Signature additivity under gluing

Van Kampen gives the glued homomorphism to $\mathrm{\Gamma }$ from the matching restrictions. Let $j:H_2^{(2)}(W_0)\oplus H_2^{(2)}(W_1)\to H$ be the sum of the inclusion maps, and let $V=\overline{\mathrm{im}j}$. The closed intersection operator $J$ on $H$ is invertible by [F2]. Under Poincaré duality, the annihilator $V^{{\perp }_J}$ identifies with the kernel of restriction $H_{(2)}^2(X)\to H_{(2)}^2(W_0)\oplus H_{(2)}^2(W_1)$: evaluation on each piece is exactly the pairing with its inclusion in $X$.

Weak cohomology Mayer–Vietoris exactness identifies that kernel with the closure of the image of the connecting map $H_{(2)}^1(M)\to H_{(2)}^2(X)$. Cap boundary naturality in [F2] identifies this connecting map, up to its harmless orientation sign, with $H_2^{(2)}(M)\to H_2^{(2)}(X)$ after cap duality on $M$ and $X$. Cap duality is a bounded isomorphism, so it preserves these closures. Hence $V^{{\perp }_J}$ is the closure of this last image. The inclusion $M\to X$ factors through either piece, so this image lies in $V$. Thus $V$ is coisotropic.

The pullback form $j^{\ast }Jj$ is the orthogonal sum of the intersection forms on $W_0$ and $-W_1$. Indeed the same-piece term is the absolute-to-relative intersection pairing by cap naturality; cross terms vanish because representative cellular chains can be pushed into the respective collar interiors, which are disjoint. This can be checked on the finite dual cellular chains before completion: each intersection contribution comes from a cell and its dual in the same piece, with its given group-ring coefficient, and there are no cross-piece intersections. The identity extends by boundedness to the reduced homology in [F2]. Reversing orientation on $W_1$ negates its form.

Discard $\ker j$; it is a null summand of the pullback form. The restriction of $j$ to its Hilbert orthogonal complement is injective with dense image in $V$. By [F3] its pullback has the signature of the restricted form on $V$. By the coisotropic conclusion of step 2.1, this is the signature of $J$ on $H$. Step 2.2 and additivity of spectral trace on an orthogonal sum give the asserted $L^2$ signature equality.

Apply the same argument to the trivial coefficient group. Reduced homology is ordinary finite-dimensional complex homology; its form is the complexification of the real intersection form. Complexification preserves both real inertia dimensions, as in [F3]. The resulting equality is the asserted ordinary signature equality.