Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that $$ \sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$
More generally, if we glue the boundaries of two such manifolds $X_1$ and $X_2$ via a orientation-reversing diffeomorphism (or even just part of the boundaries - see here) then Novikov additivity holds. Wall has famously analyzed the failure of additivity when we glue along a submanifold of the boundary that itself has boundary.
I am interested in how this story extends to intersection forms. In particular:
What can be said about the relationship between the intersection form $Q_{X}$, where $X = X_1 \cup_\phi X_2$ where $\phi$ is an orientation-reversing diffeomorphism between closed submanifolds of the boundary, and the intersection forms $Q_{X_1}$ and $Q_{X_2}$?
I know that additivity of intersection forms holds in the case where the boundaries of $X_1$ and $X_2$ are both homology spheres and they are glued along their entire boundary - so in particular, $Q_{X_1 \sharp X_2} = Q_{X_1} + Q_{X_2}$. I wonder if there is some error term in the general case that has been identified. For what it's worth, I am primarily interested in the case where $k=1$.
