K_A=\ker g_P$, a finite algebraic projection module whose Hilbert closure is the kernel $K_H$ of the completed map. Lift $K_A$ as harmonic algebraic cycles into the middle homology. Its actual $g$ image lies in the target $T^{\mathrm{\prime }}$, so $K_A\mathrm{/}\ker (g{\mathrm{\mid }}_{K_A})$ is zero-dimensional. The quotient formula [F3] makes that kernel dense in $K_H$. Algebraic exactness puts it in the actual preceding image; projecting a preimage off its $T$ part does not change its harmonic image. Thus $\mathrm{im}{f}_P$ contains this dense kernel and is contained in $K_A$. The closure of the completed preceding image is precisely $K_H$. This proves weak exactness, and proves only weak exactness, without assuming images closed.

Riesz duality of the completed finite projective complex identifies its completed reduced cohomology with the Hilbert dual of completed reduced homology: conjugate adjoint differentials have harmonic space the orthogonal complement of their boundary closures. Compose inverse relative cap, restriction, and this identification to obtain the bounded middle intersection operator $J_M$. The cochain cap/evaluation definition agrees on algebraic $A$ homology by step 3.1 and the same simplex formula, and [F4] makes its extension Hermitian. Thus $J_M$ is self-adjoint. Its Hilbert radical is exactly $\ker J_M$, a closed projection submodule. Removing that radical leaves its orthogonal complement, but its restricted operator need not be bounded below; signature is defined by the positive and negative spectral projections, which discard only the zero spectral projection. If $M$ is closed, restriction has no relative condition, so cap duality from step 5.1 and Riesz duality make $J_M$ boundedly invertible. This is the only place below where invertibility is asserted.

The cap boundary, restriction and support/excision identities persist in these completed models. Restriction and cone connectors are finite matrix maps by steps 1.1 and 3.1, and the actual cap equivalences are finite matrices by step 4.1. For the identities needed here, work directly on singular cochains with the $A$ coefficient fiber: [F2,F5] give the cap-boundary identities on the finite orientation chain, and [F4] gives the support/excision identities, with the prescribed standard form. Step 3.1 identifies this actual cohomology and homology with the finite $A$ complexes, so the identities hold on their harmonic algebraic parts. These are dense by step 6.1 and all the maps just named are bounded by step 5.1, hence the identities hold on the Hilbert completions. No assertion that an arbitrary homology identity supplies a chain homotopy is needed. A change of finite-projective model is handled by the finite chain maps $a^{\mathrm{\prime }}b$ and their actual comparison homotopies; they induce bounded inverse harmonic maps. Both models compute the same singular $A$ cap/evaluation form, so these harmonic maps preserve it on the dense algebraic parts and then everywhere. Inertia is invariant under this bounded congruence by [F3], proving model independence.

Let $X$ now be a closed oriented four-manifold decomposed into a finite graph of compact topological pieces with disjoint closed seams as in step 3.1. Put $H={\mathcal{H}}_2(X)$ and let $j:{⨁}_v{\mathcal{H}}_2(M_v)\to H$ sum the actual inclusions, with $U=\overline{\mathrm{im}j}$. Its closed intersection operator $J_X$ is invertible by step 7.2. Under cap and Riesz duality, $U^{{\perp }_{J_X}}$ is the kernel of cohomology restriction from $X$ to the pieces, since evaluation on a piece is pairing with its inclusion. Weak graph Mayer–Vietoris exactness from step 7.1 identifies this kernel with the closure of the connecting image from ${⨁}_e{\mathcal{H}}^1(N_e)$. The cap boundary identity of step 8.1 identifies its dual image, up to the fixed seam orientation signs, with the closure of the inclusion image from ${⨁}_e{\mathcal{H}}_2(N_e)$. Cap is boundedly invertible on $X$ and on each closed three-dimensional seam, so these are actual equal closed submodules. Each seam inclusion factors through a piece, hence that image lies in $U$. Therefore $U^{{\perp }_{J_X}}\subseteq U$, establishing coisotropy with image closures retained throughout.

