On the (super)additivity of simplicial volume

TL;DR

This paper demonstrates that simplicial volume exhibits superadditivity when manifolds are glued along specific boundary submanifolds, extending known additivity results and introducing new tools in bounded cohomology.

Contribution

It establishes superadditivity of simplicial volume for certain gluings and generalizes additivity results for aspherical manifolds using novel bounded cohomology techniques.

Findings

Simplicial volume is superadditive under specific boundary gluings.

New results on relative bounded cohomology are developed.

Generalization of additivity results for aspherical manifolds.

Abstract

We show that the simplicial volume is superadditive with respect to gluings along certain submanifolds of the boundary. Our criterion applies to boundary connected sums and 1-handle attachments. Moreover, we generalize a well-known additivity result in the case of aspherical manifolds. Our arguments are based on new results about relative bounded cohomology and pairs of multicomplexes, which are of independent interest.