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Yes, the signature is additive in this case. Wall gave a precise description of the failure of additivity in gluing two $4k$-manifolds along a subset of their boundaries. (I think the name of the paper is "On the non-additivity of the signature", or something like that, but I don't have it in front of me.) Let $A$ and $B$ be two $4k$-manifolds, and suppose we are gluing along $S\subset \partial A$ and and an identical $(4k{-}1)$-manifold $S\subset \partial B$. Consider the $(4k{-}2)$-manifold $\partial S$. It bounds three manifolds: $S$, $\partial A\setminus S$, and $\partial B\setminus S$. The kernels of these inclusions give rise to three lagrangian subspaces of the symplectic middle-dimensional homology of $\partial S$; call them $L_1, L_2, L_3$. Then $\sigma(A \cup_S B) = \sigma(A) + \sigma(B) + c(L_1, L_2, L_3)$, where $\sigma$ denotes signature and $c(L_1, L_2, L_3)$ is an integer which, as the notation suggests, depends only on the three lagrangians $L_1$, $L_2$, and $L_3$.

You assume that $\partial S$ is empty. In this case $c(L_1, L_2, L_3) = 0$ and the signature is additive.

One simple way to define $c(L_1, L_2, L_3)$ (not the way Wall does it) is as follows. Let $n$ be the dimension of $L_i$ and let $V$ be the symplectic vector space which contains the lagrangians. If $n=1$ we define $c(L_1, L_2, L_3) = \pm 1$ according to the cyclic order of $L_1$, $L_2$, and $L_3$, assuming the three lagrangians are distinct. If they are not distinct we define $c(L_1, L_2, L_3) =0$. For $n>1$ it is not hard to show that we can find a symplecitc direct sum decomposition $V = \oplus_\alpha V^\alpha$, with $V^\alpha$ 2-real-dimensional, and lagrangians $L_i^\alpha\subset V^\alpha$, such that $L_i = \oplus_\alpha L_i^\alpha$. We now extend linearly and define $c(L_1, L_2, L_3) = \sum_\alpha c(L_1^\alpha, L_2^\alpha, L_3^\alpha)$.

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Yes, the signature is additive in this case. Wall gave a precise description of the failure of additivity in gluing two $4k$-manifolds along a subset of their boundaries. (I think the name of the paper is "On the non-additivity of the signature", or something like that, but I don't have it in front of me.) Let $A$ and $B$ be two $4k$-manifolds, and suppose we are gluing along $S\subset \partial A$ and and identical $(4k{-}1)$-manifold $S\subset \partial B$. Consider the $(4k{-}2)$-manifold $\partial S$. It bounds three manifolds: $S$, $\partial A\setminus S$, and $\partial B\setminus S$. The kernels of these inclusions give rise to three lagrangian subspaces of the symplectic middle-dimensional homology of $\partial S$; call them $L_1, L_2, L_3$. Then $\sigma(A \cup_S B) = \sigma(A) + \sigma(B) + c(L_1, L_2, L_3)$, where $\sigma$ denotes signature and $c(L_1, L_2, L_3)$ is an integer which, as the notation suggests, depends only on the three lagrangians $L_1$, $L_2$, and $L_3$.

You assume that $\partial S$ is empty. In this case $c(L_1, L_2, L_3) = 0$ and the signature is additive.