The pullback form $j^{\ast }{J}_{X}j$ is the orthogonal sum of the pieces' relative intersection forms. Same-piece terms are [F4]'s actual collared inclusion identity. For a cross term, take a piece's inverse relative-cap class. Excision identifies it with a class on $X$ relative to the union of the other pieces: remove their closed cores off the seams and retract the remaining collars, exactly as in [F4]'s support comparison. Its cap is the inclusion image of the original piece's class by the local orientation identity. Its restriction to every other piece is zero, so its evaluation on every other piece's class is zero. This proves cross-term vanishing by relative cochains, without a geometric transverse-intersection assertion. Apply the same argument directly with $A$ coefficient fibers; steps 6.1 and 8.1 extend it continuously to the completed harmonic modules. Orientation reversal negates inverse cap and hence the corresponding piece operator.

Discard $\ker j$, a null summand of $j^{\ast }{J}_{X}j$. On its Hilbert orthogonal complement, $j$ is injective with dense image in $U$, without any assertion that this image is closed. Dense congruence in [F3] identifies the pullback signature with the signature of $J_X{\mathrm{\mid }}_U$. Coisotropic reduction in [F3] and step 9.1 identify this last signature with the signature on all of $H$. Step 10.1 and finite additivity of spectral trace therefore prove ${\mathrm{sign}}_G^{(2)}(X)={\sum }_v{\mathrm{sign}}_G^{(2)}(M_v)\mathrm{.}$ The boundary piece operators may have nonclosed ranges and radicals; the argument uses invertibility only for the closed $J_X$, and their zero spectral projections contribute zero.

For the stated possibly partial-boundary gluing $W=W_0{\cup }_N{W}_1$, take its closed double $X=W{\cup }_{\mathrm{\partial }W}(-W)$. Extend the coefficient system by the fold map $X\to W$. The reflection swapping the two copies reverses orientation and preserves the folded system, so bounded cap naturality and model independence give ${\mathrm{sign}}_G^{(2)}(X)=-{\mathrm{sign}}_G^{(2)}(X)$, hence zero. View the same $X$ as the finite graph of pieces $W_0,W_1,-W$, with seams $N$ and the remaining boundary components of $W_0,W_1$. These seams are disjoint closed whole components, with no triple intersections. Step 11.1 gives $0={\mathrm{sign}}_G^{(2)}(W_0)+{\mathrm{sign}}_G^{(2)}(W_1)-{\mathrm{sign}}_G^{(2)}(W)$, the required formula. Empty remaining boundary or disconnected pieces produce a finite disjoint union of the same constructions; empty seams give ordinary direct sums. Apply the identical proof to the trivial coefficient group to obtain ordinary complex signature additivity; complexification of a finite real symmetric form preserves both inertia dimensions, giving the ordinary real formula.

Finally take a possibly uncountable target $G$. Include the glued manifolds and their doubles among the finitely many spaces considered. A component with boundary is a retract of its closed double by the collar fold map, so its fundamental group is a quotient of that closed double's group. Thus [F6], applied to closed compact components, makes all the needed fundamental groups finitely generated, so the subgroup $H$ generated by the images of those finitely many generating lists is countable; enumerate its elements by finite words and take the first word for each element. Construct the entire finite-projective diagram first over $\mathbb{C}H$ and extend its finite matrices to $\mathbb{C}G$. For each right coset $H\gamma$, its subspace ${\mathrm{\ell }}^2(H\gamma )$ is invariant for every left-$H$ matrix; the unitary ${\delta }_h\mapsto {\delta }_{h\gamma }$ identifies the restriction with the same ${\mathrm{\ell }}^2(H)$ matrix. Kernel, closure, harmonic and positive/negative spectral projections are therefore the identical operators on these blocks, by functional calculus. Their canonical trace is the sum of identity-coordinate diagonals, so the $G$ trace equals the $H$ trace on every relevant projection, regardless of the number of cosets. This proves the stated dimension and signature equality and compatibility with the countable convention. Full AC was the standing Hilbert and projective-comparison hypothesis; all completed maps were finite matrices, and no smoothing, finite CW inference, ordinary-versus-$L^2$ closed index equality or analytic rho foundation entered the proof.