Let \(M\) be a compact, connected, orientable \(n\)-manifold with boundary. The simplical volume \( \| M, \partial M\|\) of \(M\) is defined to be the Gromov norm of the fundamental class that is the image of the generator of \(H^{n}(M, \partial M; {\mathbb Z})\) in \(H^{n}(M, \partial M; {\mathbb R})\). A natural question to ask is whether this volume is additive under a gluing along parts of the boundaries of two manifolds \(M_{1}\) and \(M_{2}\). The author shows that under certain circumstances there is an additivity. In particular if the two manifolds \(M_{1}\) and \(M_{2}\) are glued along a homeomorphism \(f: A_{1} \rightarrow A_{2}\) of submanifolds of the boundaries of \(M_{1}\) and \(M_{2}\) to give \(M = M_{1} \cup_{f} M_{2}\) such that the images \(im(\pi_{1}(A_{i}) \rightarrow \pi_{1}( M_{i}))\) are amenable for \(i =1,2\) and \(f_{*}\) restricts to an isomorphism
\[ f_{*}: \ker(\pi_{1}(A_{1}) \rightarrow \pi_{1}(M_{1})) \rightarrow \ker(\pi_{1}(A_{2}) \rightarrow \pi_{1}(M_{2})) \]
\[ f_{*}: \ker(\pi_{1} (\partial A_{1}) \rightarrow \pi_{1}(\partial M_{1})) \rightarrow \ker(\pi_{1}(\partial A_{2}) \rightarrow \pi_{1}(\partial M_{2})) \]
Then \(\| M, \partial M \| \geq \| M_{1}, \partial M_{1} \| + \| M_{2} , \partial M_{2} \| \)
If \(A_{1}\) and \(A_{2}\) are connected components of \(\partial M_{1}\) resp. \(\partial M_{2}\) then the inequality is an equality.
The author gives counter examples to show that the condition \[ \ker(\pi_{1}(A_{1}) \rightarrow \pi_{1}(M_{1})) \;\approx \ker(\pi_{1}(A_{2}) \rightarrow \pi_{1}(M_{2})) \] cannot be weakened.
The proof is quite technical, so the author provides a helpful outline of the plan of the proof.