  We study homological representations of mapping class groups, including the
braid groups. These arise from the twisted homology of certain configuration
spaces, and come in many different flavours. Our goal is to give a unified
general account of the fundamental relationships (non-degenerate pairings,
embeddings, isomorphisms) between the many different flavours of homological
representations. Our motivating examples are the Lawrence-Bigelow
representations of the braid groups, which are of central importance in the
study of the braid groups themselves, as well as their connections with quantum
invariants of knots and links.
