Finite cellular L2 duality and weak exactness

For a finite matrix differential $d_i$, its algebraic kernel over $A$ is exactly $p_Z{A}^{n_i}$, where $p_Z$ projects onto its Hilbert kernel: the matrix equality $d_i{p}_Z=0$ gives one inclusion, and applying $p_Z$ to each column gives the other. Let $p_B$ project onto the closure of the image of $d_{i+1}$. Then $p_B\le p_Z$. Orthogonal projection onto the algebraic module $P_i=(p_Z-p_B)A^{n_i}$ induces a surjection from algebraic homology $H_i^{\mathrm{a}\mathrm{l}\mathrm{g}}$, with kernel $T_i=p_B{A}^{n_i}\mathrm{/}\mathrm{im}{d}_{i+1}$. This kernel has dimension zero by [F1]. The target $P_i$ is projective, so the surjection splits algebraically; in fact inclusion of the harmonic projection module into cycles gives a splitting. Thus $H_i^{\mathrm{a}\mathrm{l}\mathrm{g}}\cong P_i\oplus T_i$. Separately, $P_i$ embeds in the Hilbert module ${\mathcal{H}}_i=(p_Z-p_B){\mathrm{\ell }}^2(G{)}^{n_i}$ by applying its columns to ${\delta }_e$. This embedding is injective, since a group von Neumann operator annihilating ${\delta }_e$ annihilates all its right translates. Its image is dense because it contains $(p_Z-p_B)(\mathbb{C}G{)}^{n_i}{\delta }_e$. Consequently the natural map from $H_i^{\mathrm{a}\mathrm{l}\mathrm{g}}$ to ${\mathcal{H}}_i$ has image this dense $P_i$, not necessarily all of ${\mathcal{H}}_i$. The same dictionary applies to the dual finite matrix complex.

The cellular cap map in [F3] is represented by a map between finite free $R$ complexes, using the dual cellular decomposition or a finite common subdivision. Its homology maps are isomorphisms: apply [F3] to the regular $R$ fiber as a complex local system; all the constructions commute with right multiplication by $R$. Cellular cochains here are the finite dual matrix complex: on each relative cell the local-coefficient cochain is its freely specified fiber value, and induction over the finite cellular filtration gives the cochain comparison, just as for the cellular chain comparison in [F2]. The cap map's cone is therefore a bounded acyclic complex of finite free $R$ modules. Such a complex is contractible: start at its lowest nonzero degree, split the surjection onto that projective term, remove the resulting two-term identity complex, and continue over the finitely many degrees. All splittings are maps between finite projective summands of finite free modules and hence are finite matrices of elements of $R$. Every such matrix acts boundedly on ${\mathrm{\ell }}^2(G)$. The cap map and its chain homotopy inverse consequently extend to bounded chain homotopy equivalences after completion.

Consider three consecutive terms of an algebraically exact homology sequence in [F2], and decompose each as $P\oplus T$ by step 1.1. Every map from $T$ to a finite algebraic projection module vanishes: its image has dimension zero, whereas every nonzero element in a finite projection module generates a positive-dimensional submodule by the spectral quotient formula and faithfulness in [F1]. Hence the sequence induces maps $f_P,g_P$ on the algebraic projective reduced parts. These are restrictions of bounded maps on their Hilbert completions: restriction and inclusion come from finite cellular matrices, and connecting maps can be computed using the degreewise free splittings of the relative or cellular union chain sequence, again by finite matrices followed by harmonic projections. Let $K_A=\ker g_P$, a finite algebraic projection module whose Hilbert closure is the kernel $K_H$ of the completed map, by the same matrix-kernel argument as in step 1.1. View $K_A$ inside the middle algebraic homology using its harmonic-cycle splitting. Its image under the actual second homology map lies in the target $T$, so $K_A\mathrm{/}\ker (g{\mathrm{\mid }}_{K_A})$ has dimension zero. The quotient formula in [F1] makes $\ker (g{\mathrm{\mid }}_{K_A})$ Hilbert-dense in $K_H$. Exactness identifies each of its elements with an actual preceding image; projecting that preimage off its $T$ summand does not change its projected image. Thus $\ker (g{\mathrm{\mid }}_{K_A})\subseteq \mathrm{im}{f}_P\subseteq K_A$. The completed preceding image consequently has closure $K_H$. This proves weak exactness on completed reduced homology without identifying $P$ with its Hilbert completion.

Bounded chain homotopies descend to completed reduced homology, because differentials and all maps are bounded and therefore preserve closures of boundaries. Thus the maps in step 1.2 induce bounded mutually inverse maps on the Hilbert completions ${\mathcal{H}}_i$ of step 1.1. Orthogonal projection identifies completed reduced cohomology of the finite dual matrix complex with the Hilbert dual of completed reduced homology: both are the orthogonal complement of boundaries within cycles, and the involution on $R$ identifies the adjoint matrices. On a closed manifold the intersection map is exactly cap duality followed by this Riesz identification; it is therefore bounded and invertible. Its usual symmetry sign follows from the cellular cup/cap identity.

The cellular identity $\mathrm{\partial }(c\cap z)=\pm ((\delta c)\cap z)\pm c\cap \mathrm{\partial }z$ proves compatibility with boundary and restriction before completion. The cup/cap and intersection identities likewise hold on cellular cycles, by their defining evaluation on the dual cells. Any chain homotopies involved are finite matrices, hence remain bounded after completion by step 1.2. They vanish on reduced homology, and every relevant map is continuous. The identities therefore persist on reduced homology with the same orientation signs. Together with steps 1.1, 2.1 and 2.2 this proves all assertions.