A theory of PWD-structures

A general study is undertaken of product-wedge-diagonal (=PWD) structures on a space. In part this concept may be viewed as arising from G.W. Whiteheadʹs fat-wedge characterization of Lusternik–Schnirelmann category. From another viewpoint PWD-structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD-structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X → Y . A principal result of the paper uses Hopf invariants to formulate a Berstein–Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U * V that has PWD-structure defined canonically via the join structure in terms of diagonal maps on U and V.