Massey products for homotopy inner products

This paper defines Massey-type products for a homotopy inner product on an $A_{\mathrm{\infty }}$ algebra, called Massey inner products. We include an explicit description of ordinary Massey products for $A_{\mathrm{\infty }}$ algebras, and for $A_{\mathrm{\infty }}$ modules, and show that Massey product sets can have interesting internal structure. We give non-trivial applications from lens spaces, links, and low dimensional manifolds, showing Massey inner products contain information beyond ordinary Massey products.