Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4661338 | Topology and its Applications | 2007 | 20 Pages |
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. Whitehead's 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.